Singular HJB equations with applications to KPZ on the real line

Abstract

This paper is devoted to studying Hamilton-Jacobi-Bellman equations with distribution-valued coefficients, which are not well-defined in the classical sense and are understood by using the paracontrolled distribution method introduced in (Gubinelli et al. in Forum Math Pi 3(6):1, 2015). By a new characterization of weighted Hölder spaces and Zvonkin’s transformation we prove some new a priori estimates, and therefore establish the global well-posedness for singular HJB equations. As applications, we obtain global well-posedness in polynomial weighted Hölder spaces for KPZ type equations on the real line, as well as modified KPZ equations for which the Cole–Hopf transformation is not applicable.

1 Introduction

Recall that the classical Kardar-Parisi-Zhang equation is given as follows:

$$\begin{aligned} {{\mathscr {L}}}h:=\left( \partial _t - \Delta \right) h=(\partial _x h)^{2}+\xi ,\quad h(0)=h_0, \end{aligned}$$

(1.1)

where $\xi$ is a Gaussian space-time white noise. This equation was introduced in [36] as a model for the growth of interfaces represented by a height function h. In [36] the authors predicted that under a 1–2–3 scaling the height function must converge to a scale invariant random field called the KPZ fixed point (see [8, 41, 47] and references therein). It is conjectured that the large scale behaviour of a large class of interface growth models is described by the KPZ fixed point. These models are said to belong to the KPZ universality class and this is referred to as the strong KPZ universality conjecture. A weaker form of universality which is now called the weak universality conjecture states that the KPZ equation is itself a universal description of the fluctuations of weakly asymmetric growth models (see e.g. [3, 32, 34] and references therein). The main difficulty in solving Eq. (1.1) comes from the singularity of space-time white noise and the nonlinearity, since $(\partial _x h)^2$ cannot be understood in the classical sense because $\partial _x h$ is not a function. This problem can be avoided by using the Cole–Hopf transformation (see [2, 3, 36] and also [7, 28]). In fact, letting $w:=\mathrm {e}^h$ and formally using Itô’s formula, one sees that

$$\begin{aligned} {{\mathscr {L}}}w=w\xi ,\quad w(0)=\mathrm {e}^{h_0}, \end{aligned}$$

(1.2)

which can be understood in Itô’s sense ([52]). In [2, 3] the solutions to (1.1) are defined by $\log w$, where w is a positive solution to (1.2), now known as the Cole–Hopf solution. It remained an open problem to clarify in what sense the Cole–Hopf solution genuinely solves the original KPZ equation.

A revolutionary step was made by Hairer [26] using methods from rough path theory. He was able to solve the classical KPZ equation on the torus. Later, Hairer introduced the theory of regularity structures in [27] and Gubinelli, Imkeller and Perkowski proposed the paracontrolled distribution method in [20, 23], which made it possible to study a large class of PDEs driven by singular noises. The key idea of these theories is to use the structure of the solution to give a meaning to the terms which are not classically well-defined. These terms are well-defined with the help of renormalization for the  “enhanced noise”, i.e. the noise and the higher order terms appearing in the decomposition of the equations. More precisely, $(\partial _xh)^2$ can be formally understood as a subtraction of an infinite correction term: $(\partial _xh)^2-\infty$.

After this breakthrough, an avalanche of excitement and intriguing results followed, proving local/global existence and uniqueness of solutions to a large class of singular SPDEs, including the generalized parabolic Anderson model, the dynamical $\Phi ^4$ model and other interesting examples ([9, 29, 30, 55] and references therein). Very recently, geometric stochastic heat equations with values in a Riemannian manifold M were studied in [4] via regularity structures theory and in [11, 49] by Dirichlet forms, which can be written in local coordinates as generalized coupled KPZ equations (see Sect. 6.1 for more details).

Up to now, most of the well-known works in the field of singular SPDEs are considered with the finite volume case. Since the large scale behavior of the KPZ equation is related to the important KPZ fixed point (see [41] and below), it is natural to consider the KPZ equation on the real line. In fact, new phenomena may occur in the infinite volume setting. For example, in [11] it was shown that solutions to geometric stochastic heat equations exhibit different long-time behavior compared to the finite volume setting (see [49]). In general, space-time white noise in infinite volume stays in weighted Besov spaces, as does the solution. Since these spaces are typically not preserved by the nonlinearity, it obstructs the use of simple fixed point arguments for constructing local solutions. The first attempt to overcome this difficulty was due to Hairer and Labbé [29, 30] for the rough linear heat equation by introducing an exponential weight depending on time. For nonlinear equations, suitable a priori estimates in weighted spaces have been established for the dynamical $\Phi ^4_d$ model by Mourrat and Weber [43, 44] and Gubinelli and Hofmanová [18], which rely on the damping term $-\phi ^3$. In [45] a priori estimates and paracontrolled solutions to the KPZ equation on the real line were obtained by using the Cole–Hopf transformation. Moreover, using the probabilistic notion of energy solutions [21, 22, 24] or studying the associated infinitesimal generator and Kolmogorov equation [25] it is possible to give a meaning to the KPZ equation on ${{\mathbb {R}}}$, but this is restricted to initial data which is absolutely continuous w.r.t. the stationary measure. We mention that in [11] martingale solutions were constructed for geometric stochastic heat equations in ${{\mathbb {R}}}$ by using the Dirichlet form approach, which relies on an integration by parts formula for the invariant measure.

In the present paper we are concerned with the following KPZ type SPDEs on the real line:

$$\begin{aligned} {{\mathscr {L}}}h= & {} ``(\partial _x h)^{2}\text {''}+g(h)+\xi ,\quad h(0)=h_0, \end{aligned}$$

(1.3)

$$\begin{aligned} {{\mathscr {L}}}h= & {} G(x)``(\partial _x h)^{2}\text {''}+K(x)\partial _x h+\xi ,\quad h(0)=h_0, \end{aligned}$$

(1.4)

where g, G, K are bounded Lipschitz functions, and $\xi$ is a Gaussian space-time white noise on ${\mathbb {R}}^+\times {\mathbb {R}}$. Equations (1.3) and (1.4) are typical examples of singular SPDEs and can be viewed as a simplified version of the generalized KPZ equations and geometric stochastic heat equations in [4, 31]. The emphasis of this article is on deriving an a priori estimate by PDE arguments and complements the local solution theory by ruling out the possibility of finite time blow-up. As directly obtaining global well-posedness to geometric stochastic heat equation by PDE arguments is still an open problem, we study the simplified version (1.3) and (1.4). Note that neither equations can be linearized by the Cole–Hopf transformation.

As mentioned above, suitable a priori estimates and global well-posedness have been established for the dynamical $\Phi ^4_d$ model by using the strong damping term $-\phi ^3$ (see [18, 43, 44]) and for the KPZ equation by the Cole–Hopf transformation (see [25]). The main aim of this paper is to obtain global well-posedness of singular SPDEs on the whole space when the strong damping is not at hand and the Cole–Hopf transformation is not applicable. We obtain global well-posedness of Eqs. (1.3) and (1.4) by suitable a priori estimates. By a renormalization and decomposition procedure, one can reduce KPZ type SPDEs (1.3), (1.4) to the following singular Hamilton-Jacobi-Bellman equation in ${{\mathbb {R}}}^d$ (abbreviated as HJB) together with some linear equations (see Sect. 6 for more details):

$$\begin{aligned} {{\mathscr {L}}}u:=\left( \partial _t - \Delta \right) u = H(\nabla u)+ b \cdot \nabla u + f, \quad u (0) = \varphi , \end{aligned}$$

(1.5)

where $H:{{\mathbb {R}}}^d\rightarrow {{\mathbb {R}}}$ is a locally Lipschitz function of at most quadratic growth, and for some $\alpha \in (\frac{1}{2},\frac{2}{3})$ and $\kappa \in (0,1)$,

$$\begin{aligned} b \in L^\infty _T {{\mathbf {C}}}^{- \alpha } (\rho _{\kappa }), \quad f \in L^\infty _T{{\mathbf {C}}}^{-\alpha } (\rho _{\kappa }). \end{aligned}$$

Here $\rho _\kappa (x) :=\langle x\rangle ^{-\kappa }:=(1 + | x |^2)^{-\kappa / 2}$ and ${{\mathbf {C}}}^{- \alpha } (\rho _{\kappa })$ stands for the weighted Hölder (or Besov) space (see Sect. 2.1). We will first derive global well-posedness of Eq. (1.5) under general assumptions on H (see Sect. 5) and then apply it to Eqs. (1.3) and (1.4) in Sect. 6.

The difficulties that arrive in solving (1.3) and (1.4) also arise in a slightly different form for (1.5). Concerning (1.5), since $b,f \in L_T^\infty {{\mathbf {C}}}^{- \alpha } (\rho _{\kappa })$ and $\alpha \in (\frac{1}{2},\frac{2}{3})$, the best regularity space for u is $L^\infty _T{{\mathbf {C}}}^{2-\alpha }$ by Schauder’s estimate. Compared to (1.3) and (1.4) there is no difficulty defining $H(\nabla u)$ for (1.5) since f is more regular than space-time white noise. However, the transport term $b\cdot \nabla u$ is not well-defined in the classical sense.

We need to use regularity structures theory or paracontrolled distributions to give a meaning to Eq. (1.5). In this paper we use PDE arguments and paracontrolled distributions to obtain the global well-posedness of (1.5). Notice that for general H, we cannot use the Cole–Hopf transformation to transform (1.5) into a linear equation. In that sense our new approach is much more robust than the previous one.

Finally we also mention that the HJB equation appears originally in optimal control theory, whose solution represents the value function of a stochastic optimal control problem (see [17, 37, 53]). More precisely, consider the following stochastic optimal control problem:

$$\begin{aligned} V(t,x):=\inf _{\gamma }{{\mathbb {E}}}\left[ \int ^T_t L(s,X^\gamma _s(x),\gamma (s)){\mathord {\mathrm{d}}}s+\psi (X^\gamma _T(x))\right] , \end{aligned}$$

(1.6)

where the infimum is taken over all controls $\gamma$ in some class of adapted processes, L is the cost function, $\psi$ is the final bequest value, and $X^\gamma _t(x)=X^\gamma _t$ is the state process which solves the following SDE:

$$\begin{aligned} {\mathord {\mathrm{d}}}X^\gamma _t=(b(t,X^\gamma _t)+\gamma _t){\mathord {\mathrm{d}}}t+\sqrt{2}{\mathord {\mathrm{d}}}W_t,\ X^\gamma _0=x, \end{aligned}$$

where W is a d-dimensional standard Brownian motion. Let

$$\begin{aligned} H(t,x,Q):=\inf _{v\in {{\mathbb {R}}}^d}(v\cdot Q+L(t,x,v)). \end{aligned}$$

By the dynamical programming principle, V solves the following backward HJB equation:

$$\begin{aligned} \partial _t V+\Delta V+b\cdot \nabla _x V+H(\nabla V)=0,\ V(T,x)=\psi (x). \end{aligned}$$

Moreover, by the verification theorem, the optimal control $\gamma$ is then given by $\nabla V(t, X^*_t)$, where $X^*_t$ solves the following SDE:

$$\begin{aligned} {\mathord {\mathrm{d}}}X^*_t=(b(t,X^*_t)+\nabla V(t, X^*_t)){\mathord {\mathrm{d}}}t+\sqrt{2}{\mathord {\mathrm{d}}}W_t,\ X^*_0=x. \end{aligned}$$

In particular, the study of singular HJB equations provides a possibility to study the singular stochastic control problem. By singular, we mean that b may be a distribution. Recently, there is some interest in studying the control problem with rough drift b (see [42] and the reference therein). Notice that our conditions on b are automatically satisfied for $b\in L^\infty _T{{\mathbf {C}}}^{-\beta }(\rho _\kappa )$ with $\beta \in (0,\frac{1}{2})$. Thus our main results can be applied to the SDEs in [42], which may give applications to the stochastic control problem considered in [42] and the references therein. For more singular $b\in L_T^\infty {{\mathbf {C}}}^{-\alpha }(\rho _\kappa )$ with $\alpha \in (\frac{1}{2},\frac{2}{3})$, it could be viewed as a random environment and our condition allows for spatial white noise in one dimension, which may be derived from averages of a sequence of i.i.d random variables (see [46, Remark 2.2]). We also mention that the solution to the classical KPZ equation can be viewed as a stochastic control problem with singular $b\in L_T^\infty {{\mathbf {C}}}^{-\alpha }(\rho _\kappa )$, $\alpha \in (\frac{1}{2},\frac{2}{3})$ (see [23, 45]), where the solution is interpreted as a value function defined as in (1.6).

1.1 Main results

As mentioned above, we concentrate on (1.5) first and to define $b\cdot \nabla u$ in (1.5) we need to perform renormalizations by probabilistic calculations. It is not the aim of this paper to discuss the renormalization terms as this has been done extensively (see e.g. [23, 26, 45]). For the main result, we suppose that $b\circ \nabla {{\mathscr {I}}}b\in L_T^\infty {{\mathbf {C}}}^{1-2\alpha }(\rho _{2\kappa })$ and $b\circ \nabla {{\mathscr {I}}}f\in L_T^\infty {{\mathbf {C}}}^{1-2\alpha }(\rho _{2\kappa })$ are well defined, where ${{\mathscr {I}}}:={{\mathscr {L}}}^{-1}$, i.e. $(b,f)\in {\mathbb {B}}^\alpha _T(\rho _\kappa )$ (see Sect. 2.3 and Sect. 2.4), which in general can be realized by a probabilistic calculation (see Sec. 6 for examples). In the following, we are mainly concerned with the analysis of the deterministic system under the above assumptions.

The following result is a special case of the main Theorem 5.1, where a more general condition on the nonlinear term H is given (see Remark 5.2 for examples of H).

Theorem 1.1

Let $\alpha \in (\frac{1}{2},\frac{2}{3})$ and $\kappa$ be small enough so that $\delta :=2(\frac{9}{2-3\alpha }+1)\kappa <1$, ${\bar{\alpha }}:=\alpha +\kappa ^{1/4}\in (\frac{1}{2},\frac{2}{3})$. Suppose that for some $c>0$,

$$\begin{aligned} |\partial _QH(Q)|\leqslant c(1+|Q|). \end{aligned}$$

If $d\geqslant 2$, we also suppose H has sub-quadratic growth, i.e., for some $\zeta \in [0,2)$,

$$\begin{aligned} |H(Q)|\leqslant c(|Q|^\zeta +1). \end{aligned}$$

Then for any $(b,f)\in {{\mathbb {B}}}^\alpha _T({\rho _\kappa })$ and initial value $\varphi \in {{\mathbf {C}}}^{1+\alpha +\varepsilon }(\rho _{\varepsilon \delta })$, where $\varepsilon \in (0,1)$, there exists a unique paracontrolled solution $u\in {\mathbb {S}}_T^{2-{\bar{\alpha }}}(\rho _\eta )$ to the HJB equation (1.5) in the sense of (5.4) and (5.5) below, where $\eta =\eta (\kappa ,\alpha ,\zeta )<\tfrac{1-\alpha }{2}$ converges to zero as $\kappa \rightarrow 0$.

As the main application, we obtain global well-posedness of (1.3) and (1.4). The regularity of the space-time white noise $\xi$ is more rough than the coefficient f given in (1.5). To apply Theorem 1.1 we need to introduce some random distributions and use the Da Prato-Debussche trick to reduce (1.3) to (1.5) (see e.g. [13]). This is the usual pathwise approach to the KPZ equation (cf. [23, 26, 45]). Let Y and be random distributions defined in Sect. 6.

Theorem 1.2

Let $g:{{\mathbb {R}}}\rightarrow {{\mathbb {R}}}$ be bounded, Lipschitz continuous, and $\kappa >0$ be small enough, $\delta :=40\kappa <1$. For $h_0=Y(0)+{\widetilde{h}}(0)$ with ${\widetilde{h}}(0)\in {{\mathbf {C}}}^{\frac{3}{2}+\varepsilon +\gamma }(\rho _{\varepsilon \delta })$, where $\varepsilon \in (0,1)$ and $0<\gamma <1/4$, there exists a unique paracontrolled solution to (1.3) in the sense that is the unique paracontrolled solution to (6.4) with $2({\kappa ^{1/4}}+80\kappa )<\eta <1/4.$

This result improves the weight for the solution space obtained in [45] for $g\equiv 0$ and is proved in Theorem 6.3. As a further application, we also establish global well-posedness of Eq. (1.4), which is presented in Theorem 6.7.

1.2 Sketch of proofs and structure of the paper

In Sect. 2 we first introduce the basic notations and the spaces used throughout the paper. The regularization effect of heat semigroups and paracontrolled calculus are recalled in Sects. 2.2 and 2.3, respectively. The conditions for the coefficient (b, f) are discussed in Sect. 2.4.

The bulk of our argument is contained in Sections 3-5 and we now proceed to explain the strategy. We decompose (1.5) into the following two equations: $u=u_1+u_2$

$$\begin{aligned}&\left( \partial _t - \Delta \right) u_1 = b \cdot \nabla u_1 + f, \end{aligned}$$

(1.7)

$$\begin{aligned}&\left( \partial _t - \Delta \right) u_2 = b \cdot \nabla u_2 + H(\nabla u_1+\nabla u_2). \end{aligned}$$

(1.8)

In Sect. 3 we first establish Schauder’s estimate for (1.7) with sublinear weights (see Theorem 3.7). This solves the conjecture proposed in [45, Remark 1.1]. The difficulty to study (1.7) lies in the drift b living in a weighted Besov space, which prevents us from using a fixed point argument in the same space. It is possible to use the technique in [30] to solve the problem, by which the solution stays in a Besov space with exponential weights. This seems not easy to deduce a uniform $L^\infty (\rho _\delta )$ estimate for the solution to (1.8). In Sect. 3 we develop a new technique to establish a sublinear growth bound for the solutions to equation (1.7). The key idea is to use a new characterization of the weighted Hölder space (see Lemma 3.8) to localize the problem with coefficients in unweighted Besov spaces, for which we obtain the Schauder estimate depends polynomially on the norm of the coefficients compared to the exponential dependence by the usual Gronwall type argument. To this end, we add a new damping term $\lambda u_1$ to (1.7) and use the classical maximum principle. We also mention that Eq. (1.7) on the torus has been studied in [10], where the difficulty with weights does not appear. In a subsequent work [35], we also apply the localization technique developed in this paper to singular kinetic equations.

In Sects. 4 and 5, we study (1.8). Compared to (1.5) the distribution-valued f becomes function-valued. To treat the distribution-valued transport term $b\cdot \nabla u$, we use Zvonkin’s transformation to kill the singular part and transform (1.8) into the following general HJB equation (see Sect. 5)

$$\begin{aligned} \partial _t v=\mathrm {tr}(a\cdot \nabla ^2 v)+B\cdot \nabla v+\widetilde{H}(v,\nabla v),\ v(0)=\varphi , \end{aligned}$$

(1.9)

where the matrix $a\in L^\infty _T{{\mathbf {C}}}^{1-\alpha }$ is symmetric and uniformly elliptic, $B\in {{\mathbb {L}}}^\infty _T(\rho _{\delta _1})$ for some $\delta _1\in (0,1]$.

Fig. 1

Steps of solving (1.5)

More precisely, assume that ${\mathbf {u}}$ solves

$$\begin{aligned} (\partial _t-\Delta +\lambda ){\mathbf {u}}=b\cdot (\nabla {\mathbf {u}}+{{\mathbb {I}}}). \end{aligned}$$

(1.10)

If $\Phi (t,x)=x+{\mathbf {u}}(t,x)$ is a diffeomorphism in the x variable, then $u_2(t,\Phi ^{-1}(t,x))$ will solve (1.9). All the coefficients of (1.9) are function-valued with the cost that (1.9) is given in a non-divergence form PDE. This procedure is usually called Zvonkin’s transformation, which was originally used for treating SDEs with irregular drifts (see [56]). However, due to the presence of the weights, this argument needs to be refined. To this end, we use [18, Lemma 2.6] to decompose b into a singular term $b_>$ in the unweighted Besov space and a function-valued term $b_\leqslant$ with polynomial growth. Then we use Zvonkin’s transformation to kill the singular part $b_>$ by subtracting a new term (see Remark 5.4 for more details on this point). The idea comes from Zvonkin’s transformation for SDEs, but our Zvonkin’s transformation is different from the normal one. To the best of our knowledge, it is the first time to use Zvonkin’s transformation to deal with the nonlinear PDE (1.8) with singular drift b.

Section 4 is devoted to the global well-posedness of Eq. (1.9) (see Theorem 4.2). We first establish a maximum principle in Sect. 4.1 with the help of Feymann-Kac’s formula. For the subcritical case,¹ the global estimate follows from the $L^\infty (\rho _\delta )$-estimate and the $L^p$ theory of PDEs. For the critical case, the proof is more involved. In this case by taking the spatial derivative on both sides of (1.9), we obtain a divergence PDE, which only holds for $d=1$. Then the $L^\infty (\rho _\delta )$-bound and energy estimate yield the ${{\mathbb {H}}}^{2,p}_T(\rho _\eta )$-estimate of the solution to Eq. (1.9). By using this and Zvonkin’s transformation we finally establish a priori global estimates for solutions to (1.8) as well as the well-posedness of (1.5) in Sect. 5.

In the above Fig. 1, we outline the main idea and steps of solving Eq. (1.5).

In Sect. 6 we apply our main result to the KPZ type Eqs. (1.3) and (1.4). Finally, in Appendix 1, we prove the uniqueness of solutions to (1.5) based on the exponential weight approach developed in [30]. Appendix 1 is then devoted to an exponential moment estimate for SDEs used in Sect. 4.

1.3 Conventions and notations

Throughout this paper, we use C or c with or without subscripts to denote an unrelated constant, whose value may change in different places. We also use $:=$ as a way of definition. By $A\lesssim _C B$ and $A\asymp _C B$ or simply $A\lesssim B$ and $A\asymp B$, we mean that for some constant $C\geqslant 1$,

$$\begin{aligned} A\leqslant C B,\ \ C^{-1} B\leqslant A\leqslant CB. \end{aligned}$$

For convenience, we collect some commonly used notations and definitions below.

${{\mathscr {C}}}^\alpha (\rho )$: weighted Hölder space (Def. 2.3)	${{\mathscr {C}}}^\alpha :={{\mathscr {C}}}^\alpha (1)$
${{\mathbf {B}}}^\alpha _{p,q}(\rho )$: weighted Besov space (Def. 2.5)	${{\mathbf {B}}}^\alpha _{p,q}:={{\mathbf {B}}}^\alpha _{p,q}(1)$
${{\mathbf {C}}}^\alpha (\rho )$: weighted Hölder-Zygmund space (Def. 2.5)	${{\mathbf {C}}}^\alpha :={{\mathbf {C}}}^\alpha (1)$
${{\mathbb {S}}}^\alpha _T(\rho )$: Paracontrolled solution space (2.3)	${{\mathbb {S}}}^\alpha _T:={{\mathbb {S}}}^\alpha _T(1)$
${{\mathbb {B}}}^\alpha _T(\rho )$: Space of renormalized pair (Def. 2.14)	${{\mathbb {B}}}^\alpha _T:={{\mathbb {B}}}^\alpha _T(1)$
$f\prec g, f\succ g, f\circ g$: Paraproduct (Sec. 2.3)	$f\succcurlyeq g:=f\succ g+f\circ g$
$f\prec \!\!\!\prec g$: Modified paraproduct (Sec. 2.3)	${{\mathscr {L}}}_\lambda :=\partial _t-\Delta +\lambda$
$\mathrm{com}(f,g,h):=(f\prec g)\circ h-f(g\circ h)$ (Sec. 2.3)	${{\mathscr {I}}}_\lambda :=(\partial _t-\Delta +\lambda )^{-1}$
${{\mathscr {V}}}_> f$, ${{\mathscr {V}}}_\leqslant f$: Localization operator (Sec. 2.3)	${{\mathscr {L}}}:={{\mathscr {L}}}_0$, ${{\mathscr {I}}}:={{\mathscr {I}}}_0$
$P_tf(x):=(4\pi t)^{-d/2}\int _{{{\mathbb {R}}}^d}f(y)\mathrm {e}^{-|x-y|^2/(4t)}{\mathord {\mathrm{d}}}y$	$B_r:=\{x:|x|\leqslant r\}$
${{\mathscr {I}}}_{s}^tf(x):=\int _s^tP_{t-r}f(r,x){\mathord {\mathrm{d}}}r$	$\langle x\rangle :=(1+|x|^2)^{1/2}$
Commutator: $[{{\mathscr {A}}}_1,{{\mathscr {A}}}_2]f:={{\mathscr {A}}}_1({{\mathscr {A}}}_2 f)-{{\mathscr {A}}}_2({{\mathscr {A}}}_1f)$	${{\mathbb {N}}}_0:={{\mathbb {N}}}\cup \{0\}$

2 Preliminaries

2.1 Weighted Besov spaces

In this section we introduce the weighted Besov spaces which will be used in the sequel. Recall the following definition of admissible weight introduced in [51].

Definition 2.1

A $C^\infty$-smooth function $\rho :{{\mathbb {R}}}^d\rightarrow (0,\infty )$ is called an admissible weight if for each $j\in {{\mathbb {N}}}$, there is a constant $C_j>0$ such that

$$\begin{aligned} |\nabla ^j\rho (x)|\leqslant C_j\rho (x),\ \ \forall x\in {{\mathbb {R}}}^d, \end{aligned}$$

and for some $C,\beta >0$,

$$\begin{aligned} \rho (x)\leqslant C\rho (y)(1+|x-y|)^\beta ,\ \ \forall x,y\in {{\mathbb {R}}}^d. \end{aligned}$$

The set of all the admissible weights is denoted by ${{\mathscr {W}}}$.

Example 2.2

Let $\rho _\delta (x)=\langle x\rangle ^{-\delta }=(1+|x|^2)^{-\delta /2}$, where $\delta \in {{\mathbb {R}}}$. It is easy to see that $\rho _\delta \in {{\mathscr {W}}}$. Such a weight is called polynomial weight.

We introduce the following weighted Hölder space for later use.

Definition 2.3

(Weighted Hölder spaces) Let $\rho \in {{\mathscr {W}}}$ and $k\in {{\mathbb {N}}}_0$. For $\alpha \in [0,1)$, we define the weighted Hölder space ${{\mathscr {C}}}^{k+\alpha }(\rho )$ by the norm

$$\begin{aligned} \Vert f\Vert _{{{\mathscr {C}}}^{k+\alpha }(\rho )}:=\sum _{j=0}^k\Vert \nabla ^j(\rho f)\Vert _{L^\infty }+\sup _{x\not =y}\frac{|\nabla ^k(\rho f)(x)-\nabla ^k(\rho f)(y)|}{|x-y|^\alpha }<\infty . \end{aligned}$$

Remark 2.4

Note that the k-order derivative of a function in ${{\mathscr {C}}}^k(\rho )$ is not necessarily continuous. By the properties of admissible weights and elementary calculations, it is easy to see that for some $C=C(d,\rho )\geqslant 1$,

$$\begin{aligned} \Vert f\Vert _{{{\mathscr {C}}}^{k+\alpha }(\rho )}&\asymp _C\sum _{j=0}^k\Vert \rho \nabla ^jf \Vert _{L^\infty } +\sup _{|x-y|\leqslant 1}\frac{|(\rho \nabla ^k f)(x)-(\rho \nabla ^k f)(y)|}{|x-y|^\alpha }\nonumber \\&\asymp _C\sum _{j=0}^k\Vert \rho \nabla ^jf \Vert _{L^\infty }+\sup _{|x-y|\leqslant 1}\frac{\rho (x)|\nabla ^k f(x)-\nabla ^k f(y)|}{|x-y|^\alpha }. \end{aligned}$$

(2.1)

Let ${\mathcal {S}}({\mathbb {R}}^{d})$ be the space of Schwartz functions on ${\mathbb {R}}^{d}$ and ${\mathcal {S}}'({\mathbb {R}}^{d})$ the space of tempered distributions, which is the dual space of ${\mathcal {S}}({\mathbb {R}}^{d})$. The Fourier transform of $f\in {\mathcal {S}}'({\mathbb {R}}^{d})$ is defined through

$$\begin{aligned} {\widehat{f}}(z):=(2\pi )^{-d/2}\int _{{\mathbb {R}}^{d}}f(x)\mathrm {e}^{-i z\cdot x}{\mathord {\mathrm{d}}}x. \end{aligned}$$

For $j\geqslant -1$, let $\Delta _j$ be the usual block operator used in the Littlewood-Paley decomposition so that for any $f\in {{\mathcal {S}}}'({{\mathbb {R}}}^d)$ (see [1]),

$$\begin{aligned} \Delta _j f\in {{\mathcal {S}}},\ \ \mathrm{supp}(\widehat{\Delta _j f})\subset B_{2^{j+2}/3}\setminus B_{2^{j-1}},\ j\in {{\mathbb {N}}}_0, \end{aligned}$$

and

$$\begin{aligned} \mathrm{supp}(\widehat{\Delta _{-1} f})\subset B_1,\ \ f=\sum _{j\geqslant -1}\Delta _j f. \end{aligned}$$

We also introduce the following weighted Besov spaces (cf. [51]):

Definition 2.5

Let $\rho \in {{\mathscr {W}}}$ and $p,q\in [1,\infty ]$ and $\alpha \in {{\mathbb {R}}}$. The weighted Besov space ${{\mathbf {B}}}^\alpha _{p,q}(\rho )$ is defined by

$$\begin{aligned} {{\mathbf {B}}}^\alpha _{p,q}(\rho ):=\left\{ f\in {{\mathcal {S}}}'({{\mathbb {R}}}^d): \Vert f\Vert _{{{\mathbf {B}}}^\alpha _{p,q}(\rho )}:=\left( \sum _j 2^{\alpha jq}\Vert \Delta _j f\Vert _{L^p(\rho )}^q\right) ^{1/q}<\infty \right\} , \end{aligned}$$

where

$$\begin{aligned} \Vert f\Vert _{L^p(\rho )}:=\Vert \rho f\Vert _p:=\left( \int _{{{\mathbb {R}}}^d}|\rho (x)f(x)|^p{\mathord {\mathrm{d}}}x\right) ^{1/p}. \end{aligned}$$

The weighted Hölder-Zygmund space is defined by

$$\begin{aligned} {{\mathbf {C}}}^\alpha (\rho ):={{\mathbf {B}}}^\alpha _{\infty ,\infty }(\rho ). \end{aligned}$$

Remark 2.6

Let $\rho \in {{\mathscr {W}}}$. For any $0<\beta \notin {{\mathbb {N}}}$ and $\alpha \in {{\mathbb {R}}}$, $p,q\in [1,\infty ]$, it is well known that (see [51, Theorem 6.5, Theorem 6.9], [1, page99])

$$\begin{aligned} \Vert f\Vert _{{{\mathbf {C}}}^\beta (\rho )}&\asymp \Vert f\Vert _{{{\mathscr {C}}}^\beta (\rho )},\ \Vert f\Vert _{{{\mathbf {B}}}^\alpha _{p,q}(\rho )}\asymp \Vert f\rho \Vert _{{{\mathbf {B}}}^\alpha _{p,q}}. \end{aligned}$$

(2.2)

For $T>0$, $\alpha \in {{\mathbb {R}}}$ and an admissible weight $\rho \in {{\mathscr {W}}}$, let $L^\infty _T{{\mathbf {C}}}^{\alpha }(\rho )$ be the space of space-time distributions with finite norm

$$\begin{aligned} \Vert f\Vert _{L^\infty _T{{\mathbf {C}}}^{\alpha }(\rho )}:=\sup _{0\leqslant t\leqslant T} \Vert f(t)\Vert _{{{\mathbf {C}}}^{\alpha }(\rho )}<\infty . \end{aligned}$$

For $\alpha \in (0,1)$, we denote by $C_T^{\alpha }L^\infty (\rho )$ the space of $\alpha$-Hölder continuous mappings $f: [0,T]\rightarrow L^\infty (\rho )$ with finite norm

$$\begin{aligned} \Vert f\Vert _{C_T^{\alpha }L^\infty (\rho )}:=\sup _{0\leqslant t\leqslant T} \Vert f(t)\Vert _{L^\infty (\rho )}+\sup _{0\leqslant s\ne t\leqslant T} \frac{\Vert f(t)-f(s)\Vert _{L^\infty (\rho )}}{|t-s|^{\alpha }}. \end{aligned}$$

The following space will be used frequently: for $\alpha \in (0,2)$,

$$\begin{aligned} {{\mathbb {S}}}^\alpha _T(\rho ):=\Big \{f: \Vert f\Vert _{{{\mathbb {S}}}^{\alpha }_T(\rho )}:=\Vert f\Vert _{L^\infty _T{{\mathbf {C}}}^\alpha (\rho )}+\Vert f\Vert _{C_T^{\alpha /2}L^\infty (\rho )}<\infty \Big \}. \end{aligned}$$

(2.3)

We have the following simple fact (see [45, Lemma 2.11]): for $\alpha \in (0,1)$,

$$\begin{aligned} \Vert \nabla f\Vert _{{{\mathbb {S}}}^\alpha _T(\rho )}\lesssim \Vert f\Vert _{{{\mathbb {S}}}^{\alpha +1}_T(\rho )}. \end{aligned}$$

(2.4)

For $p\in [1,\infty ]$, $k\in {{\mathbb {N}}}_0$ and $T>0$, we also need the following Sobolev space:

$$\begin{aligned} {{\mathbb {H}}}^{k,p}_T:=\Big \{f: \Vert f\Vert _{{{\mathbb {H}}}^{k,p}_T}:=\Vert f\Vert _{{{\mathbb {L}}}^p_T}+\Vert \nabla ^k f\Vert _{{{\mathbb {L}}}^p_T}<\infty \Big \}, \end{aligned}$$

where, with the usual modification when $p=\infty$,

$$\begin{aligned} \Vert f\Vert _{{{\mathbb {L}}}^p_T}:=\left( \int ^T_0\!\!\!\int _{{{\mathbb {R}}}^d} |f(t,x)|^p{\mathord {\mathrm{d}}}x{\mathord {\mathrm{d}}}t\right) ^{\frac{1}{p}}. \end{aligned}$$

For an admissible weight $\rho$, we also introduce the weighted Sobolev space

$$\begin{aligned} {{\mathbb {H}}}^{k,p}_T(\rho ):=\Big \{f: \Vert f\Vert _{{{\mathbb {H}}}^{k,p}_T(\rho )}:=\Vert f\rho \Vert _{{{\mathbb {H}}}^{k,p}_T}<\infty \Big \}, \end{aligned}$$

and local space ${{\mathbb {H}}}^{k,p}_{\mathrm {loc}}$:

$$\begin{aligned} {{\mathbb {H}}}^{k,p}_{\mathrm {loc}}:=\Big \{f: f\chi _R\in {{\mathbb {H}}}^{k,p}_{T},\ \ \forall T, R>0\Big \}, \end{aligned}$$

where $\chi _R(x)=\chi (x/R)$ and $\chi \in C^\infty _c({{\mathbb {R}}}^d)$ with $\chi =1$ on $B_1$.

The following interpolation inequality will be used frequently, which is an easy consequence of Hölder’s inequality and the corresponding definition. (see [19, Lemma A.3] for a discrete version).

Lemma 2.7

Let $\rho \in {{\mathscr {W}}}$ and $\theta \in [0,1]$. Let $\alpha ,\alpha _1,\alpha _2\in {{\mathbb {R}}}$ and $\delta ,\delta _1,\delta _2\in {\mathbb {R}}$ satisfy

$$\begin{aligned} \delta =\theta \delta _1+(1-\theta )\delta _2,\ \alpha =\theta \alpha _1+(1-\theta )\alpha _2, \end{aligned}$$

and $p,q,p_1,q_1,p_2,q_2\in [1,\infty ]$ satisfy

$$\begin{aligned} \tfrac{1}{p}=\tfrac{\theta }{p_1}+\tfrac{1-\theta }{p_2},\ \ \tfrac{1}{q}=\tfrac{\theta }{q_1}+\tfrac{1-\theta }{q_2}. \end{aligned}$$

Then we have

$$\begin{aligned} \Vert f\Vert _{{{\mathbf {B}}}^\alpha _{p,q}(\rho ^\delta )}\leqslant \Vert f\Vert _{{{\mathbf {B}}}^{\alpha _1}_{p_1,q_1}(\rho ^{\delta _1})}^\theta \Vert f\Vert _{{{\mathbf {B}}}^{\alpha _2}_{p_2,q_2}(\rho ^{\delta _2})}^{1-\theta }. \end{aligned}$$

(2.5)

Moreover, for any $0<\alpha<\beta <2$ with $\theta =\alpha /\beta$, we also have

$$\begin{aligned} \Vert f\Vert _{{{\mathbb {S}}}^\alpha _T(\rho ^\delta )}\lesssim \Vert f\Vert _{{{\mathbb {S}}}^\beta _T(\rho ^{\delta _1})}^{\theta } \Vert f\Vert _{{{\mathbb {L}}}^\infty _T(\rho ^{\delta _2})}^{1-\theta }. \end{aligned}$$

(2.6)

2.2 Estimates of Gaussian heat semigroups

We proceed with the Schauder estimate for the heat semigroup. For $t>0$, let $P_t$ be the Gaussian heat semigroup defined by

$$\begin{aligned} P_t f(x):=(4\pi t)^{-d/2}\int _{{{\mathbb {R}}}^d}\mathrm {e}^{-|x-y|^2/(4t)}f(y){\mathord {\mathrm{d}}}y. \end{aligned}$$

Let $\rho$ be an admissible weight. It is well know that there is a constant $C=C(\rho ,d)>0$ such that (see [43, Lemma 2.10])

$$\begin{aligned} \Vert \Delta _j P_t f\Vert _{L^\infty (\rho )}\lesssim _C \mathrm {e}^{-2^{2j}t}\Vert \Delta _jf\Vert _{L^\infty (\rho )},\ { j\geqslant 0}, t\geqslant 0. \end{aligned}$$

(2.7)

The following lemma provides some quantified estimates for the Gaussian heat semigroup in weighted Hölder spaces.

Lemma 2.8

Let $\rho$ be an admissible weight and $T>0$.

• (i) For any $\theta >0$ and $\alpha \in {{\mathbb {R}}}$, there is a constant $C=C(\rho ,d,\alpha ,\theta , T)>0$ such that for all $t\in (0,T]$,

  $$\begin{aligned} \Vert P_t f\Vert _{{{\mathbf {C}}}^{\theta +\alpha }(\rho )}\lesssim _C t^{-\theta /2}\Vert f\Vert _{{{\mathbf {C}}}^\alpha (\rho )}. \end{aligned}$$

  (2.8)

• (ii) For any $m\in {{\mathbb {N}}}_0$ and $\theta <m$, there is a constant $C=C(\rho ,d,m,\theta , T)>0$ such that for all $t\in (0,T]$,

  $$\begin{aligned} \Vert \nabla ^m P_t f\Vert _{L^\infty (\rho )}\lesssim _C t^{(\theta -m)/2}\Vert f\Vert _{{{\mathbf {C}}}^\theta (\rho )}. \end{aligned}$$

  (2.9)

• (iii) For any $0<\theta <2$, there is a constant $C=C(\rho ,d,\theta , T)>0$ such that for all $t\in [0,T]$,

  $$\begin{aligned} \Vert P_t f-f\Vert _{L^\infty (\rho )}\lesssim _C t^{\theta /2}\Vert f\Vert _{{{\mathbf {C}}}^{\theta }(\rho )}. \end{aligned}$$

  (2.10)

Proof

(i) By the definition and (2.7), we have

$$\begin{aligned}&\Vert P_t f\Vert _{{{\mathbf {C}}}^{\theta +\alpha }(\rho )} =\sup _{j\geqslant -1} 2^{(\theta +\alpha )j}\Vert \Delta _j P_t f\Vert _{L^\infty (\rho )}\\&\quad \lesssim \sup _{j\geqslant 0} 2^{(\theta +\alpha )j}\mathrm {e}^{-2^{2j}t}\Vert \Delta _jf\Vert _{L^\infty (\rho )}+\Vert \Delta _{-1} P_tf\Vert _{L^\infty (\rho )}\\&\quad \lesssim \sup _{j\geqslant 0} 2^{\theta j}\mathrm {e}^{-2^{2j}t}\Vert f\Vert _{{{\mathbf {C}}}^\alpha (\rho )}+\Vert f\Vert _{{{\mathbf {C}}}^\alpha (\rho )}\lesssim t^{-\theta /2}\Vert f\Vert _{{{\mathbf {C}}}^\alpha (\rho )}. \end{aligned}$$

(ii) For $m\in {{\mathbb {N}}}_0$ and $\theta <m$, by (2.7) we have

$$\begin{aligned}&\Vert \nabla ^m P_t f\Vert _{L^\infty (\rho )}\leqslant \sum _{j\geqslant -1}\Vert \nabla ^m \Delta _j P_t f\Vert _{L^\infty (\rho )}\\&\quad \lesssim \sum _{j\geqslant 0} 2^{mj}\mathrm {e}^{-2^{2j}t}\Vert \Delta _jf\Vert _{L^\infty (\rho )}+\Vert \Delta _{-1} f\Vert _{L^\infty (\rho )}\\&\quad \lesssim \sum _{j\geqslant 0}(2^{mj}\mathrm {e}^{-2^{2j}t}2^{-\theta j})\Vert f\Vert _{{{\mathbf {C}}}^{\theta }(\rho )}+\Vert f\Vert _{{{\mathbf {C}}}^{\theta }(\rho )} \lesssim _T t^{(\theta -m)/2}\Vert f\Vert _{{{\mathbf {C}}}^{\theta }(\rho )}. \end{aligned}$$

(iii) By (2.9), we have

$$\begin{aligned} \Vert P_t f-f\Vert _{L^\infty (\rho )}=\left\| \int ^t_0\Delta P_s f{\mathord {\mathrm{d}}}s\right\| _{L^\infty (\rho )} \lesssim \int ^t_0s^{-1+\theta /2}\Vert f\Vert _{{{\mathbf {C}}}^{\theta }(\rho )}{\mathord {\mathrm{d}}}s \lesssim t^{\theta /2}\Vert f\Vert _{{{\mathbf {C}}}^{\theta }(\rho )}. \end{aligned}$$

The proof is complete. $\square$

For given $\lambda \geqslant 0$ and $f\in L^\infty ({{\mathbb {R}}}_+; L^\infty ({{\mathbb {R}}}^d))$, we consider the following heat equation:

$$\begin{aligned} {{\mathscr {L}}}_\lambda u:=(\partial _t-\Delta +\lambda )u=f,\ \ u(0)=0. \end{aligned}$$

The unique solution to this equation is given by

$$\begin{aligned} u(t,x)=\int ^t_0\mathrm {e}^{-\lambda (t-s)}P_{t-s}f(s,x){\mathord {\mathrm{d}}}s=:{{\mathscr {I}}}_\lambda f(t,x). \end{aligned}$$

In other words, ${{\mathscr {I}}}_\lambda$ is the inverse of ${{\mathscr {L}}}_\lambda$.

The following Schauder estimate is well known for $q=\infty$ and $\theta =2$ (see [18]). Here we spell out how the implicit constants depend on $\lambda$.

Lemma 2.9

(Schauder estimates in weighted spaces) Let $\rho \in {{\mathscr {W}}}$ and

$$\begin{aligned} \alpha \in (0,1],\ \ \theta \in (\alpha ,2]. \end{aligned}$$

For any $q\in [\frac{2}{2-\theta },\infty ]$, $T>0$, there is a constant $C=C(\rho , d,\alpha ,\theta ,q, T)>0$ such that for all $\lambda \geqslant 0$ and $f\in L^q_T{{\mathbf {C}}}^{-\alpha }(\rho )$,

$$\begin{aligned} \Vert {{\mathscr {I}}}_\lambda f\Vert _{{{\mathbb {S}}}^{\theta -\alpha }_T(\rho )} \lesssim _C(\lambda \vee 1)^{\frac{\theta }{2}+\frac{1}{q}-1}\Vert f\Vert _{L^q_T{{\mathbf {C}}}^{-\alpha }(\rho )}. \end{aligned}$$

(2.11)

Proof

Let $q\in [\frac{2}{2-\theta },\infty ]$ and $\frac{1}{p}+\frac{1}{q}=1$. For $t\in (0,T]$, by (2.7) and Hölder’s inequality, we have for $j\geqslant 0$

$$\begin{aligned} 2^{j(\theta -\alpha )}\Vert \Delta _j{{\mathscr {I}}}_\lambda f(t)\Vert _{L^\infty (\rho )}&\lesssim 2^{j(\theta -\alpha )}\int ^t_0\mathrm {e}^{-(\lambda +2^{2j})(t-s)}\Vert \Delta _jf(s)\Vert _{L^\infty (\rho )}{\mathord {\mathrm{d}}}s\\&\lesssim 2^{j\theta }\left( \int ^t_0\mathrm {e}^{-p(\lambda +2^{2j})(t-s)}{\mathord {\mathrm{d}}}s\right) ^{\frac{1}{p}} \left( \int ^t_0\Vert f(s)\Vert ^q_{{{\mathbf {C}}}^{-\alpha }(\rho )}{\mathord {\mathrm{d}}}s\right) ^{\frac{1}{q}}\\&\lesssim 2^{j\theta }\left( \int ^t_0\mathrm {e}^{-p(\lambda +2^{2j}) s}{\mathord {\mathrm{d}}}s\right) ^{\frac{1}{p}}\Vert f\Vert _{L^q_T{{\mathbf {C}}}^{-\alpha }(\rho )}\\&\lesssim 2^{j\theta }(2^{2j}+\lambda )^{-\frac{1}{p}}\Vert f\Vert _{L^q_T{{\mathbf {C}}}^{-\alpha }(\rho )} \lesssim (\lambda \vee 1)^{\frac{\theta }{2}-\frac{1}{p}}\Vert f\Vert _{L^q_T{{\mathbf {C}}}^{-\alpha }(\rho )}, \end{aligned}$$

and

$$\begin{aligned} \Vert \Delta _{-1}{{\mathscr {I}}}_\lambda f(t)\Vert _{L^\infty (\rho )}&\lesssim \int ^t_0\mathrm {e}^{-\lambda (t-s)}\Vert f(s)\Vert _{{{\mathbf {C}}}^{-\alpha }(\rho )}{\mathord {\mathrm{d}}}s\\&\lesssim \Big (\int ^t_0\mathrm {e}^{-\lambda p(t-s)}{\mathord {\mathrm{d}}}s\Big )^{\frac{1}{p}}\Vert f\Vert _{L_T^q{{\mathbf {C}}}^{-\alpha }(\rho )}\\&\lesssim (\lambda \vee 1)^{-\frac{1}{p}}\Vert f\Vert _{L^q_T{{\mathbf {C}}}^{-\alpha }(\rho )}, \end{aligned}$$

which implies by the definition of Besov spaces that

$$\begin{aligned} \Vert {{\mathscr {I}}}_\lambda f\Vert _{L^\infty _T{{\mathbf {C}}}^{\theta -\alpha }(\rho )}\lesssim _C (\lambda \vee 1)^{\frac{\theta }{2}+\frac{1}{q}-1}\Vert f\Vert _{L^q_T{{\mathbf {C}}}^{-\alpha }(\rho )}. \end{aligned}$$

(2.12)

On the other hand, let $u={{\mathscr {I}}}_\lambda f$. For $0\leqslant t_1< t_2\leqslant T$, we have

$$\begin{aligned} u(t_2)-u(t_1)&=\int _0^{t_1}(\mathrm {e}^{-\lambda (t_2-s)}-\mathrm {e}^{-\lambda (t_1-s)}) P_{t_2-s}f(s){\mathord {\mathrm{d}}}s\\&\quad +(P_{t_2-t_1}-I){{\mathscr {I}}}_\lambda f(t_1)+\int _{t_1}^{t_2}\mathrm {e}^{-\lambda (t_2-s)}P_{t_2-s}f(s) {\mathord {\mathrm{d}}}s\\&=:I_1+I_2+I_3. \end{aligned}$$

For $I_1$, by (2.8) and Hölder’s inequality, we have

$$\begin{aligned} \Vert I_1\Vert _{L^\infty (\rho )}&\leqslant |\mathrm {e}^{-\lambda (t_2-t_1)}-1|\int _0^{t_1}\mathrm {e}^{-\lambda (t_1-s)}\Vert P_{t_2-s}f(s)\Vert _{L^\infty (\rho )} ds\\&\lesssim \Big ((\lambda (t_2-t_1))\wedge 1\Big )\int _0^{t_1}\mathrm {e}^{-\lambda (t_1-s)}(t_2-s)^{-\frac{\alpha }{2}}\Vert f(s)\Vert _{{{\mathbf {C}}}^{-\alpha }(\rho )} ds\\&\leqslant (\lambda (t_2-t_1))^{\frac{\theta }{2}}(t_2-t_1)^{-\frac{\alpha }{2}}\left( \int _0^{t_1} \mathrm {e}^{-\lambda (t_1-s)p}ds\right) ^{1/p}\Vert f\Vert _{L^q_T{{\mathbf {C}}}^{-\alpha }(\rho )}\\&\lesssim (t_2-t_1)^{\frac{\theta -\alpha }{2}}\lambda ^{\frac{\theta }{2}-\frac{1}{p}}\Vert f\Vert _{L^q_T{{\mathbf {C}}}^{-\alpha }(\rho )}. \end{aligned}$$

For $I_2$, by (2.10) and (2.12) we have

$$\begin{aligned} \Vert I_2\Vert _{L^\infty (\rho )}&\lesssim (t_2-t_1)^{\frac{\theta -\alpha }{2}}\Vert {{\mathscr {I}}}_\lambda f\Vert _{L^\infty _T{{\mathbf {C}}}^{\theta -\alpha }(\rho )}\\&\lesssim (t_2-t_1)^{\frac{\theta -\alpha }{2}}(\lambda \vee 1)^{\frac{\theta }{2}-\frac{1}{p}}\Vert f\Vert _{L^q_T{{\mathbf {C}}}^{-\alpha }(\rho )}. \end{aligned}$$

For $I_3$, by (2.9) and the change of variable, we have

$$\begin{aligned} \Vert I_3\Vert _{L^\infty (\rho )}&\lesssim \lambda ^{\frac{\alpha }{2}-\frac{1}{p}} \left( \int _0^{\lambda (t_2-t_1)}\mathrm {e}^{-sp}s^{-\frac{\alpha p}{2}}{\mathord {\mathrm{d}}}s\right) ^{\frac{1}{p}}\Vert f\Vert _{L^q_T{{\mathbf {C}}}^{-\alpha }(\rho )}\\&\lesssim (t_2-t_1)^{\frac{\theta -\alpha }{2}}\lambda ^{-1+\frac{\theta }{2}+\frac{1}{q}}\Vert f\Vert _{L^q_T{{\mathbf {C}}}^{-\alpha }(\rho )} , \end{aligned}$$

where we used $\mathrm {e}^{-sp}s^{-\frac{\alpha p}{2}}\leqslant s^{\frac{(\theta -\alpha )p}{2}-1}$ for all $s>0$. Therefore,

$$\begin{aligned} \Vert {{\mathscr {I}}}_\lambda f\Vert _{C^{(\theta -\alpha )/2}_TL^\infty (\rho )}\lesssim _C (\lambda \vee 1)^{\frac{\theta }{2}+\frac{1}{q}-1}\Vert f\Vert _{L^q_T{{\mathbf {C}}}^{-\alpha }(\rho )}, \end{aligned}$$

(2.13)

which together with (2.12) yields (2.11). $\square$

2.3 Paracontrolled calculus

In this subsection we recall some basic ingredients in the paracontrolled calculus developed by Bony [5] and [20]. The first important fact is that the product fg of two distributions $f\in {{\mathbf {C}}}^\alpha$ and $g\in {{\mathbf {C}}}^\beta$ is well defined if and only if $\alpha +\beta >0$. In terms of Littlewood-Paley’s block operator $\Delta _j$, the product fg of two distributions f and g can be formally decomposed as

$$\begin{aligned} fg=f\prec g+f\circ g+f\succ g, \end{aligned}$$

where

$$\begin{aligned} f\prec g=g\succ f:=\sum _{j\geqslant -1}\sum _{i<j-1}\Delta _if\Delta _jg, \quad f\circ g:=\sum _{|i-j|\leqslant 1}\Delta _if\Delta _jg. \end{aligned}$$

In the following we collect some important estimates from [18] about the paraproducts in weighted Besov spaces, that will be used below.

Lemma 2.10

Let $\rho _{1},\rho _{2}$ be two admissible weights. We have for any $\beta \in {\mathbb {R}}$,

$$\begin{aligned} \Vert f\prec g\Vert _{{{\mathbf {C}}}^\beta (\rho _{1}\rho _{2})}\lesssim \Vert f\Vert _{L^\infty (\rho _{1})}\Vert g\Vert _{{{\mathbf {C}}}^{\beta }(\rho _{2})}, \end{aligned}$$

(2.14)

and for any $\alpha <0$ and $\beta \in {{\mathbb {R}}}$,

$$\begin{aligned} \Vert f\prec g\Vert _{{{\mathbf {C}}}^{\alpha +\beta }(\rho _{1}\rho _{2})}\lesssim \Vert f\Vert _{{{\mathbf {C}}}^{\alpha }(\rho _{1})}\Vert g\Vert _{{{\mathbf {C}}}^{\beta }(\rho _{2})}. \end{aligned}$$

(2.15)

Moreover, for any $\alpha ,\beta \in {{\mathbb {R}}}$ with $\alpha +\beta >0$,

$$\begin{aligned} \Vert f\circ g\Vert _{{{\mathbf {C}}}^{\alpha +\beta }(\rho _{1}\rho _{2})}\lesssim \Vert f\Vert _{{{\mathbf {C}}}^{\alpha }(\rho _{1})}\Vert g\Vert _{{{\mathbf {C}}}^{\beta }(\rho _{2})}. \end{aligned}$$

(2.16)

In particular, if $\alpha +\beta >0$, then

$$\begin{aligned} \Vert f g\Vert _{{{\mathbf {C}}}^{\alpha \wedge \beta }(\rho _{1}\rho _{2})}\lesssim \Vert f\Vert _{{{\mathbf {C}}}^{\alpha }(\rho _{1})}\Vert g\Vert _{{{\mathbf {C}}}^{\beta }(\rho _{2})}. \end{aligned}$$

(2.17)

Proof

See [18, Lemma 2.14]. $\square$

Lemma 2.11

Let $\rho _{1}, \rho _{2}, \rho _{3}$ be three admissible weights. For any $\alpha \in (0,1)$ and $\beta ,\gamma \in {\mathbb {R}}$ with $\alpha +\beta +\gamma >0$ and $\beta +\gamma <0$, there exists a bounded trilinear operator $\mathrm {com}$ on ${{\mathbf {C}}}^\alpha (\rho _{1})\times {{\mathbf {C}}}^\beta (\rho _{2})\times {{\mathbf {C}}}^\gamma (\rho _{3})$ such that

$$\begin{aligned} \Vert \mathrm {com}(f,g,h)\Vert _{{{\mathbf {C}}}^{\alpha +\beta +\gamma }(\rho _{1}\rho _{2}\rho _{3})}\lesssim \Vert f\Vert _{{{\mathbf {C}}}^\alpha (\rho _{1})}\Vert g\Vert _{{{\mathbf {C}}}^\beta (\rho _{2})}\Vert h\Vert _{{{\mathbf {C}}}^\gamma (\rho _{3})}, \end{aligned}$$

(2.18)

where for smooth functions f, g, h,

$$\begin{aligned} \mathrm {com}(f,g,h):=(f\prec g)\circ h - f(g\circ h). \end{aligned}$$

Proof

See [18, Lemma 2.16]. $\square$

Moreover, we will make use of the time-mollified paraproducts as introduced in [20, Section 5]. Let $Q:{\mathbb {R}}\rightarrow {\mathbb {R}}_{+}$ be a smooth function with support in $[-1,1]$ and $\int _{{\mathbb {R}}}Q(s)\mathrm {d}s=1$. For $T>0$ and $j\geqslant -1$, we define an operator $Q_{j}: L^\infty _T{{\mathbf {C}}}^{\alpha }(\rho )\rightarrow L^\infty _T{{\mathbf {C}}}^{\alpha }(\rho )$ by

$$\begin{aligned} Q_{j}f(t):=\int _{{\mathbb {R}}}2^{2j}Q(2^{2j}(t-s))f( (s\wedge T)\vee 0)\mathrm {d} s, \end{aligned}$$

and the modified paraproduct of $f,g\in L^\infty _T{{\mathbf {C}}}^{\alpha }(\rho )$ by

$$\begin{aligned} f\prec \!\!\!\prec g := \sum _{j\geqslant -1}(S_{j-1}Q_{j}f)\Delta _{j} g\, \text { with } S_jf=\sum _{i\leqslant j-1}\Delta _if. \end{aligned}$$

Note that for $\alpha < 0$, $\beta \in {{\mathbb {R}}}$ and $\rho _1,\rho _2\in {{\mathscr {W}}}$,

$$\begin{aligned} \Vert f\prec \!\!\!\prec g\Vert _{L^\infty _T{{\mathbf {C}}}^{\alpha +\beta }(\rho _{1}\rho _{2})}\lesssim \Vert f\Vert _{L^\infty _T{{\mathbf {C}}}^{\alpha }(\rho _{1})}\Vert g\Vert _{L^\infty _T{{\mathbf {C}}}^{\beta }(\rho _{2})}. \end{aligned}$$

(2.19)

Lemma 2.12

Let $\rho _{1},\rho _{2}$ be two admissible weights and $T>0$. For any $\alpha \in (0,1)$ and $\beta \in {\mathbb {R}}$, there is a constant $C=C(\rho _1,\rho _2,d,\alpha ,\beta )>0$ such that for all $\lambda \geqslant 0$

$$\begin{aligned} \big \Vert [{{\mathscr {L}}}_\lambda ,f\prec \!\!\!\prec ] g\big \Vert _{L^\infty _T{{\mathbf {C}}}^{\alpha +\beta -2}(\rho _{1}\rho _{2})}&\lesssim _C \Vert f\Vert _{{{\mathbb {S}}}^{\alpha }_T(\rho _{1})}\Vert g\Vert _{L^\infty _T{{\mathbf {C}}}^{\beta }(\rho _{2})}, \end{aligned}$$

(2.20)

and

$$\begin{aligned} \Vert f\prec g-f\prec \!\!\!\prec g\Vert _{L^\infty _T{{\mathbf {C}}}^{\alpha +\beta }(\rho _{1}\rho _{2})}\lesssim _C \Vert f\Vert _{C^{\alpha /2}_TL^\infty (\rho _{1})}\Vert g\Vert _{L^\infty _T{{\mathbf {C}}}^{\beta }(\rho _{2})}. \end{aligned}$$

(2.21)

Moreover, for any $\varepsilon >0$, we also have for some $C=C(\varepsilon ,\rho _1,\rho _2,d,\alpha ,\beta , T)$,

$$\begin{aligned} \Vert [\nabla {{\mathscr {I}}}_\lambda , f \prec ]g\Vert _{L^\infty _T{{\mathbf {C}}}^{\alpha +\beta +1-\varepsilon }(\rho _1\rho _2)}&\lesssim _C\Vert f\Vert _{{{\mathbb {S}}}^{\alpha }_T(\rho _1)}\Vert g\Vert _{L^\infty _T{{\mathbf {C}}}^\beta (\rho _2)}. \end{aligned}$$

(2.22)

Proof

The estimates (2.20) and (2.21) can be found in [18, Lemma 2.17]. We only prove (2.22). By definition, we have

$$\begin{aligned}&[\nabla {{\mathscr {I}}}_\lambda , f \prec ]g(t)\\&\quad =\int ^t_0\mathrm {e}^{-\lambda (t-s)}P_{t-s}\nabla (f(s)\prec g(s)){\mathord {\mathrm{d}}}s-f(t)\prec \int ^t_0\mathrm {e}^{-\lambda (t-s)}\nabla P_{t-s} g(s){\mathord {\mathrm{d}}}s\\&\quad =\int _0^t \mathrm {e}^{-\lambda (t-s)}P_{t-s} (\nabla f(s)\prec g(s)){\mathord {\mathrm{d}}}s+\int _0^t\mathrm {e}^{-\lambda (t-s)}[P_{t-s},f(s)\prec ] \nabla g(s) {\mathord {\mathrm{d}}}s\\&\quad +\int _0^t\mathrm {e}^{-\lambda (t-s)}(f(s)-f(t))\prec P_{t-s}\nabla g(s){\mathord {\mathrm{d}}}s=:I_1(t)+I_2(t)+I_3(t). \end{aligned}$$

For $I_1$, by (2.12) with $\theta =2$ and $q=\infty$ and (2.15), we have

$$\begin{aligned} \Vert I_1(t)\Vert _{L^\infty _T{{\mathbf {C}}}^{\alpha +\beta +1}(\rho _1\rho _2)}&\lesssim \Vert \nabla f\prec g\Vert _{L^\infty _T{{\mathbf {C}}}^{\alpha +\beta -1}(\rho _1\rho _2)} \lesssim \Vert f\Vert _{L^\infty _T{{\mathbf {C}}}^\alpha (\rho _1)}\Vert g\Vert _{L^\infty _T{{\mathbf {C}}}^\beta (\rho _2)}. \end{aligned}$$

For $I_2$, by a modification of [9, Lemma A.1] we have by $\mathrm {e}^{-\lambda (t-s)}\leqslant 1$

$$\begin{aligned} \Vert I_2(t)\Vert _{{{\mathbf {C}}}^{\alpha +\beta +1-\varepsilon }(\rho _1\rho _2)}&\lesssim \int _0^t (t-s)^{-1+\frac{\varepsilon }{2}}\Vert f(s)\Vert _{{{\mathbf {C}}}^\alpha (\rho _1)}\Vert g(s)\Vert _{{{\mathbf {C}}}^\beta (\rho _2)}{\mathord {\mathrm{d}}}s\\&\lesssim \Vert f\Vert _{L^\infty _T{{\mathbf {C}}}^\alpha (\rho _1)}\Vert g\Vert _{L^\infty _T{{\mathbf {C}}}^\beta (\rho _2)}. \end{aligned}$$

For $I_3$, by (2.14) and (2.8) we have by $\mathrm {e}^{-\lambda (t-s)}\leqslant 1$

$$\begin{aligned} \Vert I_3(t)\Vert _{{{\mathbf {C}}}^{\alpha +\beta +1-\varepsilon }(\rho _1\rho _2)}&\lesssim \int _0^t\Vert f(s)-f(t)\Vert _{L^\infty (\rho _1)}\Vert \nabla P_{t-s}g(s)\Vert _{{{\mathbf {C}}}^{\alpha +\beta +1-\varepsilon }(\rho _2)}{\mathord {\mathrm{d}}}s\\&\lesssim \Vert f\Vert _{C_T^{\alpha /2}L^\infty (\rho _1)}\Vert g\Vert _{L^\infty _T{{\mathbf {C}}}^\beta (\rho _2)}\int ^t_0(t-s)^{-1+\frac{\varepsilon }{2}}{\mathord {\mathrm{d}}}s. \end{aligned}$$

The proof is complete. $\square$

Finally we recall the localization operators from [18]. Let $\sum _{k\geqslant -1} w_k = 1$ be a smooth dyadic partition of unity on ${\mathbb {R}}^d$, where $w_{-1}$ is supported in a ball containing zero and there exists an annulus ${{\mathscr {A}}}$ such that for each $k\geqslant 0$, $w_k$ is supported on the annulus $2^k{{\mathscr {A}}}$. Let $(v_{m})_{m\geqslant -1}$ be a smooth dyadic partition of unity on $[0,\infty )$ such that $v_{-1}$ is supported in a ball containing zero and for each $m\geqslant 0$, $v_m$ is supported on the annulus of size $2^m$. For a given sequence $(L_{k,m})_{k,m\geqslant -1}$, we define localization operators ${\mathscr {V}}_{>}, {\mathscr {V}}_{\leqslant }$ as in [18]

$$\begin{aligned} \begin{aligned} {\mathscr {V}}_{>} f(t,x) = \sum _{k,m}w_k(x) v_{m}(t)\sum _{j>L_{k,m}}\Delta _j f(t,\cdot )(x), \\ {\mathscr {V}}_{\leqslant } f(t,x) = \sum _{k,m} w_k(x) v_{m}(t)\sum _{j\leqslant L_{k,m}}\Delta _j f(t,\cdot )(x). \end{aligned} \end{aligned}$$

(2.23)

Lemma 2.13

Let $\rho$ be an admissible weight. For given $L>0, T>0$, there exists a (universal) choice of parameters $(L_{k,m})_{k,m\geqslant -1}$ such that for all $\alpha ,\beta ,\kappa \in {{\mathbb {R}}}$

and $\gamma , \delta >0$

$$\begin{aligned}&\Vert {\mathscr {V}}_{>} f \Vert _{L^\infty _T{{\mathbf {C}}}^{- \alpha - \delta } (\rho ^{\beta -\delta })} \lesssim 2^{- \delta L} \Vert f \Vert _{L^\infty _T{{\mathbf {C}}}^{- \alpha }(\rho ^{\beta })},\\&\Vert {\mathscr {V}}_{\leqslant } f \Vert _{L^\infty _T{{\mathbf {C}}}^{\gamma -\alpha } (\rho ^{\beta +\gamma })} \lesssim 2^{\gamma L} \Vert f \Vert _{L^\infty _T{{\mathbf {C}}}^{- \alpha } (\rho ^{\beta })}, \end{aligned}$$

where the proportional constant depends on $\alpha ,\beta ,\delta ,\gamma$ but is independent of f.

Proof

The proof is exactly the same as in [18, Lemma 2.6] although $\alpha >0$ is required therein. $\square$

2.4 Renormalized pairs

In this section we introduce the renormalized pairs, which correspond to the stochastic objects in the theory of singular SPDEs. Fix $\alpha \in (\frac{1}{2},\frac{2}{3})$ and an admissible weight $\rho \in {{\mathscr {W}}}$. For $T>0$, let $b=(b_1,\cdots , b_d)$ and f be $(d+1)$-distributions in $L^\infty _T{{\mathbf {C}}}^{-\alpha }(\rho )$. First of all, we introduce two quantities for later use

$$\begin{aligned} \ell ^b_T(\rho ):=\sup _{\lambda \geqslant 0}\Vert b\circ \nabla {{\mathscr {I}}}_\lambda b\Vert _{L^\infty _T{{\mathbf {C}}}^{1-2\alpha }(\rho ^2)}+\Vert b\Vert ^2_{L_T^\infty {{\mathbf {C}}}^{-\alpha }(\rho )}+1, \end{aligned}$$

(2.24)

and for $q\in [1,\infty ]$,

$$\begin{aligned} {{\mathbb {A}}}^{b,f}_{T,q}(\rho ):=\sup _{\lambda \geqslant 0}\Vert b\circ \nabla {{\mathscr {I}}}_\lambda f\Vert _{L^q_T{{\mathbf {C}}}^{1-2\alpha }(\rho ^2)}+(\Vert b\Vert _{L_T^\infty {{\mathbf {C}}}^{-\alpha }(\rho )}+1)\Vert f\Vert _{L^q_T{{\mathbf {C}}}^{-\alpha }(\rho )}. \end{aligned}$$

(2.25)

By (2.16), except for $\alpha <\frac{1}{2}$, in general, $b(t)\circ \nabla {{\mathscr {I}}}_\lambda f(t)$ is not well-defined since by Schauder’s estimate, we only have (see Lemma 2.9)

$$\begin{aligned} \nabla {{\mathscr {I}}}_\lambda f\in L^\infty _T{{\mathbf {C}}}^{1-\alpha }(\rho ). \end{aligned}$$

However, in the probabilistic sense, it is possible to give a meaning for $b\circ \nabla {{\mathscr {I}}}_\lambda f$ when b, f belong to the chaos of Gaussian noise (see Sect. 6 below). This motivates us to introduce the following notion.

Definition 2.14

We call the above $(b,f)\in L^\infty _T{{\mathbf {C}}}^{-\alpha }(\rho )$ a renormalized pair if there exist $b_n, f_n\in L^\infty _T{{\mathscr {C}}}^\infty (\rho )$ with $\sup _{n\in {{\mathbb {N}}}}\big (\ell ^{b_n}_T(\rho )+{{\mathbb {A}}}^{b_n,f_n}_{T,\infty }(\rho )\big )<\infty$ and such that $(b_n,f_n)$ converges to (b, f) in $L_T^\infty {{\mathbf {C}}}^{-\alpha }(\rho )$. Moreover, for each $\lambda \geqslant 0$, there are functions $g_\lambda , h_\lambda \in L_T^\infty {{\mathbf {C}}}^{1-2\alpha }(\rho ^2)$ such that

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert b_n\circ \nabla {{\mathscr {I}}}_\lambda f_n-g_\lambda \Vert _{L_T^\infty {{\mathbf {C}}}^{1-2\alpha }(\rho ^2)}=0 \end{aligned}$$

(2.26)

and

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert b_n\circ \nabla {{\mathscr {I}}}_\lambda b_n-h_\lambda \Vert _{L_T^\infty {{\mathbf {C}}}^{1-2\alpha }(\rho ^2)}=0. \end{aligned}$$

(2.27)

For notational convenience, we shall write

$$\begin{aligned} g_\lambda =:b\circ \nabla {{\mathscr {I}}}_\lambda f,\ h_\lambda =:b\circ \nabla {{\mathscr {I}}}_\lambda b. \end{aligned}$$

The set of all the above renormalized pair is denoted by ${{\mathbb {B}}}^\alpha _T(\rho )$.

Remark 2.15

(i) Let $b\in {{\mathbb {L}}}^\infty _T(\rho )$ and $f\in L^\infty _T{{\mathbf {C}}}^{-\alpha }(\rho )$. Define $b_n(t,x):=b(t,\cdot )*\varrho _n(x)$ and $f_n(t,x):=f(t,\cdot )*\varrho _n(x)$ with $\varrho _n$ being the usual mollifier. By Definition 2.14 and (2.16), it is easy to see that $(b,f)\in {{\mathbb {B}}}^\alpha _T(\rho )$. Moreover, if $(b,f)\in {{\mathbb {B}}}^\alpha _T(\rho )$ and $b'\in {{\mathbb {L}}}^\infty _T(\rho )$, then $(b+b', f)\in {{\mathbb {B}}}^\alpha _T(\rho )$.

(ii) To make the convergence in (2.26) and (2.27) hold, we need to subtract some terms containing renormalization constants in the approximation $b_n\circ \nabla {{\mathscr {I}}}_\lambda f_n$ and $b_n\circ \nabla {{\mathscr {I}}}_\lambda b_n$. In Definition 2.14, we suppose that the renormalization constants are zero for simplicity. Indeed in concrete examples we can choose symmetric mollifiers for approximations, which make the renormalization constants vanish (see Sect. 6). In general we only use the uniform bounds $\sup _{n\in {{\mathbb {N}}}}\big (\ell ^{b_n}_T(\rho )+{{\mathbb {A}}}^{b_n,f_n}_{T,\infty }(\rho )\big )<\infty$ and the convergence in (2.26), (2.27). In particular, the renormalization constants do not affect our analysis and calculations.

An integration by parts allows to eliminate the parameter $\lambda$ in (2.26) and (2.27) as shown in the following lemma, where the right hand side can be estimated by some probabilistic calculations (see Sect. 6).

Lemma 2.16

Let ${{\mathscr {I}}}^{t}_s(f)=\int _s^t P_{t-r}f(r){\mathord {\mathrm{d}}}r$. For any $t>0$, we have

$$\begin{aligned} \sup _{\lambda \geqslant 0}\Vert b(t)\circ \nabla {{\mathscr {I}}}_\lambda f(t)\Vert _{{{\mathbf {C}}}^{1-2\alpha }(\rho )}\leqslant 2\sup _{s\in [0,t]}\Vert b(t)\circ \nabla {{\mathscr {I}}}^{t}_s(f)\Vert _{{{\mathbf {C}}}^{1-2\alpha }(\rho )}. \end{aligned}$$

(2.28)

Proof

Note that by integration by parts formula,

$$\begin{aligned}&\int _0^t \mathrm {e}^{-\lambda (t-s)}P_{t-s}f(s){\mathord {\mathrm{d}}}s =\int ^t_0P_{t-s}f(s){\mathord {\mathrm{d}}}s-\lambda \int ^t_0\mathrm {e}^{-\lambda (t-s)}\int ^s_0P_{t-r}f(r){\mathord {\mathrm{d}}}r{\mathord {\mathrm{d}}}s\\&\qquad =\mathrm {e}^{-\lambda t}\int ^t_0P_{t-s}f(s){\mathord {\mathrm{d}}}s+\lambda \int ^t_0\mathrm {e}^{-\lambda (t-s)}\int ^t_{s}P_{t-r}f(r){\mathord {\mathrm{d}}}r{\mathord {\mathrm{d}}}s. \end{aligned}$$

Thus,

$$\begin{aligned}&b(t)\circ \nabla {{\mathscr {I}}}_\lambda f(t)= \mathrm {e}^{-\lambda t}b(t)\circ \nabla {{\mathscr {I}}}^t_0f+\lambda \int ^t_0\mathrm {e}^{-\lambda (t-s)}b(t)\circ \nabla {{\mathscr {I}}}^t_s(f){\mathord {\mathrm{d}}}s. \end{aligned}$$

From this we get the desired estimate. $\square$

The following lemma is used to deal with the localization of renormalized terms.

Lemma 2.17

Let $T>0$, $\rho ,{\bar{\rho }}\in {{\mathscr {W}}}$, $\varepsilon \in (0,1)$ and $\alpha \in (\frac{1}{2},\frac{2}{3})$. Suppose that

$$\begin{aligned} \phi \in {{\mathbf {C}}}^{\alpha +\varepsilon }({\bar{\rho }}\rho ^{-2}),\ \psi \in {{\mathbb {S}}}^{\alpha +\varepsilon }_T,\ (b,f)\in {{\mathbb {B}}}^\alpha _T(\rho ). \end{aligned}$$

Then there is a constant $C>0$ depending only on $T, \varepsilon ,\alpha ,d,\rho ,{\bar{\rho }}$ such that for all $\lambda \geqslant 0$ and $t\in [0,T]$,

$$\begin{aligned}&\Vert ((b\phi )\circ \nabla {{\mathscr {I}}}_\lambda (f\psi ))(t)\Vert _{{{\mathbf {C}}}^{1-2\alpha }({\bar{\rho }})} \lesssim _C\Vert \phi \Vert _{{{\mathbf {C}}}^{\alpha +\varepsilon }({\bar{\rho }}\rho ^{-2})}\Vert \psi \Vert _{{{\mathbb {S}}}^{\alpha +\varepsilon }_t}{{\mathbb {A}}}^{b,f}_{t,\infty }(\rho ). \end{aligned}$$

(2.29)

Proof

We only prove the estimate (2.29). For simplicity, we drop the time variable. By using paraproduct, we have

$$\begin{aligned} (b\phi )\circ \nabla {{\mathscr {I}}}_\lambda (f\psi )&=(b\phi )\circ \nabla {{\mathscr {I}}}_\lambda (\psi \succcurlyeq f)+(b\phi )\circ \nabla {{\mathscr {I}}}_\lambda (\psi \prec f)\\&=(b\phi )\circ \nabla {{\mathscr {I}}}_\lambda (\psi \succcurlyeq f)+(b\phi )\circ [\nabla {{\mathscr {I}}}_\lambda ,\psi \prec ]f\\&\quad +\text {com}(\psi ,\nabla {{\mathscr {I}}}_\lambda f,b\phi )+\psi ((b\phi )\circ \nabla {{\mathscr {I}}}_\lambda f)\\&=(b\phi )\circ \nabla {{\mathscr {I}}}_\lambda (\psi \succcurlyeq f)+(b\phi )\circ [\nabla {{\mathscr {I}}}_\lambda ,\psi \prec ]f\\&\quad +\text {com}(\psi ,\nabla {{\mathscr {I}}}_\lambda f,b\phi )+\psi ((\phi \succcurlyeq b)\circ \nabla {{\mathscr {I}}}_\lambda f)\\&\quad +\psi \text {com}(\phi ,b,\nabla {{\mathscr {I}}}_\lambda f)+\psi \phi ( b\circ \nabla {{\mathscr {I}}}_\lambda f). \end{aligned}$$

Let $\varepsilon >0$ small enough. We estimate each term as follows:

• By (2.16), (2.11) and (2.15), we have

  $$\begin{aligned} \Vert (b\phi )\circ \nabla {{\mathscr {I}}}_\lambda (\psi \succcurlyeq f)\Vert _{{{\mathbf {C}}}^0({\bar{\rho }})}&\lesssim \Vert b\phi \Vert _{{{\mathbf {C}}}^{-\alpha }({\bar{\rho }}\rho ^{-1})}\Vert \nabla {{\mathscr {I}}}_\lambda (\psi \succcurlyeq f)\Vert _{L^\infty _t{{\mathbf {C}}}^{\alpha +\varepsilon }(\rho )}\\&\lesssim \Vert b\phi \Vert _{{{\mathbf {C}}}^{-\alpha }({\bar{\rho }}\rho ^{-1})}\Vert \psi \succ f+\psi \circ f\Vert _{L^\infty _t{{\mathbf {C}}}^{\alpha -1+\varepsilon }(\rho )}\\&\lesssim \Vert \phi \Vert _{{{\mathbf {C}}}^{\alpha +\varepsilon }({\bar{\rho }}\rho ^{-2})}\Vert b\Vert _{{{\mathbf {C}}}^{-\alpha }(\rho )}\Vert f\Vert _{L^\infty _t{{\mathbf {C}}}^{-\alpha }(\rho )}\Vert \psi \Vert _{L^\infty _t{{\mathbf {C}}}^{\alpha +\varepsilon }}. \end{aligned}$$

• By (2.16), (2.17) and (2.22), we have

  $$\begin{aligned} \Vert (b\phi )\circ [\nabla {{\mathscr {I}}}_\lambda ,\psi \prec ]f\Vert _{{{\mathbf {C}}}^0({\bar{\rho }})}&\lesssim \Vert b\phi \Vert _{{{\mathbf {C}}}^{-\alpha }({\bar{\rho }}\rho ^{-1})}\Vert [\nabla {{\mathscr {I}}}_\lambda ,\psi \prec ]f\Vert _{L_t^\infty {{\mathbf {C}}}^{\alpha +\varepsilon }(\rho )}\\&\lesssim \Vert \phi \Vert _{{{\mathbf {C}}}^{\alpha +\varepsilon }({\bar{\rho }}\rho ^{-2})}\Vert b\Vert _{{{\mathbf {C}}}^{-\alpha }(\rho )}\Vert \psi \Vert _{{{\mathbb {S}}}_t^{2\alpha -1+2\varepsilon }}\Vert f\Vert _{L_t^\infty {{\mathbf {C}}}^{-\alpha }(\rho )}. \end{aligned}$$

• By (2.18), (2.11) and (2.17), we have

  $$\begin{aligned} \Vert \text {com}(\psi ,\nabla {{\mathscr {I}}}_\lambda f,b\phi )\Vert _{{{\mathbf {C}}}^0({\bar{\rho }})}&\lesssim \Vert \psi \Vert _{{{\mathbf {C}}}^{2\alpha -1+\varepsilon }}\Vert \nabla {{\mathscr {I}}}_\lambda f\Vert _{L^\infty _t{{\mathbf {C}}}^{1-\alpha }(\rho )}\Vert b\phi \Vert _{{{\mathbf {C}}}^{-\alpha }({\bar{\rho }} \rho ^{-1})}\\&\lesssim \Vert \psi \Vert _{{{\mathbf {C}}}^{2\alpha -1+\varepsilon }}\Vert f\Vert _{L^\infty _t{{\mathbf {C}}}^{-\alpha }(\rho )}\Vert b\Vert _{{{\mathbf {C}}}^{-\alpha }(\rho )}\Vert \phi \Vert _{{{\mathbf {C}}}^{\alpha +\varepsilon }({\bar{\rho }}\rho ^{-2})}. \end{aligned}$$

• By (2.17), (2.16), (2.11) and (2.15), we have

  $$\begin{aligned} \Vert \psi ((\phi \succcurlyeq b)\circ \nabla {{\mathscr {I}}}_\lambda f)\Vert _{{{\mathbf {C}}}^0({\bar{\rho }})}&\lesssim \Vert \psi \Vert _{L^\infty }\Vert \phi \succcurlyeq b\Vert _{{{\mathbf {C}}}^{\alpha -1+\varepsilon }({\bar{\rho }}\rho ^{-1})}\Vert \nabla {{\mathscr {I}}}_\lambda f\Vert _{{{\mathbf {C}}}^{1-\alpha }(\rho )}\\&\lesssim \Vert \psi \Vert _{L^\infty }\Vert \phi \Vert _{{{\mathbf {C}}}^{\alpha +\varepsilon }({\bar{\rho }}\rho ^{-2})}\Vert b\Vert _{{{\mathbf {C}}}^{-\alpha }(\rho )}\Vert f\Vert _{L_t^\infty {{\mathbf {C}}}^{-\alpha }(\rho )}. \end{aligned}$$

• By (2.17) and (2.18), we have

  $$\begin{aligned} \Vert \psi \text {com}(\phi ,b,\nabla {{\mathscr {I}}}_\lambda f)\Vert _{{{\mathbf {C}}}^0({\bar{\rho }})} \lesssim \Vert \psi \Vert _{L^\infty }\Vert \phi \Vert _{{{\mathbf {C}}}^{2\alpha -1+\varepsilon }({\bar{\rho }}\rho ^{-2})}\Vert b\Vert _{{{\mathbf {C}}}^{-\alpha }(\rho )}\Vert f\Vert _{L^\infty _t{{\mathbf {C}}}^{-\alpha }(\rho )}. \end{aligned}$$

• By (2.17), we have

  $$\begin{aligned} \Vert \psi \phi ( b\circ \nabla {{\mathscr {I}}}_\lambda f)\Vert _{{{\mathbf {C}}}^{1-2\alpha }({\bar{\rho }})} \lesssim \Vert \psi \phi \Vert _{{{\mathbf {C}}}^{2\alpha -1+\varepsilon }({\bar{\rho }}\rho ^{-2})}\Vert b\circ \nabla {{\mathscr {I}}}_\lambda f\Vert _{{{\mathbf {C}}}^{1-2\alpha }(\rho ^2)}. \end{aligned}$$

Combining the above calculations, we obtain the desired estimate. $\square$

3 A study of linear parabolic equations in weighted Hölder spaces

In this section we consider the following linear parabolic equation:

$$\begin{aligned} {{\mathscr {L}}}_\lambda u=(\partial _t-\Delta +\lambda ) u=b\cdot \nabla u+f,\quad u(0)=u_0{\in \cup _{\epsilon >0}{{\mathbf {C}}}^{1+\alpha +\epsilon }}, \end{aligned}$$

(3.1)

where $\lambda \geqslant 0$, $b=(b_1,\cdots , b_d)$ is a vector-valued distribution and f is a scalar-valued distribution. Suppose that for some $\alpha \in (\frac{1}{2},\frac{2}{3})$ and admissible weight $\rho \in {{\mathscr {W}}}$,

$$\begin{aligned} (b, f)\in {{\mathbb {B}}}^\alpha _T(\rho ),\ \ T>0. \end{aligned}$$

(3.2)

The aim of this section is to show the well-posedness of PDE (3.1) under (3.2). We first give the definition of the paracontrolled solutions to (3.1). We then establish the Schauder estimate with the coefficients in unweighted Besov spaces by choosing $\lambda$ large enough. Then by a classical maximum principle, we obtain the Schauder estimate for (3.1) depending polynomially on the coefficients. In Sect. 3.3 we establish global well-posedness of equation (3.1) under (3.2) and obtain a uniform estimate of solution to (3.1) in Besov spaces with sublinear weights, where the proofs are based on a new characterization of weighted Hölder spaces and a localization argument.

3.1 Paracontrolled solutions

To introduce the paracontrolled solution of PDE (3.1), by Bony’s decomposition, we make the following paracontrolled ansatz as in [20]:

$$\begin{aligned} u= \nabla u\prec \!\!\!\prec {{\mathscr {I}}}_\lambda b+u^\sharp +{{\mathscr {I}}}_\lambda f+\mathrm {e}^{-\lambda t}P_tu_0, \end{aligned}$$

(3.3)

where $u^\sharp$ solves the following PDE in the weak sense

$$\begin{aligned} {{\mathscr {L}}}_\lambda u^\sharp&=\nabla u\prec b-\nabla u\prec \!\!\!\prec b+\nabla u\succ b+b\circ \nabla u-[{{\mathscr {L}}}_\lambda , \nabla u\prec \!\!\!\prec ]{{\mathscr {I}}}_\lambda b \end{aligned}$$

(3.4)

with $u^\sharp (0)=0$. Note that $b\circ \nabla u$ is not well-defined in the classical sense. By the paracontrolled ansatz (3.3), we have

$$\begin{aligned} b\circ \nabla u&=b\circ \nabla (\nabla u\prec \!\!\!\prec {{\mathscr {I}}}_\lambda b)+b\circ \nabla u^\sharp +b\circ \nabla {{\mathscr {I}}}_\lambda f+b\circ \nabla P_tu_0\mathrm {e}^{-\lambda t}\nonumber \\&=b\circ \nabla (\nabla u\prec {{\mathscr {I}}}_\lambda b)+\text {com}_1+b\circ \nabla u^\sharp +b\circ \nabla {{\mathscr {I}}}_\lambda f+b\circ \nabla P_tu_0\mathrm {e}^{-\lambda t}\nonumber \\&=b\circ (\nabla ^2 u\prec {{\mathscr {I}}}_\lambda b)+(b\circ \nabla {{\mathscr {I}}}_\lambda b)\cdot \nabla u+\text {com}\nonumber \\&\quad +\text {com}_1+b\circ \nabla u^\sharp +b\circ \nabla {{\mathscr {I}}}_\lambda f+b\circ \nabla P_tu_0\mathrm {e}^{-\lambda t}, \end{aligned}$$

(3.5)

where

$$\begin{aligned} \text {com}_1:=b\circ \nabla [\nabla u\prec \!\!\!\prec {{\mathscr {I}}}_\lambda b-\nabla u\prec {{\mathscr {I}}}_\lambda b] \end{aligned}$$

and

$$\begin{aligned} \text {com}:=\mathrm {com}(\nabla u, \nabla {{\mathscr {I}}}_\lambda b,b). \end{aligned}$$

Definition 3.1

Let $\rho ,{\bar{\rho }}\in {{\mathscr {W}}}$ be two bounded admissible weights and $\varepsilon \geqslant 0$. For given $(b,f)\in {{\mathbb {B}}}^\alpha _T(\rho )$ and $u_0\in \cup _{\epsilon >0}{{\mathbf {C}}}^{1+\alpha +\epsilon }$ for $\epsilon >0$, with notation (2.3), a pair of functions

$$\begin{aligned} (u, u^\sharp )\in {{\mathbb {S}}}^{2-\alpha }_T({\bar{\rho }})\times {{\mathbb {S}}}_T^{3-2\alpha }(\rho ^{2+\varepsilon }{\bar{\rho }}) \end{aligned}$$

(3.6)

is called a paracontrolled solution of PDE (3.1) if $(u,u^\sharp )$ satisfies (3.3) and (3.4) in the weak sense, where $b\circ \nabla u$ is well-defined by (3.5) and Lemma 3.3 below.

Remark 3.2

For $b, f\in L^\infty _T{{\mathscr {C}}}^2(\rho )$ with $\rho (x)=\langle x\rangle ^{-1}$, it is well known that PDE (3.1) has a unique classical solution. From Definition 3.1, it is not hard to see that classical solutions are paracontrolled solutions.

The following lemma makes the above definition more transparent.

Lemma 3.3

Let $T, \varepsilon \geqslant 0$ and $(u,u^\sharp )$ be a paracontrolled solution of (3.1) in the sense of Definition 3.1. For any $\gamma ,\beta \in (\alpha ,2-2\alpha ]$, there is a constant $C>0$ depending only on $T,\varepsilon ,\alpha ,\gamma ,\beta ,d, \rho ,{\bar{\rho }}$ such that for all $\lambda \geqslant 0$ and $t\in [0,T]$,

$$\begin{aligned} \Vert (b\circ \nabla u)(t)\Vert _{{{\mathbf {C}}}^{1-2\alpha }(\rho ^{2+\varepsilon }{\bar{\rho }})}&\lesssim _C \ell ^b_t(\rho )\Vert u\Vert _{{{\mathbb {S}}}^{\alpha +\gamma }_t({\bar{\rho }})} +\sqrt{\ell ^b_t(\rho )}\Vert u^\sharp (t)\Vert _{{{\mathbf {C}}}^{\beta +1}(\rho ^{1+\varepsilon }{\bar{\rho }})}\nonumber \\&\quad +\Vert (b\circ \nabla {{\mathscr {I}}}_\lambda f)(t)\Vert _{{{\mathbf {C}}}^{1-2\alpha }(\rho ^{2+\varepsilon }{\bar{\rho }})}+\Vert u_0\Vert _{{{\mathbf {C}}}^{1+\alpha +\epsilon }}. \end{aligned}$$

(3.7)

Proof

Below we drop the time variable t and fix

$$\begin{aligned} \gamma ,\beta \in (\alpha ,2-2\alpha ]. \end{aligned}$$

Recall $1-2\alpha <0$. We now estimate each term in (3.5) as follows:

• By (2.16) and (2.9), we have

  $$\begin{aligned} \Vert b\circ \nabla P_tu_0\Vert _{{{\mathbf {C}}}^{1-2\alpha }(\rho ^{2+\varepsilon }{\bar{\rho }})}\lesssim \Vert u_0\Vert _{{{\mathbf {C}}}^{1+\alpha +\epsilon }}\Vert b\Vert _{{{\mathbf {C}}}^{-\alpha }(\rho )}. \end{aligned}$$

• Since $\gamma >\alpha$, by (2.15), (2.16) and (2.11), we have

  $$\begin{aligned} \Vert b\circ (\nabla ^2 u\prec {{\mathscr {I}}}_\lambda b)\Vert _{{{\mathbf {C}}}^{1-2\alpha }(\rho ^2{\bar{\rho }})}&\lesssim \Vert b\Vert _{{{\mathbf {C}}}^{-\alpha }(\rho )}\Vert \nabla ^2 u\prec {{\mathscr {I}}}_\lambda b\Vert _{{{\mathbf {C}}}^{\gamma }(\rho {\bar{\rho }})}\\&\lesssim \Vert b\Vert _{{{\mathbf {C}}}^{-\alpha }(\rho )}\Vert \nabla ^2 u\Vert _{{{\mathbf {C}}}^{\gamma +\alpha -2}({\bar{\rho }})} \Vert {{\mathscr {I}}}_\lambda b\Vert _{{{\mathbf {C}}}^{2-\alpha }(\rho )}\\&\lesssim \Vert b\Vert ^2_{L^\infty _t{{\mathbf {C}}}^{-\alpha }(\rho )}\Vert u\Vert _{{{\mathbf {C}}}^{\gamma +\alpha }({\bar{\rho }})} \lesssim \ell ^b_t(\rho )\Vert u\Vert _{{{\mathbf {C}}}^{\alpha +\gamma }({\bar{\rho }})}. \end{aligned}$$

• By (2.17), we have

  $$\begin{aligned} \Vert \nabla u(b\circ \nabla {{\mathscr {I}}}_\lambda b)\Vert _{{{\mathbf {C}}}^{1-2\alpha }(\rho ^2{\bar{\rho }})}&\lesssim \Vert \nabla u\Vert _{{{\mathbf {C}}}^{\gamma +\alpha -1}({\bar{\rho }})}\Vert b\circ \nabla {{\mathscr {I}}}_\lambda b\Vert _{{{\mathbf {C}}}^{1-2\alpha }(\rho ^2)}\\&\lesssim \ell ^b_t(\rho )\Vert u\Vert _{{{\mathbf {C}}}^{\alpha +\gamma }({\bar{\rho }})}. \end{aligned}$$

• Since $\gamma >\alpha$, by (2.18) and (2.11), we have

  $$\begin{aligned} \Vert \mathrm{com}\Vert _{{{\mathbf {C}}}^{1-2\alpha }(\rho ^2{\bar{\rho }})}&\lesssim \Vert b\Vert _{{{\mathbf {C}}}^{-\alpha }(\rho )} \Vert \nabla u\Vert _{{{\mathbf {C}}}^{\gamma +\alpha -1}({\bar{\rho }})}\Vert \nabla {{\mathscr {I}}}_\lambda b\Vert _{{{\mathbf {C}}}^{1-\alpha }(\rho )}\\&\lesssim \Vert b\Vert ^2_{L^\infty _t{{\mathbf {C}}}^{-\alpha }(\rho )}\Vert u\Vert _{{{\mathbf {C}}}^{\gamma +\alpha }({\bar{\rho }})} \lesssim \ell ^b_t(\rho )\Vert u\Vert _{{{\mathbf {C}}}^{\alpha +\gamma }({\bar{\rho }})}. \end{aligned}$$

• By Lemma 2.10, (2.4) (2.21) and (2.11), we have

  $$\begin{aligned} \Vert \text {com}_1\Vert _{{{\mathbf {C}}}^{1-2\alpha }(\rho ^2{\bar{\rho }})}&\lesssim \Vert b\Vert _{{{\mathbf {C}}}^{-\alpha }(\rho )} \Vert \nabla u\prec \!\!\!\prec {{\mathscr {I}}}_\lambda b-\nabla u\prec {{\mathscr {I}}}_\lambda b\Vert _{{{\mathbf {C}}}^{\gamma +1}(\rho {\bar{\rho }})}\\&\lesssim \Vert b\Vert _{{{\mathbf {C}}}^{-\alpha }(\rho )} \Vert \nabla u\Vert _{C^{(\gamma +\alpha -1)/2}_tL^\infty ({\bar{\rho }})}\Vert {{\mathscr {I}}}_\lambda b\Vert _{L^\infty _t{{\mathbf {C}}}^{2-\alpha }(\rho )}\\&\lesssim \Vert b\Vert _{L^\infty _t{{\mathbf {C}}}^{-\alpha }(\rho )}^2\Vert u\Vert _{{{\mathbb {S}}}^{\alpha +\gamma }_t({\bar{\rho }})} \lesssim \ell ^b_t(\rho )\Vert u\Vert _{{{\mathbb {S}}}^{\alpha +\gamma }_t({\bar{\rho }})}. \end{aligned}$$

• Since $\beta >\alpha$, by (2.16), we have

  $$\begin{aligned} \Vert b\circ \nabla u^\sharp \Vert _{L^\infty (\rho ^{2+\varepsilon }{\bar{\rho }})}&\lesssim \Vert b\Vert _{{{\mathbf {C}}}^{-\alpha }(\rho )}\Vert \nabla u^\sharp \Vert _{{{\mathbf {C}}}^{\beta }(\rho ^{1+\varepsilon }{\bar{\rho }})} \leqslant \sqrt{\ell ^b_t(\rho )}\Vert u^\sharp \Vert _{{{\mathbf {C}}}^{\beta +1}(\rho ^{1+\varepsilon }{\bar{\rho }})}. \end{aligned}$$

Combining the above calculations and by (3.5), we obtain the estimate. $\square$

3.2 Schauder’s estimate for paracontrolled solutions without weights

As the first step towards the Schauder estimate for (3.1), we assume that the coefficients are in an unweighted Besov space. More precisely, we assume $(b,f)\in {{\mathbb {B}}}^\alpha _T:={{\mathbb {B}}}^\alpha _T(1)$, and for simplicity, we shall write

$$\begin{aligned} \ell ^b_T=\ell ^b_T(1),\ \ {{\mathbb {A}}}^{b,f}_{T,q}={{\mathbb {A}}}^{b,f}_{T,q}(1). \end{aligned}$$

The proof will be divided into two steps. We first prove a Schauder estimate depending polynomially on the coefficients for $\lambda$ large enough. Then by a classical maximum principle we extend the result to all $\lambda \geqslant 0$. The following Schauder estimate is a consequence of Lemmas 2.9 and 3.3.

Lemma 3.4

Assume $u_0=0$. For any $\theta \in (1+\tfrac{3\alpha }{2},2)$, $q\in (\frac{2}{2-\theta },\infty ]$ and $T>0$, there exist constants $c_0, c_1>0$ only depending on $\theta ,\alpha ,d,q,T$ such that for all $\lambda \geqslant c_0(\ell ^b_T)^{1/(1-\frac{\theta }{2}-\frac{1}{q})}$ and any paracontrolled solution $u_\lambda =u$ to PDE (3.1),

$$\begin{aligned} \Vert u_\lambda \Vert _{{{\mathbb {S}}}^{\theta -\alpha }_T}\leqslant c_1{{\mathbb {A}}}^{b,f}_{T,q}. \end{aligned}$$

(3.8)

Moreover, there is a constant $c_2>0$ such that for all $\lambda \geqslant 0$,

$$\begin{aligned} \Vert u_\lambda \Vert _{{{\mathbb {S}}}^{2-\alpha }_T}+\Vert u^\sharp _\lambda \Vert _{{{\mathbb {S}}}^{3-2\alpha }_T} \leqslant c_2(\ell ^b_T)^{\frac{4}{2-3\alpha }}\Big (\Vert u_\lambda \Vert _{{{\mathbb {L}}}_T^\infty }+{{\mathbb {A}}}^{b,f}_{T,\infty }\Big ). \end{aligned}$$

(3.9)

Proof

Below we fix

$$\begin{aligned} \theta \in (1+\tfrac{3\alpha }{2},2],\ \ q\in [\tfrac{2}{2-\theta },\infty ],\ \gamma ,\beta \in (\alpha , \theta -2\alpha ]. \end{aligned}$$

By (2.11), (2.15) and (2.14), we clearly have

$$\begin{aligned} \begin{aligned} (\lambda \vee 1)^{1-\frac{\theta }{2}-\frac{1}{q}}\Vert u\Vert _{{{\mathbb {S}}}^{\theta -\alpha }_T}&\lesssim \Vert b\prec \nabla u+b\succ \nabla u+b\circ \nabla u+f\Vert _{L^q_T{{\mathbf {C}}}^{-\alpha }}\\&\lesssim \Vert b\Vert _{L^\infty _T{{\mathbf {C}}}^{-\alpha }}\Vert \nabla u\Vert _{L^q_TL^\infty } +\Vert b\circ \nabla u\Vert _{L^q_T{{\mathbf {C}}}^{-\alpha }}+{{\mathbb {A}}}^{b,f}_{T,q}, \end{aligned} \end{aligned}$$

(3.10)

and by Lemma 2.12,

$$\begin{aligned} (\lambda \vee 1)^{1-\frac{\theta }{2}-\frac{1}{q}}\Vert u^\sharp \Vert _{{{\mathbb {S}}}^{\theta +\gamma -1}_T}&\lesssim \Vert \nabla u\prec b-\nabla u\prec \!\!\!\prec b\Vert _{L^\infty _T{{\mathbf {C}}}^{\gamma -1}}+\Vert \nabla u\succ b\Vert _{L^\infty _T{{\mathbf {C}}}^{\gamma -1}}\\&\quad +\Vert [{{\mathscr {L}}}_\lambda , \nabla u\prec \!\!\!\prec ]{{\mathscr {I}}}_\lambda b\Vert _{L^\infty _T{{\mathbf {C}}}^{\gamma -1}}+\Vert b\circ \nabla u\Vert _{L^q_T{{\mathbf {C}}}^{\gamma -1}}\\&\lesssim \Vert u\Vert _{{{\mathbb {S}}}^{\gamma +\alpha }_T}\Vert b\Vert _{L^\infty _T{{\mathbf {C}}}^{-\alpha }}+\Vert b\circ \nabla u\Vert _{L^q_T{{\mathbf {C}}}^{1-2\alpha }}, \end{aligned}$$

where we used (2.4), (2.21), (2.22) and (2.15) in the second inequality. Moreover, by (3.7), we also have

$$\begin{aligned} \Vert b\circ \nabla u\Vert _{L^q_T{{\mathbf {C}}}^{1-2\alpha }}\lesssim \ell ^b_T\Vert u\Vert _{{{\mathbb {S}}}^{\gamma +\alpha }_T} +\sqrt{\ell ^b_T}\Vert u^\sharp \Vert _{L^q_T{{\mathbf {C}}}^{\beta +1}}+{{\mathbb {A}}}^{b,f}_{T,q}. \end{aligned}$$

Thus, we obtain that for all $\lambda \geqslant 0$,

$$\begin{aligned} \begin{aligned}&(\lambda \vee 1)^{1-\frac{\theta }{2}-\frac{1}{q}}\left( \Vert u\Vert _{{{\mathbb {S}}}^{\theta -\alpha }_T}+\Vert u^\sharp \Vert _{{{\mathbb {S}}}^{\theta +\gamma -1}_T}\right) \\&\lesssim \ell ^b_T\Vert u\Vert _{{{\mathbb {S}}}^{\gamma +\alpha }_T} +\sqrt{\ell ^b_T}\Vert u^\sharp \Vert _{L^\infty _T{{\mathbf {C}}}^{\beta +1}}+{{\mathbb {A}}}^{b,f}_{T,q}. \end{aligned} \end{aligned}$$

(3.11)

In particular, letting $\gamma =\theta -2\alpha$ and $\beta =2\theta -2\alpha -2$, we get for some $c=c(\theta ,\alpha ,d,q,T)$,

$$\begin{aligned} (\lambda \vee 1)^{1-\frac{\theta }{2}-\frac{1}{q}}\left( \Vert u\Vert _{{{\mathbb {S}}}^{\theta -\alpha }_T}+\Vert u^\sharp \Vert _{{{\mathbb {S}}}^{2\theta -2\alpha -1}_T}\right) \lesssim _c\ell ^b_T\Big (\Vert u\Vert _{{{\mathbb {S}}}^{\theta -\alpha }_T}+\Vert u^\sharp \Vert _{{{\mathbb {S}}}^{2\theta -2\alpha -1}_T}\Big ) +{{\mathbb {A}}}^{b,f}_{T,q}. \end{aligned}$$

Choosing $\lambda$ such that $\lambda ^{1-\frac{\theta }{2}-\frac{1}{q}}\geqslant c\ell ^b_T$, we obtain (3.8).

On the other hand, letting $\theta =2$ and $q=\infty$ in (3.11), we obtain that for any $\gamma ,\beta \in (\alpha , 2-2\alpha ]$,

$$\begin{aligned} \Vert u\Vert _{{{\mathbb {S}}}^{2-\alpha }_T}+\Vert u^\sharp \Vert _{{{\mathbb {S}}}^{1+\gamma }_T}&\lesssim \ell ^b_T\Vert u\Vert _{{{\mathbb {S}}}^{\gamma +\alpha }_T} +\sqrt{\ell ^b_T}\Vert u^\sharp \Vert _{L^\infty _T{{\mathbf {C}}}^{\beta +1}}+{{\mathbb {A}}}^{b,f}_{T,\infty }. \end{aligned}$$

(3.12)

If $\alpha<\beta<\gamma <2-2\alpha$, then by (2.6) and Young’s inequality, we have for any $\varepsilon \in (0,1)$,

$$\begin{aligned} \begin{aligned} \Vert u\Vert _{{{\mathbb {S}}}^{2-\alpha }_T}+\Vert u^\sharp \Vert _{{{\mathbb {S}}}^{1+\gamma }_T}&\leqslant \varepsilon \Big (\Vert u\Vert _{{{\mathbb {S}}}^{2-\alpha }_T}+\Vert u^\sharp \Vert _{{{\mathbb {S}}}_T^{1+\gamma }}\Big ) +C_\varepsilon (\ell ^b_T)^{\frac{2-\alpha }{2-\gamma -2\alpha }}\Vert u\Vert _{{{\mathbb {L}}}^\infty _T}\\&\quad +C_\varepsilon (\ell ^b_T)^{\frac{1+\gamma }{2(\gamma -\beta )}}\Vert u^\sharp \Vert _{{{\mathbb {L}}}^\infty _T}+C{{\mathbb {A}}}^{b,f}_{T,\infty }. \end{aligned} \end{aligned}$$

(3.13)

Note that by (3.3),

$$\begin{aligned} \Vert u^\sharp \Vert _{{{\mathbb {L}}}^\infty _T}&=\Vert u-\nabla u\prec \!\!\!\prec {{\mathscr {I}}}_\lambda b-{{\mathscr {I}}}_\lambda f\Vert _{{{\mathbb {L}}}^\infty _T}\\&\lesssim \Vert u\Vert _{{{\mathbb {L}}}^\infty _T}(1+\Vert b\Vert _{L^\infty _T{{\mathbf {C}}}^{-\alpha }})+\Vert f\Vert _{L_T^\infty {{\mathbf {C}}}^{-\alpha }}\lesssim \Vert u\Vert _{{{\mathbb {L}}}^\infty _T}\sqrt{\ell ^b_T}+{{\mathbb {A}}}^{b,f}_{T,\infty }. \end{aligned}$$

Substituting it into (3.13) and taking $\varepsilon =1/2$, we obtain

$$\begin{aligned} \Vert u\Vert _{{{\mathbb {S}}}^{2-\alpha }_T}+\Vert u^\sharp \Vert _{{{\mathbb {S}}}^{1+\gamma }_T}\lesssim (\ell ^b_T)^{\frac{2-\alpha }{2-\gamma -2\alpha }\vee (\frac{1+\gamma }{2(\gamma -\beta )}+\frac{1}{2})} \Big (\Vert u\Vert _{{{\mathbb {L}}}^\infty _T}+{{\mathbb {A}}}^{b,f}_{T,\infty }\Big ), \end{aligned}$$

which, by choosing $\gamma =2/3$ and $\beta$ close to $\alpha$, yields that

$$\begin{aligned} \Vert u\Vert _{{{\mathbb {S}}}^{2-\alpha }_T}+\Vert u^\sharp \Vert _{{{\mathbb {S}}}^{5/3}_T}\lesssim (\ell ^b_T)^{\frac{8-3\alpha }{2(2-3\alpha )}} \Big (\Vert u\Vert _{{{\mathbb {L}}}_T^\infty }+{{\mathbb {A}}}^{b,f}_{T,\infty }\Big ). \end{aligned}$$

Moreover, by (3.12) with $\gamma =2-2\alpha$ and $\beta =2/3$, we get

$$\begin{aligned} \Vert u^\sharp \Vert _{{{\mathbb {S}}}^{3-2\alpha }_T} \lesssim \ell ^b_T\Vert u\Vert _{{{\mathbb {S}}}^{2-\alpha }_T} +\sqrt{\ell ^b_T}\Vert u^\sharp \Vert _{{{\mathbb {S}}}^{5/3}_T}+{{\mathbb {A}}}^{b,f}_{T,\infty } \lesssim (\ell ^b_T)^{\frac{4}{2-3\alpha }}\Big (\Vert u\Vert _{{{\mathbb {L}}}_T^\infty }+{{\mathbb {A}}}^{b,f}_{T,\infty }\Big ). \end{aligned}$$

The proof is complete. $\square$

Now we can show the main result of this section, where the key point is to obtain an estimate depending polynomially on the quantity $\ell ^b_T$. Note that a simple Gronwall argument will lead to the exponential dependence on $\ell ^b_T$.

Theorem 3.5

Let $T>0$ and $u_0=0$. For any $(b,f)\in {{\mathbb {B}}}^\alpha _T$, there is a unique paracontrolled solution $u_\lambda =u$ to PDE (3.1) in the sense of Definition 3.1. Moreover, there exist $q=q(\alpha )>1$ and a constant $c_3=c_3(\alpha ,d,T)>0$ such that for all $\lambda \geqslant 0$,

$$\begin{aligned}&\Vert u_\lambda \Vert _{{{\mathbb {L}}}^\infty _T}\leqslant c_3(\ell ^b_T)^{\frac{5}{2-3\alpha }}{{\mathbb {A}}}^{b,f}_{T,q},\\&\Vert u_\lambda \Vert _{{{\mathbb {S}}}^{2-\alpha }_T}+\Vert u_\lambda ^\sharp \Vert _{{{\mathbb {S}}}^{3-2\alpha }_T}\leqslant c_3(\ell ^b_T)^{\frac{9}{2-3\alpha }}{{\mathbb {A}}}^{b,f}_{T,\infty }. \end{aligned}$$

Proof

We first assume that

$$\begin{aligned} b,f\in L^\infty _T {{\mathscr {C}}}^2,\ \ \forall T>0. \end{aligned}$$

Fix $\lambda \geqslant 0$. For any $\lambda '>0$, it is well known that there is a unique classical solution w to the following PDE:

$$\begin{aligned} \partial _tw=\Delta w-(\lambda '+\lambda ) w+b\cdot \nabla w+f,\ \ w(0)=0. \end{aligned}$$

(3.14)

In particular, for any $\theta \in (1+\frac{3}{2}\alpha ,2)$ and $q\in (\frac{2}{2-\theta },\infty ]$, by (3.8), we have for $\lambda '\geqslant c_0(\ell ^b_T)^{1/(1-\frac{\theta }{2}-\frac{1}{q})}$,

$$\begin{aligned} \Vert w\Vert _{{{\mathbb {L}}}^\infty _T}\leqslant \Vert w\Vert _{L_T^\infty {{\mathbf {C}}}^{\theta -\alpha }}\leqslant c_1 \cdot {{\mathbb {A}}}^{b,f}_{T,q}. \end{aligned}$$

Now let u be the unique classical solution to PDE (3.1) with $u_0=0$. Let ${\bar{u}}=u-w$. Then ${\bar{u}}$ solves the following PDE:

$$\begin{aligned} \partial _t{\bar{u}}=\Delta {\bar{u}}-\lambda \bar{u}+b\cdot \nabla {\bar{u}}+\lambda ' w,\ \ {\bar{u}}(0)=0. \end{aligned}$$

By the classical maximum principle, we have

$$\begin{aligned} \Vert {\bar{u}}\Vert _{{{\mathbb {L}}}^\infty _T}\leqslant \lambda ' T\Vert w\Vert _{{{\mathbb {L}}}^\infty _T}. \end{aligned}$$

Hence, by taking $\theta$ close to $1+\frac{3\alpha }{2}$ and q large enough, we obtain

$$\begin{aligned} \Vert u\Vert _{{{\mathbb {L}}}^\infty _T}\leqslant (\lambda ' T+1)\Vert w\Vert _{{{\mathbb {L}}}^\infty _T}\leqslant (c_0T (\ell ^b_T)^{1/(1-\frac{\theta }{2}-\frac{1}{q})}+1)c_1\cdot {{\mathbb {A}}}^{b,f}_{T,q} \lesssim (\ell ^b_T)^{\frac{5}{2-3\alpha }}\cdot {{\mathbb {A}}}^{b,f}_{T,q}, \end{aligned}$$

which together with (3.9) yields

$$\begin{aligned} \Vert u\Vert _{{{\mathbb {S}}}^{2-\alpha }_T}+\Vert u^\sharp \Vert _{{{\mathbb {S}}}^{3-2\alpha }_T}\lesssim (\ell ^b_T)^{\frac{9}{2-3\alpha }}{{\mathbb {A}}}^{b,f}_{T,\infty }. \end{aligned}$$

(3.15)

(Existence) Let $b_n$ and $f_n$ be the smoothing approximations of b and f in ${{\mathbb {B}}}^\alpha _T$. We consider the following approximation equation:

$$\begin{aligned} \partial _t u_n=\Delta u_n-\lambda u_n+b_n\cdot \nabla u_n+f_n,\quad u_n(0)=0. \end{aligned}$$

By the assumption and (3.15), we have the following uniform estimate:

$$\begin{aligned} \sup _{n\in {{\mathbb {N}}}}\Big (\Vert u_n\Vert _{{{\mathbb {S}}}^{2-\alpha }_T}+\Vert u^\sharp _n\Vert _{{{\mathbb {S}}}^{3-2\alpha }_T}\Big )\lesssim 1. \end{aligned}$$

Using this uniform estimate and by a standard compact and weak convergence method, we can show the existence of a paracontrolled solution (see [18]).

(Uniqueness) Let $u_1$ and $u_2$ be two paracontrolled solution of PDE (3.1). Let ${\bar{u}}:=u_1-u_2$. Clearly, $\bar{u}$ is a paracontrolled solution of

$$\begin{aligned} \partial _t{\bar{u}}=\Delta {\bar{u}}-\lambda {\bar{u}}+b\cdot \nabla {\bar{u}},\ \ u(0)=0. \end{aligned}$$

Let $\theta \in (1+\alpha ,2)$ and $q=\frac{2}{2-\theta }$. By (2.11), we have

$$\begin{aligned} \Vert {\bar{u}}\Vert ^q_{{{\mathbb {S}}}^{\theta -\alpha }_T}\leqslant C\int ^T_0\Vert (b\cdot \nabla {\bar{u}})(t)\Vert ^q_{{{\mathbf {C}}}^{-\alpha }}{\mathord {\mathrm{d}}}t. \end{aligned}$$

(3.16)

On the other hand, by (2.14), (2.15) and Lemma 3.3 we have

$$\begin{aligned} \Vert (b\cdot \nabla {\bar{u}})(t)\Vert _{{{\mathbf {C}}}^{-\alpha }}&\leqslant \Vert (b\prec \nabla {\bar{u}})(t)\Vert _{{{\mathbf {C}}}^{-\alpha }}+\Vert (b\succ \nabla \bar{u})(t)\Vert _{{{\mathbf {C}}}^{-\alpha }}+\Vert (b\circ \nabla {\bar{u}})(t)\Vert _{{{\mathbf {C}}}^{-\alpha }} \\ {}&\lesssim \Vert b(t)\Vert _{{{\mathbf {C}}}^{-\alpha }}\Vert \nabla {\bar{u}}(t)\Vert _{L^\infty }+\Vert (b\circ \nabla {\bar{u}})(t)\Vert _{{{\mathbf {C}}}^{1-2\alpha }} \\ {}&\lesssim \Vert \nabla {\bar{u}}(t)\Vert _{L^\infty }+ \Vert {\bar{u}}\Vert _{{{\mathbb {S}}}^{2-\alpha }_t}+\Vert \bar{u}^\sharp \Vert _{L^\infty _t{{\mathbf {C}}}^{3-2\alpha }} {\mathop {\lesssim }\limits ^{(3.9)}} \Vert \nabla \bar{u}\Vert _{{{\mathbb {L}}}^\infty _t}+\Vert {\bar{u}}\Vert _{{{\mathbb {L}}}^\infty _t}. \end{aligned}$$

Substituting this into (3.16) and by $\theta -\alpha >1$, we obtain

$$\begin{aligned} \Vert {\bar{u}}\Vert ^q_{L^\infty _T{{\mathbf {C}}}^{\theta -\alpha }}\leqslant C\int ^T_0\Vert \bar{u}\Vert ^q_{L_t^\infty {{\mathbf {C}}}^{\theta -\alpha }}{\mathord {\mathrm{d}}}t, \end{aligned}$$

which in turn implies that ${\bar{u}}=0$. The uniqueness is proven. $\square$

Remark 3.6

The polynomial dependence on $\ell ^b_T$ in Theorem 3.5 together with a new characterization for weighted Hölder spaces in Lemma 3.8 below shall be used to establish the Schauder estimate in sublinear weighted Hölder spaces (see [45, Remark 1.1]).

3.3 Schauder’s estimate for paracontrolled solutions with weights

In this section we show the well-posedness of PDE (3.1) in weighted Hölder spaces. Recall that for $\delta \in {{\mathbb {R}}}$,

$$\begin{aligned} \rho _\delta (x):=(1+|x|^2)^{-\delta /2}=:\langle x\rangle ^{-\delta }. \end{aligned}$$

Now we give the main result of this section.

Theorem 3.7

Let $\alpha \in (\frac{1}{2},\frac{2}{3})$ and $\vartheta :=\frac{9}{2-3\alpha }$. Choose $\kappa >0$ so that

$$\begin{aligned} \delta :=(2\vartheta +2)\kappa \leqslant 1,\ \ \delta _0:=(\frac{55}{27}\vartheta +4)\kappa . \end{aligned}$$

For any $T>0$ and $\lambda \geqslant 0$, $(b,f)\in {{\mathbb {B}}}^\alpha _T(\rho _\kappa )$ and $u_0\in \cup _{\epsilon >0}{{\mathbf {C}}}^{1+\alpha +\epsilon }$, there exists a unique paracontrolled solution $(u, u^\sharp )$ to PDE (3.1) in the sense of Definition 3.1 with

$$\begin{aligned} \Vert u\Vert _{{{\mathbb {S}}}^{2-\alpha }_T(\rho _\delta )}+\Vert u^\sharp \Vert _{{{\mathbb {S}}}^{3-2\alpha }_T(\rho _{\delta _0})} \lesssim _C{{\mathbb {A}}}^{b,f}_{T,\infty }(\rho _\kappa ), \end{aligned}$$

(3.17)

where $C=C(T,d,\alpha ,\kappa ,\ell ^b_T(\rho _\kappa ))>0$.

To prove the result we first prove a characterization of weighted Hölder spaces. To this end, we introduce the following notations. Let $\chi \in C^\infty _c({{\mathbb {R}}}^d)$ with

$$\begin{aligned} \chi (x)=1,\ \ |x|\leqslant 1/8,\ \ \ \chi (x)=0,\ \ |x|>1/4, \end{aligned}$$

and for $r>0$ and $z\in {{\mathbb {R}}}^d$,

$$\begin{aligned} \chi ^z_r(x):=\chi ((x-z)/r),\ \ \phi ^z_r(x):=\chi ^z_{r(1+|z|)}(x). \end{aligned}$$

To show the existence of a paracontrolled solution, we need the following simple characterization of weighted Hölder spaces.

Lemma 3.8

Let $\alpha \geqslant 0$ and $r\in (0,1]$. For any $\delta ,\kappa \in {{\mathbb {R}}}$, there is a constant $C=C(r,\alpha ,d,\delta ,\kappa )>0$ such that

$$\begin{aligned} \Vert f\Vert _{{{\mathscr {C}}}^\alpha (\rho _\delta \rho _\kappa )} \asymp _C\sup _{z}\left( \rho _\delta (z)\Vert \phi ^z_r f\Vert _{{{\mathscr {C}}}^\alpha (\rho _\kappa )}\right) . \end{aligned}$$

(3.18)

Moreover, for any $m\in {{\mathbb {N}}}_0$, we also have

$$\begin{aligned} \sup _z\Vert \nabla \phi _r^z\Vert _{{{\mathscr {C}}}^m(\rho ^{-1}_1)}<\infty . \end{aligned}$$

(3.19)

Proof

Without loss of generality, we may assume $\kappa =0$ by noting that

$$\begin{aligned} \sup _{z}\left( \rho _\delta (z)\Vert \phi ^z_r f\Vert _{{{\mathscr {C}}}^\alpha (\rho _\kappa )}\right) \asymp \sup _{z}\left( \rho _\delta (z)\Vert \phi ^z_r\rho _\kappa f\Vert _{{{\mathscr {C}}}^\alpha }\right) \asymp \Vert \rho _\delta \rho _\kappa f\Vert _{{{\mathscr {C}}}^\alpha }. \end{aligned}$$

First of all, for fixed $r\in (0,1]$, $\delta \in {{\mathbb {R}}}$ and any $m\in {{\mathbb {N}}}$, we have for some $C=C(m,r,\delta ,d)>0$,

$$\begin{aligned} \Vert \rho _\delta \phi ^z_r\Vert _{{{\mathscr {C}}}^m}\lesssim _C \rho _\delta (z),\ \ \forall z\in {{\mathbb {R}}}^d. \end{aligned}$$

(3.20)

Indeed, let $B_r(z)$ be the ball with radius r centered at z. Noting that for any $\delta \in {{\mathbb {R}}}$ and $x\in B_{(1+|z|)/2}(z)$,

$$\begin{aligned} \rho _\delta (x)\leqslant 2^{|\delta |}(1+|x|)^{-\delta }\leqslant 4^{|\delta |}(1+|z|)^{-\delta }=4^{|\delta |}\rho _\delta (z), \end{aligned}$$

(3.21)

we have by definition and the chain rule,

$$\begin{aligned} \begin{aligned} \Vert \rho _\delta \phi ^z_r\Vert _{{{\mathscr {C}}}^m}&=\sum _{k=0}^m\Vert \nabla ^k(\rho _\delta \phi ^z_r)\Vert _{L^\infty }\lesssim \sum _{k=0}^m\sum _{j=0}^k\Vert \nabla ^{k-j}\rho _\delta \nabla ^{j}\phi ^z_r\Vert _{L^\infty }\\&\lesssim \sum _{k=0}^m\sum _{j=0}^k\Vert \rho _\delta \nabla ^{j}\phi ^z_r\Vert _{L^\infty } \leqslant 4^{|\delta |}\sum _{k=0}^m\sum _{j=0}^k\rho _\delta (z)\Vert \nabla ^{j}\phi ^z_r\Vert _{L^\infty }\lesssim \rho _\delta (z). \end{aligned} \end{aligned}$$

(3.22)

Moreover, since the definition of ${{\mathscr {C}}}^\alpha$ is local,

$$\begin{aligned} \Vert f\Vert _{{{\mathscr {C}}}^\alpha }\lesssim \sup _z\Vert f\Vert _{{{\mathscr {C}}}^\alpha (B_{r/16}(z))}\lesssim \sup _{z}\Vert \chi ^z_{r/2} f\Vert _{{{\mathscr {C}}}^\alpha }. \end{aligned}$$

Thus by (3.20) and $\chi ^z_{r/2} \phi ^z_r=\chi ^z_{r/2}$, we have

$$\begin{aligned} \Vert f\Vert _{{{\mathscr {C}}}^\alpha (\rho _\delta )}&=\Vert \rho _\delta f\Vert _{{{\mathscr {C}}}^\alpha }\lesssim \sup _{z}\Vert \chi ^z_{r/2} \rho _\delta f\Vert _{{{\mathscr {C}}}^\alpha }=\sup _{z}\Vert \chi ^z_{r/2} \phi ^z_r\rho _\delta \phi ^z_r f\Vert _{{{\mathscr {C}}}^\alpha }\\&\lesssim \sup _{z}\left( \Vert \chi ^z_{r/2}\phi ^z_r \rho _\delta \Vert _{{{\mathscr {C}}}^{[\alpha ]+1}}\Vert \phi ^z_r f\Vert _{{{\mathscr {C}}}^\alpha }\right) \lesssim \sup _{z}\left( \rho _\delta (z)\Vert \phi ^z_r f\Vert _{{{\mathscr {C}}}^\alpha }\right) , \end{aligned}$$

and

$$\begin{aligned} \sup _{z}\left( \rho _\delta (z)\Vert \phi ^z_r f\Vert _{{{\mathscr {C}}}^\alpha }\right)&\lesssim \sup _{z}\left( \rho _\delta (z)\Vert \phi ^z_r \rho _\delta ^{-1}\Vert _{{{\mathscr {C}}}^{[\alpha ]+1}}\Vert \rho _\delta f\Vert _{{{\mathscr {C}}}^\alpha }\right) \lesssim \Vert \rho _\delta f\Vert _{{{\mathscr {C}}}^\alpha }. \end{aligned}$$

So, (3.18) is proven. On the other hand, note that for any $m\in {{\mathbb {N}}}_0$,

$$\begin{aligned} \Vert \nabla \phi _r^z\Vert _{{{\mathscr {C}}}^m}\lesssim (1+|z|)^{-1}. \end{aligned}$$

As in (3.22), by (3.21) we have (3.19). $\square$

Remark 3.9

Estimate (3.19) provides an extra weight $\rho _1$ and helps us to obtain the a priori estimate for the solutions in Besov spaces with polynomial weights.

Now we give the proof of Theorem 3.7.

Proof of Theorem 3.7

(Existence). Without loss of generality we may assume $\lambda =0$ and $u_0=0$. In fact, for general initial data $u_0\in \cup _{\epsilon >0}{{\mathbf {C}}}^{1+\alpha +\epsilon }$, by considering ${\bar{u}}=u-u_0$, we can reduce the nonzero initial value to zero initial value with f replaced by ${\bar{f}}=f+\Delta u_0+b\cdot \nabla u_0\in {{\mathbf {C}}}^{-\alpha }(\rho _\kappa )$. In this case, by Lemma 2.10,

$$\begin{aligned} \Vert b\circ \nabla {{\mathscr {I}}}(\Delta u_0)\Vert _{L^\infty _T{{\mathbf {C}}}^\epsilon (\rho _\kappa )}\lesssim 1, \end{aligned}$$

and by Lemma 2.17 with $\psi =\nabla u_0, f=b, \phi =1, \bar{\rho }=\rho _{2\kappa }, \rho =\rho _\kappa$,

$$\begin{aligned}&\Vert b\circ \nabla {{\mathscr {I}}}(b\cdot \nabla u_0)\Vert _{L^\infty _T{{\mathbf {C}}}^{1-2\alpha }(\rho _{2\kappa })} \lesssim 1. \end{aligned}$$

Hence, we still have

$$\begin{aligned} (b,{\bar{f}})\in {{\mathbb {B}}}^\alpha _T(\rho _\kappa ). \end{aligned}$$

Now, let $T>0$ and $b_n, f_n\in L^\infty _T{{\mathscr {C}}}^\infty (\rho _\kappa )$ be as in the definition of ${{\mathbb {B}}}^\alpha _T(\rho _\kappa )$. For every n, define

$$\begin{aligned} {\bar{b}}_n(t,x):=b_n(t,x)\chi _n(x),\ \ \bar{f}_n(t,x):=f_n(t,x)\chi _n(x), \end{aligned}$$

where $\chi _n(x)=\chi (x/n)$ and $\chi \in C^\infty _c({{\mathbb {R}}}^d)$ equals to 1 on $B_1$. It is well known that there is a unique classical solution $u_n\in L^\infty _T{{\mathscr {C}}}^2$ solving (3.1) with $(b,f)=({\bar{b}}_n,{\bar{f}}_n)$. Our main aim is to show that there is a constant $C>0$ independent of n such that

$$\begin{aligned} \Vert u_n\Vert _{{{\mathbb {S}}}^{2-\alpha }_T(\rho _\delta )}+\Vert u^\sharp _n\Vert _{{{\mathbb {S}}}^{3-2\alpha }_T(\rho _{\delta _0})} \lesssim _C {{\mathbb {A}}}^{{\bar{b}}_n,{\bar{f}}_n}_{T,\infty }(\rho _\kappa ) \end{aligned}$$

(3.23)

On the other hand, by (2.29) with ${\bar{\rho }}=\rho ^2=\rho _{2\kappa }$ and $\phi =\psi =\chi _n$, we also have for some C independent of n,

$$\begin{aligned} {{\mathbb {A}}}^{{\bar{b}}_n,\bar{f}_n}_{T,\infty }(\rho _\kappa )\lesssim _C{{\mathbb {A}}}^{b_n,f_n}_{T,\infty }(\rho _\kappa ),\quad \ell ^{{\bar{b}}_n}_{T}(\rho _\kappa )\lesssim _C\ell _T^{b_n}(\rho _\kappa ). \end{aligned}$$

Hence,

$$\begin{aligned} \sup _n\Big (\Vert u_n\Vert _{{{\mathbb {S}}}^{2-\alpha }_T(\rho _\delta )}+\Vert u^\sharp _n\Vert _{{{\mathbb {S}}}^{3-2\alpha }_T(\rho _{\delta _0})}\Big )<\infty . \end{aligned}$$

Thus, by a standard compact argument, we can show the existence of a paracontrolled solution (see [18]).

Now we prove (3.23) by a localization technique. For simplicity, we drop the bar and subscript n and assume $b,f\in L^\infty _T{{\mathscr {C}}}^2$. We fix $0<r<1/2$. Note that $\phi ^z_{2r}=1$ on the support of $\phi ^z_r$. For each $z\in {{\mathbb {R}}}^d$, it is easy to see that $u_z:=u\phi ^z_r$ satisfies the following PDE:

$$\begin{aligned} \partial _t u_z=\Delta u_z+b_z\cdot \nabla u_z+F_z,\ \ u_z(0)=0, \end{aligned}$$

where $b_z:=b\phi ^z_{2r}$ and

$$\begin{aligned} F_z:=f\phi ^z_r-2\nabla u\cdot \nabla \phi ^z_r-u\Delta \phi ^z_r-b\cdot \nabla \phi ^z_ru. \end{aligned}$$

Let q be the same as in Theorem 3.5. By Theorem 3.5, there is constant $C>0$ such that for all $z\in {{\mathbb {R}}}^d$,

$$\begin{aligned} \Vert u_z\Vert _{{{\mathbb {S}}}^{2-\alpha }_T} \leqslant C(\ell ^{b_z}_T)^{\vartheta }{{\mathbb {A}}}^{b_z,F_z}_{T,\infty },\ \Vert u_z\Vert _{{{\mathbb {L}}}^\infty _T}\leqslant C(\ell ^{b_z}_T)^{\vartheta }{{\mathbb {A}}}^{b_z,F_z}_{T,q}. \end{aligned}$$

(3.24)

Let $\varepsilon >0$ be small enough. By the definition of $F_z$, using $\phi _{2r}^z\nabla \phi _r^zu=\nabla \phi _r^zu$ and (2.17), we have

$$\begin{aligned} \Vert F_z\Vert _{{{\mathbf {C}}}^{-\alpha }}&\leqslant \Vert f\phi ^z_r\Vert _{{{\mathbf {C}}}^{-\alpha }}+2\Vert \nabla u\cdot \nabla \phi ^z_r\Vert _{L^\infty } +\Vert u\Delta \phi ^z_r\Vert _{L^\infty }+\Vert b\phi ^z_{2r}\cdot \nabla \phi ^z_ru\Vert _{{{\mathbf {C}}}^{-\alpha }} \nonumber \\&\lesssim \Vert f\Vert _{{{\mathbf {C}}}^{-\alpha }(\rho _\kappa )}\Vert \phi ^z_r\Vert _{{{\mathbf {C}}}^{\alpha +\varepsilon }(\rho ^{-1}_\kappa )}+\Vert \nabla u\Vert _{L^\infty (\rho _1)} \Vert \nabla \phi ^z_r\Vert _{L^\infty (\rho ^{-1}_1)} \nonumber \\&\quad +\Vert u\Vert _{L^\infty (\rho _1)}\Vert \Delta \phi ^z_r\Vert _{L^\infty (\rho ^{-1}_1)} +\Vert b\Vert _{{{\mathbf {C}}}^{-\alpha }(\rho _\kappa )}\Vert \phi ^z_{2r}\nabla \phi ^z_ru\Vert _{{{\mathbf {C}}}^{\alpha +\varepsilon }(\rho _\kappa ^{-1})} \nonumber \\&\lesssim \Vert f\Vert _{{{\mathbf {C}}}^{-\alpha }(\rho _\kappa )}\Vert \phi ^z_r\Vert _{{{\mathscr {C}}}^1(\rho ^{-1}_\kappa )}+\Vert u\Vert _{{{\mathscr {C}}}^1(\rho _1)}\Vert \nabla \phi ^z_r\Vert _{{{\mathscr {C}}}^1(\rho ^{-1}_1)}\nonumber \\&\quad +\Vert b\Vert _{{{\mathbf {C}}}^{-\alpha }(\rho _\kappa )}\Vert u\Vert _{{{\mathscr {C}}}^1(\rho _1)}\Vert \nabla \phi ^z_r\Vert _{{{\mathscr {C}}}^1(\rho ^{-1}_1)} \Vert \phi ^z_{2r}\Vert _{{{\mathscr {C}}}^1(\rho _\kappa ^{-1})},\nonumber \\&{{\mathop {\lesssim }\limits ^{(3.19) }}\Vert f\Vert _{{{\mathbf {C}}}^{-\alpha }(\rho _\kappa )}\Vert \phi ^z_r\Vert _{{{\mathscr {C}}}^1(\rho ^{-1}_\kappa )}+\Vert u\Vert _{{{\mathscr {C}}}^1(\rho _1)} \Big (1+\Vert b\Vert _{{{\mathbf {C}}}^{-\alpha }(\rho _\kappa )} \Vert \phi ^z_{2r}\Vert _{{{\mathscr {C}}}^1(\rho _\kappa ^{-1})}\Big ) }. \end{aligned}$$

(3.25)

In particular,

$$\begin{aligned} \Vert F_z\Vert _{L^q_T{{\mathbf {C}}}^{-\alpha }}\lesssim & {} \Vert f\Vert _{L_T^q{{\mathbf {C}}}^{-\alpha }(\rho _\kappa )}\Vert \phi _r^z\Vert _{{{\mathscr {C}}}^1(\rho _\kappa ^{-1})}\nonumber \\&+\Big (1+\Vert b\Vert _{L_T^\infty {{\mathbf {C}}}^{-\alpha }(\rho _\kappa )} \Vert \phi _{2r}^z\Vert _{{{\mathscr {C}}}^1(\rho _\kappa ^{-1})}\Big )\left( \int _0^T\Vert u(t)\Vert _{{{\mathscr {C}}}^1(\rho _1)}^q{\mathord {\mathrm{d}}}t\right) ^{1/q}.\nonumber \\ \end{aligned}$$

(3.26)

Similarly, we also have

$$\begin{aligned} \Vert (b_z\circ \nabla {{\mathscr {I}}}_\lambda F_z)\Vert _{{{\mathbf {C}}}^{1-2\alpha }}&\leqslant \Vert b_z\circ \nabla {{\mathscr {I}}}_\lambda ({f}\phi ^z_r)\Vert _{{{\mathbf {C}}}^{1-2\alpha }}+\Vert b_z\circ \nabla {{\mathscr {I}}}_\lambda (b\cdot \nabla \phi ^z_ru )\Vert _{{{\mathbf {C}}}^{1-2\alpha }}\\&+\Vert b_z\circ \nabla {{\mathscr {I}}}_\lambda (u\Delta \phi ^z_r+2\nabla u\cdot \nabla \phi ^z_r)\Vert _{L^\infty }=:I_1^z+I_2^z+I_3^z. \end{aligned}$$

For $I^z_1$, by (2.29) with ${\bar{\rho }}\equiv 1$, $\rho =\rho _\kappa$ and $\psi =\phi ^z_r$, we have

$$\begin{aligned} I^z_1\lesssim \Vert \phi ^z_{2r}\Vert _{{{\mathbf {C}}}^{\alpha +\varepsilon }(\rho _\kappa ^{-2})}\Vert \phi ^z_r\Vert _{{{\mathbf {C}}}^{\alpha +\varepsilon }}{{\mathbb {A}}}^{b,f}_{t,\infty }(\rho _\kappa ) \lesssim \Vert \phi ^z_{2r}\Vert _{{{\mathscr {C}}}^1(\rho _\kappa ^{-2})}{{\mathbb {A}}}^{b,f}_{t,\infty }(\rho _\kappa ). \end{aligned}$$

For $I^z_2$, by (2.29) with ${\bar{\rho }}\equiv 1$, $\rho =\rho _\kappa$ and $\psi =\nabla \phi ^z_r u$, we have

$$\begin{aligned} I^z_2&\lesssim \Vert \phi ^z_{2r}\Vert _{{{\mathbf {C}}}^{\alpha +\varepsilon }(\rho _\kappa ^{-2})}\Vert \nabla \phi ^z_r u\Vert _{{{\mathbb {S}}}_t^{\alpha +\varepsilon }}{{\mathbb {A}}}^{b,b}_{t,\infty }(\rho _\kappa )\\&\lesssim \Vert \phi ^z_{2r}\Vert _{{{\mathscr {C}}}^1(\rho _\kappa ^{-2})}\Vert \nabla \phi ^z_r\Vert _{{{\mathscr {C}}}^1(\rho ^{-1}_1)}\Vert u\Vert _{{{\mathbb {S}}}_t^1(\rho _1)}\ell ^{b}_t(\rho _\kappa )\\&{\mathop {\lesssim }\limits ^{(3.19) }} \Vert \phi ^z_{2r}\Vert _{{{\mathscr {C}}}^1(\rho _\kappa ^{-2})}\Vert u\Vert _{{{\mathbb {S}}}_t^1(\rho _1)}\ell ^{b}_t(\rho _\kappa ). \end{aligned}$$

For $I^z_3$, as in (3.25), since

$$\begin{aligned} \Vert b_z\Vert _{{{\mathbf {C}}}^{-\alpha }}\lesssim \Vert b\Vert _{{{\mathbf {C}}}^{-\alpha }(\rho _\kappa )}\Vert \phi _{2r}^z\Vert _{{{\mathscr {C}}}^1(\rho _\kappa ^{-1})}, \end{aligned}$$

(3.27)

by (2.17), we have

$$\begin{aligned} I^z_3&\lesssim \Vert b_z\Vert _{{{\mathbf {C}}}^{-\alpha }}\Vert \nabla {{\mathscr {I}}}_\lambda (u\Delta \phi ^z_r+2\nabla u\cdot \nabla \phi ^z_r)\Vert _{{{\mathbf {C}}}^{\alpha +\varepsilon }}\\&\lesssim \Vert b\Vert _{{{\mathbf {C}}}^{-\alpha }(\rho _\kappa )}\Vert \phi ^z_{2r}\Vert _{{{\mathscr {C}}}^{1}(\rho ^{-1}_\kappa )}\Vert u\Delta \phi ^z_r+2\nabla u\cdot \nabla \phi ^z_r\Vert _{L_t^\infty L^\infty }\\&\lesssim \Vert b\Vert _{{{\mathbf {C}}}^{-\alpha }(\rho _\kappa )}\Vert \phi ^z_{2r}\Vert _{{{\mathscr {C}}}^1(\rho ^{-1}_\kappa )} \Vert u\Vert _{L_t^\infty {{\mathscr {C}}}^1(\rho _1)}\Vert \nabla \phi ^z_r\Vert _{{{\mathscr {C}}}^1(\rho ^{-1}_1)}\\&{\mathop {\lesssim }\limits ^{(3.19) }}\Vert b\Vert _{{{\mathbf {C}}}^{-\alpha }(\rho _\kappa )}\Vert \phi ^z_{2r}\Vert _{{{\mathscr {C}}}^1(\rho ^{-1}_\kappa )} \Vert u\Vert _{L_t^\infty {{\mathscr {C}}}^1(\rho _1)}. \end{aligned}$$

Combining the above calculations, by the definition of ${{\mathbb {A}}}^{b_z,F_z}_{T,q}$, (3.19), (3.26) and (3.27), we get

$$\begin{aligned}&{{\mathbb {A}}}^{b_z,F_z}_{T,q} =\sup _\lambda \Vert b_z\circ \nabla {{\mathscr {I}}}_\lambda F_z\Vert _{L^q_T{{\mathbf {C}}}^{1-2\alpha }}+(\Vert b_z\Vert _{L^\infty _T{{\mathbf {C}}}^{-\alpha }}+1)\Vert F_z\Vert _{L^q_T{{\mathbf {C}}}^{-\alpha }}\\&\lesssim \Big (\Vert \phi ^z_{2r}\Vert _{{{\mathscr {C}}}^1(\rho _\kappa ^{-2})}+\Vert \phi _{2r}^z\Vert _{{{\mathscr {C}}}^1(\rho _\kappa ^{-1})}(\Vert \phi _r^z\Vert _{{{\mathscr {C}}}^1(\rho _\kappa ^{-1})}+\Vert \phi _{2r}^z\Vert _{{{\mathscr {C}}}^1(\rho _\kappa ^{-1})})+\Vert \phi _r^z\Vert _{{{\mathscr {C}}}^1(\rho _\kappa ^{-1})}+1\Big )\\&\qquad \times \Big ({{\mathbb {A}}}^{b,f}_{T,\infty }(\rho _\kappa )+\ell _T^b(\rho _\kappa )\Big (\int ^T_0\Vert u\Vert _{{{\mathbb {S}}}^{2\alpha }_t(\rho _1)}^q{\mathord {\mathrm{d}}}t\Big )^{1/q}\Big ). \end{aligned}$$

By Lemma 3.8, we have

$$\begin{aligned} \sup _z\rho _\kappa (z)\Vert \phi _{2r}^z\Vert _{{{\mathscr {C}}}^1(\rho _\kappa ^{-1})}\lesssim 1. \end{aligned}$$

On the other hand, by Lemma 3.8 and (2.29) with ${\bar{\rho }}=1, {\bar{\rho }}=\rho _\kappa$, we have

$$\begin{aligned}\sup _z\rho ^2_{\kappa }(z)\ell _T^{b_z} \lesssim \sup _z\rho ^2_{\kappa }(z)(\Vert \phi _{2r}^z\Vert _{{{\mathscr {C}}}^1(\rho _\kappa ^{-2})}+\Vert \phi _{2r}^z\Vert _{{{\mathscr {C}}}^1(\rho _\kappa ^{-1})}^2)\ell _T^b(\rho _\kappa ) \lesssim \ell _T^b(\rho _\kappa ),\end{aligned}$$

which together with the above estimate implies that for $\delta =(2\vartheta +2)\kappa \leqslant 1$,

$$\begin{aligned}&\sup _z\rho _\delta (z)(\ell _T^{b_z})^{\vartheta }{{\mathbb {A}}}^{b_z,F_z}_{T,q} \leqslant \Big (\sup _z\rho _\kappa ^2(z)\ell ^{b_z}_T\Big )^{\vartheta }\sup _z\rho ^2_\kappa (z){{\mathbb {A}}}^{b_z,F_z}_{T,q}\\&\quad \leqslant \big (\ell ^{b}_T(\rho _\kappa )\big )^{\vartheta +1} \left( {{\mathbb {A}}}^{b,f}_{T,\infty }(\rho _\kappa )+\left( \int ^T_0\Vert u\Vert _{{{\mathbb {S}}}^{2\alpha }_t(\rho _1)}^q{\mathord {\mathrm{d}}}t\right) ^{1/q}\right) . \end{aligned}$$

Note that by (2.6) and Young’s inequality,

$$\begin{aligned} \Vert u\Vert _{{{\mathbb {S}}}^{2\alpha }_t(\rho _1)} \leqslant \varepsilon \Vert u\Vert _{{{\mathbb {S}}}^{2-\alpha }_t(\rho _1)}+C_\varepsilon \Vert u\Vert _{{{\mathbb {L}}}^\infty _t(\rho _1)}. \end{aligned}$$

Hence, multiplying both sides of (3.24) by $\rho _\delta (z)$ we arrive at

$$\begin{aligned} \Vert u\Vert _{{{\mathbb {S}}}^{2-\alpha }_T(\rho _\delta )} \leqslant \varepsilon \Vert u\Vert _{{{\mathbb {S}}}^{2-\alpha }_T(\rho _\delta )}+C_\varepsilon \Vert u\Vert _{{{\mathbb {L}}}^\infty _T(\rho _1)}+C_\varepsilon {{\mathbb {A}}}^{b,f}_{T,\infty }(\rho _\kappa ), \end{aligned}$$

and

$$\begin{aligned} \Vert u\Vert _{{{\mathbb {L}}}^\infty _T(\rho _\delta )}&\lesssim {{\mathbb {A}}}^{b,f}_{T,\infty }(\rho _\kappa )+\left( \int ^T_0\Vert u\Vert ^q_{{{\mathbb {S}}}^{2-\alpha }_t(\rho _1)} {\mathord {\mathrm{d}}}t\right) ^{1/q}. \end{aligned}$$

The above two estimates imply that

$$\begin{aligned} \Vert u\Vert _{{{\mathbb {L}}}^\infty _T(\rho _1)}\leqslant \Vert u\Vert _{{{\mathbb {L}}}^\infty _T(\rho _\delta )}\lesssim {{\mathbb {A}}}^{b,f}_{T,\infty }(\rho _\kappa )+\left( \int ^T_0\Vert u\Vert ^q_{{{\mathbb {L}}}^\infty _t(\rho _1)}{\mathord {\mathrm{d}}}t\right) ^{1/q}. \end{aligned}$$

Finally, we use Gronwall’s inequality to deduce the first estimate in (3.23).

By (3.3), (2.19) and (2.12) we have for weight $\rho , {\bar{\rho }}\in {{\mathscr {W}}}$

$$\begin{aligned} \Vert u^\sharp \Vert _{L^\infty _T{{\mathbf {C}}}^{2-\alpha }(\rho {\bar{\rho }})}&\lesssim \Vert u\Vert _{L^\infty _T{{\mathbf {C}}}^{2-\alpha }(\rho {\bar{\rho }})}+\Vert \nabla u\prec \!\!\!\prec {{\mathscr {I}}}_\lambda b\Vert _{L^\infty _T{{\mathbf {C}}}^{2-\alpha }(\rho {\bar{\rho }})} +\Vert {{\mathscr {I}}}_\lambda f\Vert _{L^\infty _T{{\mathbf {C}}}^{2-\alpha }(\rho {\bar{\rho }})}\nonumber \\&\lesssim \Vert u\Vert _{L^\infty _T{{\mathbf {C}}}^{2-\alpha }({\bar{\rho }})}+\Vert \nabla u\Vert _{{{\mathbb {L}}}^\infty _T({\bar{\rho }})}\Vert b\Vert _{L^\infty _T{{\mathbf {C}}}^{-\alpha }(\rho )} +\Vert f\Vert _{L^\infty _T{{\mathbf {C}}}^{-\alpha }(\rho )}\nonumber \\&\lesssim \sqrt{\ell ^b_T(\rho )}\Vert u\Vert _{L^\infty _T{{\mathbf {C}}}^{2-\alpha }({\bar{\rho }})}+\Vert f\Vert _{L^\infty _T{{\mathbf {C}}}^{-\alpha }(\rho )}. \end{aligned}$$

(3.28)

Next we estimate each term on the right hand side of (3.4) by using Lemma 2.10.

• By (2.21), (2.4) we have

  $$\begin{aligned} \Vert \nabla u\prec b-\nabla u\prec \!\!\!\prec b\Vert _{L^\infty _T{{\mathbf {C}}}^{1-2\alpha }(\rho {\bar{\rho }})}\lesssim \Vert u\Vert _{{{\mathbb {S}}}_T^{2-\alpha }({\bar{\rho }})}\Vert b_{L^\infty _T{{\mathbf {C}}}^{-\alpha }(\rho )}. \end{aligned}$$

• By (2.15), we have

  $$\begin{aligned} \Vert \nabla u\succ b\Vert _{L_T^\infty {{\mathbf {C}}}^{1-2\alpha }(\rho {\bar{\rho }})}\lesssim \Vert u\Vert _{L_T^\infty {{\mathbf {C}}}^{2-\alpha }({\bar{\rho }})}\Vert b\Vert _{L_T^\infty {{\mathbf {C}}}^{-\alpha }(\rho )}. \end{aligned}$$

• By (2.20) and (2.12) we have

  $$\begin{aligned} \Vert [{{\mathscr {L}}},\nabla u \prec \!\!\!\prec ]{{\mathscr {I}}}b\Vert _{L_T^\infty {{\mathbf {C}}}^{1-2\alpha }(\rho {\bar{\rho }})}\lesssim \Vert u\Vert _{{{\mathbb {S}}}_T^{2-\alpha }({\bar{\rho }})}\Vert b\Vert _{L_T^\infty {{\mathbf {C}}}^{-\alpha }(\rho )}. \end{aligned}$$

• By Lemma 3.3 with $\gamma =2-2\alpha$, $\beta \in (\alpha ,2-2\alpha )$, we have

  $$\begin{aligned} \Vert b\circ \nabla u\Vert _{L_T^\infty {{\mathbf {C}}}^{1-2\alpha }(\rho ^{2+\varepsilon }{\bar{\rho }})}\lesssim \Vert u\Vert _{{{\mathbb {S}}}_T^{2-\alpha }({\bar{\rho }})} +\Vert u^\sharp \Vert _{L_T^\infty {{\mathbf {C}}}^{\beta +1}(\rho ^{1+\varepsilon }{\bar{\rho }})}+{{\mathbb {A}}}^{b,f}_{T,\infty }(\rho ). \end{aligned}$$

Combining the above calculations and by (3.4) and (2.11) with $\theta =2$ and $q=\infty$, we obtain

$$\begin{aligned} \Vert u^\sharp \Vert _{{\mathbb {S}}_T^{3-2\alpha }(\rho ^{2+\varepsilon }{\bar{\rho }})}&\lesssim \Vert u\Vert _{{{\mathbb {S}}}_T^{2-\alpha }({\bar{\rho }})} +\Vert u^\sharp \Vert _{L_T^\infty {{\mathbf {C}}}^{\beta +1}(\rho ^{1+\varepsilon }{\bar{\rho }})}+{{\mathbb {A}}}^{b,f}_{T,\infty }(\rho ). \end{aligned}$$

(3.29)

On the other hand, for $\varepsilon >\frac{2\alpha -1}{2-3\alpha }$, one can choose $\beta$ close to $\alpha$ so that

$$\begin{aligned} \theta :=\tfrac{\varepsilon }{1+\varepsilon }=\tfrac{\alpha +\beta -1}{1-\alpha }. \end{aligned}$$

Thus by interpolation inequality (2.5), Young’s inequality and (3.28), for any $\delta >0$,

$$\begin{aligned} \Vert u^\sharp \Vert _{L_T^\infty {{\mathbf {C}}}^{\beta +1}(\rho ^{1+\varepsilon }{\bar{\rho }})}&\lesssim \Vert u^\sharp \Vert _{L_T^\infty {{\mathbf {C}}}^{3-2\alpha }(\rho ^{2+\varepsilon }{\bar{\rho }})}^{\theta } \Vert u^\sharp \Vert _{L^\infty _T{{\mathbf {C}}}^{2-\alpha }(\rho {\bar{\rho }})}^{1-\theta }\\&\leqslant \delta \Vert u^\sharp \Vert _{L_T^\infty {{\mathbf {C}}}^{3-2\alpha }(\rho ^{2+\varepsilon }{\bar{\rho }})} +C_\delta \Big (\Vert u\Vert _{{{\mathbb {S}}}_T^{2-\alpha }({\bar{\rho }})}+{{\mathbb {A}}}^{b,f}_{T,\infty }(\rho )\Big ). \end{aligned}$$

Substituting this into (3.29), we obtain the second estimate in (3.23) by taking $\rho =\rho _\kappa , {\bar{\rho }}=\rho _\delta$.

(Uniqueness). It follows by Theorem A.2 in the appendix. $\square$

4 Hamilton–Jacobi–Bellman equations

The next two sections are devoted to a priori estimates on solutions to Eq. (1.8). The proof is divided into two steps. First we construct a $C^1$-diffeomorphism and perform a Zvonkin transformation through this diffeomorphism. After this transform the singular part in (1.8) disappears and we obtain an HJB equation in non-divergence form. We then obtain a priori estimates for this HJB equation, which leads to global uniform bounds for solutions to (1.8). To this end, in this section we consider the following general HJB equation:

$$\begin{aligned} \partial _t v=\mathrm {tr}(a\cdot \nabla ^2 v)+B\cdot \nabla v+H(v,\nabla v),\ v(0)=v_0, \end{aligned}$$

(4.1)

where $a:{{\mathbb {R}}}_+\times {{\mathbb {R}}}^d\rightarrow {{\mathbb {R}}}^d\otimes {{\mathbb {R}}}^d$ is a symmetric matrix-valued measurable function, and $B:{{\mathbb {R}}}_+\times {{\mathbb {R}}}^d\rightarrow {{\mathbb {R}}}^d$ is a vector-valued measurable function, and

$$\begin{aligned} H(t,x,v,Q): {{\mathbb {R}}}^+\times {{\mathbb {R}}}^d\times {{\mathbb {R}}}\times {{\mathbb {R}}}^d\rightarrow {{\mathbb {R}}}\end{aligned}$$

is a real-valued measurable function, and continuous in v, Q for each t, x.

For instance, for any $\zeta \in [1,2]$, the equation

$$\begin{aligned} {{\mathscr {L}}}v= |\nabla v|^\zeta +B \cdot \nabla v + f \end{aligned}$$

(4.2)

is a typical HJB equation. Note that for $\lambda >0$, if we define

$$\begin{aligned} v_\lambda (t,x):=v(\lambda ^2 t,\lambda x),\ B_\lambda (t,x):=\lambda B(\lambda ^2 t,\lambda x),\ f_\lambda (t,x):=\lambda ^2f(\lambda ^2 t,\lambda x), \end{aligned}$$

then

$$\begin{aligned} {{\mathscr {L}}}v_\lambda =\lambda ^{2-\zeta } |\nabla v_\lambda |^\zeta +B_\lambda \cdot \nabla v_\lambda + f_\lambda . \end{aligned}$$

In particular, if $\zeta =2$, then the nonlinear term has the same order as the Laplacian term in scaling level. In this case, we say that HJB Eq. (4.2) is critical. While for $\zeta <2$, the nonlinear term can be controlled well by the Laplacian term. In this case, we say that HJB equation (4.2) is subcritical².

Throughout this section we use the following polynomial weight function

$$\begin{aligned} \rho _\delta (x):=\langle x\rangle ^{-\delta }=(1+|x|^2)^{-\delta /2}\Rightarrow \rho ^\gamma _\delta =\rho _{\gamma \delta },\ \delta ,\gamma \in {{\mathbb {R}}}, \end{aligned}$$

and make the following elliptic assumption on a:

($\mathbf{H} ^\alpha _1$):

$a:{{\mathbb {R}}}_+\times {{\mathbb {R}}}^d\rightarrow {{\mathbb {R}}}^d\otimes {{\mathbb {R}}}^d$ is a symmetric $d\times d$-matrix-valued measurable function and satisfies that for some $c_0\in (0,1)$,

$$\begin{aligned} c_0|\xi |^2\leqslant \sum _{i,j=1}^da_{ij}(t,x)\xi _i\xi _j\leqslant c_0^{-1}|\xi |^2,\ \ \forall \xi \in {{\mathbb {R}}}^d, \end{aligned}$$

(4.3)

and for some $\alpha \in (0,1)$ and $c_1\geqslant 1$,

$$\begin{aligned} |a(t,x)-a(t,y)|\leqslant c_1|x-y|^\alpha . \end{aligned}$$

About the nonlinear term H, we separately consider two cases: subcritical case for all $d\in {{\mathbb {N}}}$ and critical case only for $d=1$, and assume

($\mathbf{H} ^{\delta ,\zeta }_{\mathrm{sub}}$):

Suppose that for some $\delta ,\zeta \in [0,2)$ and $c_2>0$,

$$\begin{aligned} |H(t,x,v,Q)|\lesssim _{c_2}\langle x\rangle ^\delta +|Q|^\zeta . \end{aligned}$$

(4.4)

($\mathbf{H} ^{\delta ,\beta }_{\mathrm{crit}}$):

Suppose that $d=1$ and H can be decomposed as $H_s+H_c$ with $H_s$ satisfying ($\mathbf{H} ^{\delta ,\zeta }_{\mathrm{sub}}$) and $H_c$ satisfying for some $\delta \in [0,2)$ and $c_2>0$,

$$\begin{aligned} |H_c(t,x,v,Q)|\lesssim _{c_2}\langle x\rangle ^\delta +|Q|^2,\ \ |\partial _vH_c(t,x,v,Q)|\lesssim _{c_2}\langle x\rangle ^\delta +|v|^2+|Q|, \end{aligned}$$

(4.5)

and for some $\beta \in (0,1]$ and all $|x-y|\leqslant 1$,

$$\begin{aligned} |H_c(t,x,v,Q)-H_c(t,y,v,Q)|\lesssim _{ c_2}|x-y|^\beta (\langle x\rangle ^\delta +\langle y\rangle ^\delta +|v|^2+|Q|^2). \end{aligned}$$

(4.6)

We introduce the following definition of strong solution to HJB Eq. (4.1).

Definition 4.1

We call a function $v\in \cap _{p\geqslant 2}{{\mathbb {H}}}^{2,p}_{loc}$ strong solution to (4.1) if for all $\psi \in C^\infty _c({{\mathbb {R}}}^d)$ and $t\geqslant 0$,

$$\begin{aligned} \langle v(t),\psi \rangle =\langle v_0,\psi \rangle +\int ^t_0\Big \langle \big (\mathrm {tr}(a\cdot \nabla ^2 v)+B\cdot \nabla v+H(v,\nabla v)\big )(s),\psi \Big \rangle {\mathord {\mathrm{d}}}s, \end{aligned}$$

where $\langle v_0,\psi \rangle :=\int v_0\psi$. In particular, for all $t\geqslant 0$ and Lebesgue almost all $x\in {{\mathbb {R}}}^d$,

$$\begin{aligned} v(t,x)=v_0(x)+\int ^t_0\Big (\mathrm {tr}(a\cdot \nabla ^2 v)+B\cdot \nabla v+H(v,\nabla v)\Big )(s,x){\mathord {\mathrm{d}}}s. \end{aligned}$$

The aim of this section is to establish the following well-posedness for HJB Eq. (4.1). For simplicity of notation, we introduce the following parameter set for the dependence of constants:

$$\begin{aligned} \Theta :=(T, d,\alpha ,\beta ,\zeta ,\delta ,c_0,c_1,c_2). \end{aligned}$$

Theorem 4.2

Let $T>0$, $\delta \in (0,2)$ and $\alpha ,\beta ,\delta _1\in (0,1]$. Suppose that ($\mathbf{H} ^\alpha _1$), $B\in {{\mathbb {L}}}_T^\infty (\rho _{\delta _1})$ and ($\mathbf{H} ^{\delta ,\zeta }_\mathrm{sub}$) or ($\mathbf{H} ^{\delta ,\beta }_{\mathrm{crit}}$) hold. Let

$$\begin{aligned} \left\{ \begin{aligned}&\eta>\tfrac{\zeta \delta }{2-\zeta }\vee [2\delta _1+\delta ],\&\mathrm{under }\, (\mathbf{H} ^{\delta ,\zeta }_{\mathrm{sub}});\\&\eta >2\left( \tfrac{(1+2\beta )\delta }{\beta }\vee (\delta _1+\delta ){\vee \tfrac{\delta \vee (2\delta -1)}{2-\zeta }}\right) ,&\mathrm{under }\, (\mathbf{H} ^{\delta ,\beta }_{\mathrm{crit}}). \end{aligned} \right. \end{aligned}$$

(4.7)

(Existence) For any initial value $v_0\in {{\mathscr {C}}}^2(\rho _{\delta })$, there exist $p_0$ large enough and strong solution v to HJB Eq. (4.1), which satisfies the following estimate: for any $p\geqslant p_0$, there is a constant $C=C(\Theta , p,\eta ,\delta _1,\Vert B\Vert _{{{\mathbb {L}}}^\infty _T(\rho _{\delta _1})},\Vert v_0\Vert _{{{\mathscr {C}}}^2(\rho _\delta )})>0$ such that

$$\begin{aligned} \Vert v\Vert _{{{\mathbb {L}}}^\infty _T(\rho _{\delta })}+\Vert \partial _tv\Vert _{{{\mathbb {L}}}^{p}_T(\rho _\eta )}+\Vert v\Vert _{{{\mathbb {H}}}^{2,p}_T(\rho _\eta )}\leqslant C. \end{aligned}$$

(4.8)

In particular, for any $0\leqslant \varepsilon '<\varepsilon \leqslant 2$,

$$\begin{aligned} \Vert v\Vert _{C^{\varepsilon '/2}_T{{\mathbf {C}}}^{2-\varepsilon }(\rho _\eta )}\leqslant C. \end{aligned}$$

(Uniqueness) If, in addition, for some $C>0$,

$$\begin{aligned} |\partial _v H(t,x,v,Q)|^{1/2}+|\partial _Q H(t,x,v,Q)|\lesssim _C \langle x\rangle +|v|^{1/\delta }+|Q|^{1/\eta }, \end{aligned}$$

(4.9)

then there is a unique strong solution with regularity (4.8).

Remark 4.3

(i) When $a\in L^\infty _T{{\mathscr {C}}}^1$, the above regularity result could be obtained by De-Giorgi’s iteration method since it can be written in the divergence form (cf. [39]). However, there seems no literature studying this problem when a is only Hölder continuous. Moreover, the unbounded B and H cause some difficulties for obtaining the global estimates, which is crucial for a-priori estimate for (1.8) and KPZ type equations. We believe that the above theorem is of its own interest.

(ii) The condition in (4.7) on $\eta$ comes from the energy estimate and the integrability of the weights in ${{\mathbb {R}}}^d$ (see Theorems 4.6 and 4.7 below).

In the following we first establish a maximum principle in Sect. 4.1. The subcritical case is treated in Sect. 4.2 by using $L^\infty (\rho _\delta )$-estimate and $L^p$-theory for PDEs. For the critical case, we take spatial derivative on both sides of (4.1) and obtain a PDE in divergence form. Then using the $L^\infty (\rho _\delta )$-bound and energy estimate we obtain the ${{\mathbb {H}}}^{2,p}_T(\rho _\eta )$-estimate in Sect. 4.3.

4.1 Maximum principle in weighted spaces

We first show the following maximum principle in weighted spaces by an exponential transform and a probabilistic method.

Theorem 4.4

(Maximum principle) Let $T>0$ and $\delta \in (0,2)$. Suppose (4.3) and for some $c_2, c_3>0$,

$$\begin{aligned} |H(t,x,v,Q)|\leqslant c_2 \langle x\rangle ^\delta +c_3|Q|^2,\ \ B\in {{\mathbb {L}}}^\infty _T(\rho _1). \end{aligned}$$

For any $v_0\in L^\infty (\rho _{\delta })$, there is a function $C(r)=C_\Theta (r)>0$ with $C(0)=0$ such that for any strong solution $v\in \cap _{p\geqslant 2} {{\mathbb {H}}}^{2,p}_{loc}\cap {{\mathbb {L}}}^\infty _T(\rho _\delta )$ to (4.1) with initial value $v_0$,

$$\begin{aligned} \Vert v\Vert _{{{\mathbb {L}}}^\infty _T(\rho _{\delta })}\leqslant C(c_2+\Vert v_0\Vert _{L^\infty (\rho _{\delta })}). \end{aligned}$$

(4.10)

Proof

We use a probabilistic method. For $\lambda >0$, define

$$\begin{aligned} w(t,x):=\mathrm {e}^{\lambda v(t,x)}. \end{aligned}$$

By the chain rule, it is easy to see that w satisfies

$$\begin{aligned} \partial _t w=\mathrm {tr}(a\cdot \nabla ^2 w)+B\cdot \nabla w+\lambda w \Big (H(v,\nabla v)-\lambda \mathrm {tr}(a\cdot \nabla v\otimes \nabla v)\Big ). \end{aligned}$$

For simplicity of notations, we write

$$\begin{aligned} F_\delta (x):=c_2\langle x\rangle ^\delta ,\ U_\lambda :=\lambda w \Big (H(v,\nabla v)-\lambda \mathrm {tr}(a\cdot \nabla v\otimes \nabla v)-F_\delta \Big ). \end{aligned}$$

Next we reverse the time variable. For a space-time function f, we set

$$\begin{aligned} f^T(t,x):=f(T-t,x). \end{aligned}$$

It is easy to see that $w^T(t,x)=w(T-t,x)$ solves the following backward equation:

$$\begin{aligned} \partial _t w^T+\mathrm {tr}(a^T\cdot \nabla ^2 w^T)+B^T\cdot \nabla w^T+U_\lambda ^T+\lambda w^TF_\delta =0, \end{aligned}$$

(4.11)

with subjected to the final condition

$$\begin{aligned} w^T(T,x)=w(0,x)=\mathrm {e}^{\lambda v_0(x)}. \end{aligned}$$

(4.12)

Under (4.3) and $B\in {{\mathbb {L}}}^\infty _T(\rho _1)$, for each $(t,x)\in [0,T]\times {{\mathbb {R}}}^d$, it is well known that the following SDE has a (probabilistically) weak solution starting from x at time t (see [37, page 87, Theorem 1])

$$\begin{aligned} X^{t,x}_s=x+\int ^s_t\sqrt{2a^T}(r,X^{t,x}_r){\mathord {\mathrm{d}}}W_r+\int ^s_tB^T(r,X^{t,x}_r){\mathord {\mathrm{d}}}r,\ \ \forall s\in [t,T], \end{aligned}$$

where W is a d-dimensional Brownian motion on some stochastic basis $(\Omega ',{\mathcal {F}}',{\mathbb {P}})$. For $R>0$, define a stopping time

$$\begin{aligned} \tau _R:=\inf \{s\geqslant t: |X^{t,x}_s|> R\}. \end{aligned}$$

It is well known that the following Krylov estimate holds ([37, page 52, Theorem 2]): for any $p\geqslant d+1$,

$$\begin{aligned} {{\mathbb {E}}}\left( \int ^{T\wedge \tau _R}_t f(s, X^{t,x}_s){\mathord {\mathrm{d}}}s\right) \leqslant C_R\left( \int ^T_t\!\!\!\int _{B_R}|f(s, x)|^p{\mathord {\mathrm{d}}}x{\mathord {\mathrm{d}}}s\right) ^{1/p}. \end{aligned}$$

Since $v\in \cap _{p\geqslant 2}{{\mathbb {H}}}^{2,p}_{loc}\cap {{\mathbb {L}}}^\infty _T(\rho _\delta )$, it is easy to see that

$$\begin{aligned} w^T\in \cap _{p\geqslant 2}{{\mathbb {H}}}^{2,p}_{\mathrm{{loc}}},\ \ \partial _t w^T\in \cap _{p\geqslant 2}{{\mathbb {L}}}^p_{\mathrm{{loc}}}. \end{aligned}$$

Thus, for each fixed (t, x), by generalized Itô’s formula (see [37, page 122, Theorem 1]), we have

$$\begin{aligned} {\mathord {\mathrm{d}}}_s w^T(s,X^{t,x}_s)&=(\partial _sw^T+\mathrm {tr}(a^T\cdot \nabla ^2w^T)+B^T\cdot \nabla w^T)(s,X^{t,x}_s){\mathord {\mathrm{d}}}s\\&\quad +(\sqrt{2a^T}\cdot \nabla w^T)(s,X^{t,x}_s){\mathord {\mathrm{d}}}W_s, \end{aligned}$$

and by (4.11) and (4.12),

$$\begin{aligned}&\mathrm {e}^{\int ^{t'}_{t} \lambda F_\delta (X^{t,x}_s){\mathord {\mathrm{d}}}s} w^T(t',X^{t,x}_{t'})\\&=w^T(t,x)+\int ^{t'}_t\mathrm {e}^{\int ^s_{t} \lambda F_\delta (X^{t,x}_r){\mathord {\mathrm{d}}}r}{\mathord {\mathrm{d}}}_s w^T(s,X^{t,x}_s)\\&\quad +\int ^{t'}_t\mathrm {e}^{\int ^s_{t} \lambda F_\delta (X^{t,x}_r){\mathord {\mathrm{d}}}r}(\lambda F_\delta w^T)(s,X^{t,x}_s){\mathord {\mathrm{d}}}s\\&=w^T(t,x)-\int ^{t'}_t\mathrm {e}^{\int ^s_{t} \lambda F_\delta (X^{t,x}_r){\mathord {\mathrm{d}}}r}U^T_\lambda (s,X^{t,x}_s){\mathord {\mathrm{d}}}s+M_{t'}, \end{aligned}$$

where

$$\begin{aligned} M_{t'}:=\int ^{t'}_t\mathrm {e}^{\int ^s_{t} \lambda F_\delta (X^{t,x}_r){\mathord {\mathrm{d}}}r}(\sqrt{2a^T}\cdot \nabla w^T)(s,X^{t,x}_s){\mathord {\mathrm{d}}}W_s. \end{aligned}$$

By (4.3) and $|H(v,Q)|\leqslant F_\delta +c_3|Q|^2$, one can choose $\lambda =c_3/c_0$ so that

$$\begin{aligned} U^T_\lambda \leqslant \lambda w \Big (c_3|\nabla v|^2-\lambda c_0 |\nabla v|^2\Big )=0. \end{aligned}$$

Hence, for $\lambda =(c_3/c_0)\vee 1$,

$$\begin{aligned} \mathrm {e}^{\lambda v(T-t,x)}=w^T(t,x)\leqslant \mathrm {e}^{\int ^{t'}_{t} \lambda F_\delta (X^{t,x}_s){\mathord {\mathrm{d}}}s}w^T(t',X^{t,x}_{t'})-M_{t'}. \end{aligned}$$

Since $t'\mapsto M_{t'\wedge \tau _R}$ is a martingale, we have

$$\begin{aligned} \mathrm {e}^{\lambda v(T-t,x)}\leqslant {{\mathbb {E}}}\left( \mathrm {e}^{\int ^{T\wedge \tau _R}_{t} \lambda F_\delta (X^{t,x}_s){\mathord {\mathrm{d}}}s} w^T(T\wedge \tau _R, X^{t,x}_{T\wedge \tau _R})\right) . \end{aligned}$$

On the other hand, by Lemma B.1 in appendix, for any $\gamma \geqslant 0$ and $\alpha \in [0,2)$,

$$\begin{aligned} {{\mathbb {E}}}\left( \mathrm {e}^{\gamma \sup _{s\in [t,T]}\langle X^{t,x}_s\rangle ^\alpha }\right) \leqslant C(\gamma )\mathrm {e}^{C_2\gamma \langle x\rangle ^\alpha }. \end{aligned}$$

Since $w^T(t,x)\leqslant \mathrm {e}^{\lambda \Vert v\Vert _{{{\mathbb {L}}}^\infty _T(\rho _\delta )}\langle x\rangle ^\delta }$, letting $R\rightarrow \infty$ and by the dominated convergence theorem, we get

$$\begin{aligned} \mathrm {e}^{\lambda v(T-t,x)}&\leqslant {{\mathbb {E}}}\left( \mathrm {e}^{\int ^{T}_{t} \lambda F_\delta (X^{t,x}_s){\mathord {\mathrm{d}}}s}w^T(T, X^{t,x}_T)\right) ={{\mathbb {E}}}\left( \mathrm {e}^{\int ^{T}_{t} \lambda F_\delta (X^{t,x}_s){\mathord {\mathrm{d}}}s+\lambda v_0(X^{t,x}_T)}\right) \\&\leqslant {{\mathbb {E}}}\left( \mathrm {e}^{\ell _0\sup _{s\in [t,T]}\langle X^{t,x}_s\rangle ^{\delta }}\right) \leqslant C(\ell _0)\mathrm {e}^{\ell _0 \langle x\rangle ^{\delta }}, \end{aligned}$$

where $\ell _0:=\lambda (c_2T+\Vert v_0\Vert _{L^\infty (\rho _{\delta })})$. Hence,

$$\begin{aligned} v(T-t,x)\leqslant C(\ell _0)\langle x\rangle ^{\delta }. \end{aligned}$$

By applying the above estimate to $-v$, we obtain the desired estimate. $\square$

4.2 Subcritical case

In this section we consider the subcritical case ($\mathbf{H} ^{\delta ,\zeta }_{\mathrm{sub}}$) and prove some a priori regularity estimates. To this end, we first show the following interpolation inequalities in weighted spaces, which will play important roles in dealing with the weights.

Lemma 4.5

(i) For any $p\geqslant 2$ and $r,p\in [1,\infty ]$ satisfying $\frac{2}{p}=\frac{1}{r}+\frac{1}{q}$, and $\delta ,\delta _1,\delta _2\in {{\mathbb {R}}}$ with $\delta _1+\delta _2=2\delta$, there is a constant $C=C(p,r,q,\delta ,\delta _1,\delta _2)>0$ such that

$$\begin{aligned} \Vert \nabla v\rho _\delta \Vert _{L^p}\lesssim _C\Vert \nabla ^2 v\rho _{\delta _1}\Vert ^{1/2}_{L^q}\Vert v\rho _{\delta _2}\Vert ^{1/2}_{L^r} +\Vert v\rho _{\delta +1}\Vert _{L^p}. \end{aligned}$$

(4.13)

(ii) For any $p,q\in [2,\infty ), r\in [2,\infty ]$ satisfying $\frac{q+2}{p}=1+\frac{2}{r}$, and $\delta ,\delta _1,\delta _2\in {{\mathbb {R}}}$ with $\delta =\frac{q\delta _1}{q+2}+\frac{2\delta _2}{q+2}$, there is a constant $C=C(p,q,r,\delta ,\delta _1,\delta _2)>0$ such that

$$\begin{aligned} \Vert \nabla v\rho _\delta \Vert _{L^p}\lesssim _C \left( \int |\nabla ^2 v|^2||\nabla v|^{q-2}\rho ^q_{\delta _1}\right) ^{\frac{1}{q+2}} \Vert v\rho _{\delta _2}\Vert _{L^r}^{\frac{2}{q+2}}+\Vert v\rho _{\delta +1}\Vert _{L^p}. \end{aligned}$$

(4.14)

Proof

By definition and the integration by parts, we have

$$\begin{aligned} \Vert \nabla v\rho _\delta \Vert _{L^p}^p&=\int |\nabla v|^p\rho _{\delta p}=\int \langle \nabla v,\nabla v|\nabla v|^{p-2}\rho _{\delta p}\rangle \nonumber \\&\lesssim \int |v|\Big (|\nabla ^2 v||\nabla v|^{p-2}\rho _{\delta p}+|\nabla v|^{p-1}|\nabla \rho _{\delta p}|\Big ). \end{aligned}$$

(4.15)

(i) By Hölder’s inequality we have

$$\begin{aligned} \int |v||\nabla ^2 v||\nabla v|^{p-2}\rho _{\delta p}&\leqslant \Vert v\rho _{\delta _2}\Vert _{L^r}\Vert \nabla ^2 v\rho _{\delta _1}\Vert _{L^q}\Vert \nabla v\rho _\delta \Vert _{L^p}^{p-2}, \end{aligned}$$

and by $|\nabla \rho _{\delta }|\lesssim \rho _{\delta +1}$,

$$\begin{aligned} \int |v| |\nabla v|^{p-1}|\nabla \rho _{\delta p}|\leqslant \Vert \nabla v\rho _\delta \Vert _{L^p}^{p-1}\Vert v\rho _{\delta +1}\Vert _{L^p}. \end{aligned}$$

(4.16)

Therefore,

$$\begin{aligned} \Vert \nabla v\rho _\delta \Vert _{L^p}^p\lesssim \Vert v\rho _{\delta _2}\Vert _{L^r}\Vert \nabla ^2 v\rho _{\delta _1}\Vert _{L^q}\Vert \nabla v\rho _\delta \Vert _{L^p}^{p-2} +\Vert \nabla v\rho _\delta \Vert _{L^p}^{p-1}\Vert v\rho _{\delta +1}\Vert _{L^p}. \end{aligned}$$

Thus by Young’s inequality, we obtain (4.13).

(ii) On the other hand, by Hölder’s inequality we have

$$\begin{aligned}&\int |v||\nabla ^2 v||\nabla v|^{p-2}\rho _{\delta p} \leqslant \left( \int |\nabla ^2 v|^2||\nabla v|^{q-2}\rho _{\delta _1 q}\right) ^{1/2}\Vert \nabla v\rho _{\delta }\Vert _{L^p}^{p-\frac{q}{2}-1}\Vert v\rho _{\delta _2}\Vert _{L^r}, \end{aligned}$$

which together with (4.15) and (4.16) yields (4.14). $\square$

We now prove the following a priori $L^p$-regularity estimate by the ${{\mathbb {L}}}_T^\infty (\rho _\delta )$ estimate obtained in Theorem 4.4.

Theorem 4.6

Let $T>0$, $\delta \in (0,2)$ and $\alpha ,\delta _1\in (0,1]$. Suppose ($\mathbf{H} ^\alpha _1$), $B\in {{\mathbb {L}}}_T^\infty (\rho _{\delta _1})$ and ($\mathbf{H} ^{\delta ,\zeta }_{\mathrm{sub}}$). Then for any $\eta >(2\delta _1+\delta )\vee \frac{\zeta \delta }{2-\zeta }$ and $v_0\in {{\mathscr {C}}}^2(\rho _{\delta })$, there is a $p_0$ large enough so that for all $p>p_0$ and any strong solution v of HJB (4.1),

$$\begin{aligned} \Vert \partial _t (v\rho _\eta )\Vert _{{{\mathbb {L}}}^{p}_T}+\Vert v\rho _\eta \Vert _{{{\mathbb {H}}}^{2,p}_T}\leqslant C, \end{aligned}$$

where $C=C(\Theta ,\eta ,p,\delta _1,\Vert B\Vert _{{{\mathbb {L}}}^\infty _T(\rho _{\delta _1})},\Vert v_0\Vert _{{{\mathscr {C}}}^2(\rho _{\delta })})$.

Proof

Multiplying both sides of (4.1) by $\rho _\eta$, we get

$$\begin{aligned} \partial _t (v\rho _\eta )=\mathrm {tr}(a\cdot \nabla ^2(v\rho _\eta ))-\Gamma _\rho +(B\cdot \nabla v)\rho _\eta +H(v,\nabla v)\rho _\eta , \end{aligned}$$

(4.17)

where

$$\begin{aligned} \Gamma _\rho =\mathrm {tr}(a\cdot (2\nabla v\otimes \nabla \rho _\eta +v\nabla ^2\rho _\eta )). \end{aligned}$$

Fix

$$\begin{aligned} p>\frac{(2-\zeta )d}{(2-\zeta )\eta -\zeta \delta }\vee \frac{d}{\eta -2\delta _1-\delta }=:p_0. \end{aligned}$$

By the $L^p$-theory of PDEs (see [38]), there is a constant $C=C(\Theta , p)$ such that

$$\begin{aligned} \Vert \partial _t(v\rho _\eta )\Vert _{{{\mathbb {L}}}^{p}_T}+\Vert v\rho _\eta \Vert _{{{\mathbb {H}}}^{2,p}_T}\lesssim _C\Vert H(v,\nabla v)\rho _\eta +(B\cdot \nabla v)\rho _\eta -\Gamma _\rho \Vert _{{{\mathbb {L}}}^p_T}+\Vert v_0\rho _\eta \Vert _{H^{2,p}}. \end{aligned}$$

Since $p(\eta -\delta )>d$, we have

$$\begin{aligned} \Vert v_0\rho _\eta \Vert _{H^{2,p}}\lesssim \Vert v_0\rho _{\delta }\Vert _{{{\mathscr {C}}}^2}\left( \int _{{{\mathbb {R}}}^d}\rho ^p_{\eta -\delta }(x){\mathord {\mathrm{d}}}x\right) ^{1/p} \lesssim \Vert v_0\Vert _{{{\mathscr {C}}}^2(\rho _{\delta })}, \end{aligned}$$

and by (4.4),

$$\begin{aligned} \Vert H(v,\nabla v)\rho _\eta \Vert _{{{\mathbb {L}}}^p_T}\lesssim \Vert \rho _{\eta -\delta }\Vert _{L^p}+\Vert |\nabla v|^\zeta \rho _\eta \Vert _{{{\mathbb {L}}}^p_T} \lesssim 1+\Vert \nabla v\rho _{\eta /\zeta }\Vert _{{{\mathbb {L}}}^{\zeta p}_T}^\zeta . \end{aligned}$$

By interpolation inequality (4.13) and using $|\nabla \rho _\delta |\lesssim \rho _{\delta +1}$, we have

$$\begin{aligned} \Vert \nabla v\rho _{\eta /\zeta }\Vert ^\zeta _{{{\mathbb {L}}}^{\zeta p}_T}&\leqslant \Vert \nabla ^2 v\rho _\eta \Vert _{{{\mathbb {L}}}^p_T}^{\zeta /2}\Vert v\rho _{\eta (2/\zeta -1)}\Vert _{{{\mathbb {L}}}^q_T}^{\zeta /2} +\Vert v\rho _{\eta /\zeta +1}\Vert ^\zeta _{{{\mathbb {L}}}^{\zeta p}_T}, \end{aligned}$$

where $q=p\zeta /(2-\zeta )$. Since $p(\eta -\zeta \delta /(2-\zeta ))>d$, by (4.10), we have

$$\begin{aligned} \Vert v\rho _{2\eta /\zeta -\eta }\Vert _{{{\mathbb {L}}}^q_T}^q&=\int ^T_0\!\!\!\int _{{{\mathbb {R}}}^d} |v(t,x)|^q\rho _{\eta p}(x){\mathord {\mathrm{d}}}x{\mathord {\mathrm{d}}}t\\&\lesssim \int _{{{\mathbb {R}}}^d}\rho _{\delta }(x)^{-p\zeta /(2-\zeta )}\rho _{\eta p}(x){\mathord {\mathrm{d}}}x\\&\lesssim \int _{{{\mathbb {R}}}^d}(1+|x|)^{\frac{p\zeta \delta }{2-\zeta }-\eta p}{\mathord {\mathrm{d}}}x\lesssim 1, \end{aligned}$$

and also,

$$\begin{aligned} \Vert v\rho _{\eta /\zeta +1}\Vert ^\zeta _{{{\mathbb {L}}}^{\zeta p}_T}\lesssim \Vert \rho _{\eta /\zeta +1-\delta }\Vert ^\zeta _{{{\mathbb {L}}}^{\zeta p}_T}\lesssim 1. \end{aligned}$$

Thus, for any $\varepsilon \in (0,1)$, by Young’s inequality,

$$\begin{aligned} \Vert H(v,\nabla v)\rho _\eta \Vert _{{{\mathbb {L}}}^p_T}\lesssim \varepsilon \Vert \nabla ^2 v\rho _\eta \Vert _{{{\mathbb {L}}}^p_T}+1. \end{aligned}$$

Since $B\in {{\mathbb {L}}}^\infty _T(\rho _{\delta _1})$ and $\eta >2\delta _1+\delta$ and $p(\eta -2\delta _1-\delta )>d$, we also have by (4.13) and (4.10)

$$\begin{aligned} \Vert (B\cdot \nabla v)\rho _\eta \Vert _{{{\mathbb {L}}}^p_T}&\lesssim \Vert \rho _{\eta -\delta _1}|\nabla v|\Vert _{{{\mathbb {L}}}^p_T} \lesssim \Vert \nabla ^2 v\rho _\eta \Vert _{{{\mathbb {L}}}^p_T}^{1/2}\Vert v\rho _{\eta -2\delta _1}\Vert ^{1/2}_{{{\mathbb {L}}}^p_T}+\Vert v\rho _{\eta +1}\Vert _{{{\mathbb {L}}}^p_T} \\ {}&\lesssim \varepsilon \Vert \nabla ^2 v\rho _\eta \Vert _{{{\mathbb {L}}}^p_T}+1. \end{aligned}$$

Moreover, noting that

$$\begin{aligned} |\Gamma _\rho |&\lesssim |\nabla v||\nabla \rho _\eta |+|v||\nabla ^2\rho _\eta |\lesssim \rho _{\eta }|\nabla v|+\rho _{\eta }|v|, \end{aligned}$$

we have by (4.13) and (4.10)

$$\begin{aligned} \Vert \Gamma _\rho \Vert _{{{\mathbb {L}}}^p_T}\lesssim \Vert \nabla v\rho _\eta \Vert _{{{\mathbb {L}}}^p_T}+\Vert v\rho _\eta \Vert _{{{\mathbb {L}}}^p_T}\lesssim \Vert \nabla ^2v\rho _\eta \Vert _{{{\mathbb {L}}}^p_T}^{1/2}+1. \end{aligned}$$

Combining the above calculations, by Young’s inequality, we get

$$\begin{aligned} \Vert \partial _t (v\rho _\eta )\Vert _{{{\mathbb {L}}}^{p}_T}+\Vert v\rho _\eta \Vert _{{{\mathbb {H}}}^{2,p}_T}\lesssim 1. \end{aligned}$$

The result now follows. $\square$

4.3 Critical one dimensional case

In this section we consider the critical one dimensional case and prove the following a priori estimate.

Theorem 4.7

Let $T>0$ and $\alpha ,\delta _1\in (0,1], \delta \in (0,2)$. Suppose ($\mathbf{H} ^\alpha _1$), $B\in {{\mathbb {L}}}_T^\infty (\rho _{\delta _1})$ and ($\mathbf{H} ^{\delta ,\beta }_{\mathrm{crit}}$) . For any $\eta >2\big (\frac{(1+2\beta )\delta }{\beta }\vee (\delta _1+\delta )\vee \frac{\delta \vee (2\delta -1)}{2-\zeta }\big )$ and $v_0\in {{\mathscr {C}}}^2(\rho _{\delta })$, there is a $p_0$ large enough so that for all $p>p_0$ and any strong solution v of HJB (4.1),

$$\begin{aligned} \Vert \partial _t (v\rho _\eta )\Vert _{{{\mathbb {L}}}^{p}_T}+\Vert v\rho _\eta \Vert _{{{\mathbb {H}}}^{2,p}_T}\leqslant C, \end{aligned}$$

where $C=C(\Theta ,\eta ,p,\delta _1,\Vert B\Vert _{{{\mathbb {L}}}^\infty _T(\rho _{\delta _1})}, \Vert v_0\Vert _{{{\mathscr {C}}}^2(\rho _{\delta })})$.

The key point of the proof of this theorem is that we can use the Hölder regularity of H in x and integration by parts to treat the quadratic growth of H in Q for the equation obtained in divergence form (see (4.20) below) by taking partial derivatives on both sides of the Eq. (4.1).

Lemma 4.8

Under the assumptions of Theorem 4.7, for any $\eta >\frac{(1+2\beta )\delta }{\beta }\vee (\delta _1+\delta )\vee \frac{\delta \vee (2\delta -1)}{2-\zeta }$, there is a $p_0$ large enough so that for all $p>p_0$ and any strong solution v of HJB (4.1),

$$\begin{aligned} \Vert \partial _x v\rho _\eta \Vert _{L^\infty _TL^p}+\int ^T_0\!\!\!\int |\partial ^2_x v|^2|\partial _x v|^{p-2}\rho ^p_{\eta }\leqslant C. \end{aligned}$$

(4.18)

Proof

Let $p\geqslant 2$ be fixed, whose value will be determined below. Define

$$\begin{aligned} w(t,x):=\partial _x v(t,x),\ \ {{\mathbb {A}}}^w_p:=\int |\partial _x w|^2|w|^{p-2}\rho ^p_{\eta }. \end{aligned}$$

For given $q\in [\frac{p}{2}+1,p+2]$ and $\gamma \in {{\mathbb {R}}}$, by (4.14) and (4.10) and $|\nabla \rho _\delta |\lesssim \rho _{\delta +1}$ we have

$$\begin{aligned} \left( \int |w|^q\rho _{p\eta +\gamma }\right) ^{1/q}&\lesssim \left( \int |\partial _x w|^2|w|^{p-2}\rho _{p\eta }\right) ^{\frac{1}{p+2}} \Vert v\rho _{\delta _2}\Vert ^{\frac{2}{p+2}}_{L^r}+\Vert v\rho _{\frac{p\eta +\gamma }{q}+1}\Vert _{L^q}\\&\lesssim \left( {{\mathbb {A}}}^w_p\right) ^{\frac{1}{p+2}}\Vert \rho _{\delta _2-\delta }\Vert ^{\frac{2}{p+2}}_{L^r} +\Vert \rho _{\frac{p\eta +\gamma }{q}+1-\delta }\Vert _{L^q}, \end{aligned}$$

where

$$\begin{aligned} \delta _2:=\tfrac{(p+2-q)p\eta }{2q}+\tfrac{(p+2)\gamma }{2q},\ \ r:=\tfrac{2q}{p+2-q}\in [2,\infty ]. \end{aligned}$$

Recalling $\rho _\delta (x)=\langle x\rangle ^{-\delta }$ and $d=1$, we have for $q=p+2$ and $\gamma =2\delta$, or $q\in [\frac{p}{2}+1,p+2)$ and $\gamma >\frac{2q\delta }{p+2}+(1-p\eta )(1-\frac{q}{p+2})=:\gamma _0$,

$$\begin{aligned} \Vert \rho _{\delta _2-\delta }\Vert _{L^r}+\Vert \rho _{\frac{p\eta +\gamma }{q}+1-\delta }\Vert _{L^q}<\infty . \end{aligned}$$

Thus we always have

$$\begin{aligned} \int |w|^q\rho _{p\eta +\gamma } \lesssim \left\{ \begin{aligned}&{{\mathbb {A}}}^w_p+1,\qquad \qquad q=p+2,\gamma =2\delta ,\\&({{\mathbb {A}}}^w_p)^{\frac{q}{p+2}}+1,\quad q\in [\tfrac{p}{2}+1,p+2), \gamma >\gamma _0.&\end{aligned} \right. \end{aligned}$$

(4.19)

Now by (4.1), one sees that

$$\begin{aligned} \partial _t w=\partial _x\left( a\cdot \partial _xw\right) +\partial _x(Bw)+\partial _xH(v,w). \end{aligned}$$

(4.20)

Since $\eta >\big (\frac{1+2\beta }{\beta }\big )\delta \vee (\delta _1+\delta )\vee \frac{\delta \vee (2\delta -1)}{2-\zeta }$, we can choose p large enough such that

$$\begin{aligned} \begin{aligned} \eta >&\left( \Big [2\tfrac{p+1}{p}+\tfrac{p+2}{\beta p}\Big ]\delta +\tfrac{1}{p}\right) \vee \left( (1+\tfrac{2}{p})\delta _1+\tfrac{1}{p}+\delta \right) \\ {}&{\vee \left( \tfrac{p+2\zeta -2}{(2-\zeta )p}\delta +\tfrac{1}{p}\right) \vee \left( \tfrac{p+\zeta }{(2-\zeta )p}(2\delta -1)\right) }.\end{aligned} \end{aligned}$$

(4.21)

Multiplying both sides of (4.20) by $w|w|^{p-2}\rho _{p\eta }$ and integrating on ${{\mathbb {R}}}$, we obtain

$$\begin{aligned} \frac{1}{p}\partial _t\int |w\rho _\eta |^p&=-\int a\partial _x w\partial _x(w|w|^{p-2}\rho _{p\eta })-\int Bw\partial _x(w|w|^{p-2}\rho _{p\eta })\\&\quad -\int H_s(v,w)\partial _x(w|w|^{p-2}\rho _{p\eta })-\int H_c(v,w)\partial _x(w|w|^{p-2}\rho _{p\eta })\\&=:I_1+I_2+I_3+I_4. \end{aligned}$$

For $I_1$, since $a\geqslant c_0$ and $\eta >\frac{1}{p}+\delta$, by (4.19) with $q=p$ and $\gamma =0$, we have

$$\begin{aligned} I_1&\leqslant -c_0\int |\partial _x w|^2|w|^{p-2}\rho _{p\eta }+C\int |\partial _xw||w|^{p-1}\rho _{p\eta }\\&\leqslant -\frac{c_0}{2}{{\mathbb {A}}}^w_p+C\int |w|^p\rho _{p\eta }\leqslant -\frac{c_0}{4}{{\mathbb {A}}}^w_p+C. \end{aligned}$$

For $I_2$, since $|B|\leqslant \Vert B\Vert _{{{\mathbb {L}}}^\infty _T(\rho _{\delta _1})}\rho ^{-1}_{\delta _1}$ and $\eta >(1+\frac{2}{p})\delta _1+\tfrac{1}{p}+\delta$, by (4.19) with $q=p$ and $\gamma =-2\delta _1$, we have

$$\begin{aligned} I_2&\lesssim \int |\partial _x w||w|^{p-1}\rho _{p\eta -\delta _1}+\int |w|^p\rho _{p\eta +1-\delta _1}\\&\lesssim \left( {{\mathbb {A}}}^w_p\right) ^{1/2}\left( \int |w|^p\rho _{p\eta -2\delta _1}\right) ^{1/2}+\int |w|^p\rho _{p\eta }\\&\lesssim \left( {{\mathbb {A}}}^w_p\right) ^{(p+1)/(p+2)}+1. \end{aligned}$$

For $I_3$ since $\eta > \left( \tfrac{p+2\zeta -2}{(2-\zeta )p}\delta +\tfrac{1}{p}\right) \vee \left( \tfrac{p+\zeta }{(2-\zeta )p}(2\delta -1)\right)$, by (4.19) with $q=p-2+2\zeta$, $\gamma =0$ and $q=p+\zeta$, $\gamma =1$

$$\begin{aligned} I_3\lesssim & {} \int \rho _{-\delta } |\partial _x (w|w|^{p-2}\rho _{p\eta })|+\int |w|^{\zeta } |\partial _x (w|w|^{p-2}\rho _{p\eta })| \\\lesssim & {} \left( {{\mathbb {A}}}^w_p\right) ^{1/2}\left( \int |w|^{p-2}\rho _{p\eta -2\delta }\right) ^{1/2}+\int |w|^{p-1}\rho _{p\eta +1-\delta } \\&+\left( {{\mathbb {A}}}^w_p\right) ^{1/2}\left( \int |w|^{p-2+2\zeta }\rho _{p\eta }\right) ^{1/2}+\int |w|^{p+\zeta }\rho _{p\eta +1}\\\lesssim & {} \left( {{\mathbb {A}}}^w_p\right) ^{(p+\zeta )/(p+2)}+1. \end{aligned}$$

Now we treat the most difficult term $I_4$. The key idea is to use regularity of H w.r.t the spatial variable and integration by parts. To balance the weight, we use a convolution approximation. Let $\phi _\varepsilon (y)=\varepsilon ^{-1}\phi (y/\varepsilon )$, where $\phi \in C^\infty _c((-1,1))$ is a smooth density function. Define for given $t\in [0,T]$ and $v,Q\in {{\mathbb {R}}}$,

$$\begin{aligned} H_\varepsilon (t,x,v,Q)=\int H_c(t,y,v,Q)\phi _{\varepsilon \rho _{\delta /\beta }(x)}(x-y){\mathord {\mathrm{d}}}y. \end{aligned}$$

(4.22)

We make the following decomposition for $I_4$:

$$\begin{aligned} I_4&=\int (H_\varepsilon (v,w)-H_c(v,w))\partial _x(w|w|^{p-2}\rho _{p\eta }) \\ {}&\quad -(p-1)\int H_\varepsilon (v,w)\partial _xw|w|^{p-2}\rho _{p\eta } \\ {}&\quad -\int H_\varepsilon (v,w)w |w|^{p-2}\partial _x\rho _{p\eta } \\ {}&:=I_{41}-I_{42}-I_{43}. \end{aligned}$$

For $I_{41}$, noting that by (4.6), (4.22) and (4.10),

$$\begin{aligned}&|H_\varepsilon (x,v,w)-H_c(x,v,w)|\leqslant \int |H(y,v,w)-H_c(x,v,w)|\phi _{\varepsilon \rho _{\delta /\beta }(x)}(x-y){\mathord {\mathrm{d}}}y \\ {}&\qquad \qquad \lesssim \varepsilon ^\beta \rho _{\delta }(x)\int (\langle x\rangle ^\delta +\langle y\rangle ^\delta +|v|^2+|w|^2)\phi _{\varepsilon \rho _{\delta /\beta }(x)}(x-y){\mathord {\mathrm{d}}}y \\ {}&\qquad \qquad \lesssim \varepsilon ^\beta \rho _\delta (x)\Big (\langle x\rangle ^\delta +\langle x\rangle ^{2\delta }+|w|^2\Big ) \lesssim \rho ^{-1}_{\delta }(x)+\varepsilon ^\beta \rho _{\delta }(x)|w|^2, \end{aligned}$$

we have

$$\begin{aligned} I_{41}&\lesssim \int \rho ^{-1}_{\delta }|\partial _x(w|w|^{p-2}\rho _{p\eta })| +\varepsilon ^\beta \int \rho _{\delta }w^2 |\partial _x(w|w|^{p-2}\rho _{p\eta })|=:I_{311}+I_{312}. \end{aligned}$$

For $I_{411}$, noting that by the chain rule and $|\nabla \rho _{p\eta }|\lesssim \rho _{p\eta +1}$,

$$\begin{aligned} |\partial _x(w|w|^{p-2}\rho _{p\eta })|\lesssim |w|^{p-2}|\partial _x w|\rho _{p\eta }+|w|^{p-1}\rho _{p\eta +1}, \end{aligned}$$

(4.23)

since $\eta >\frac{1}{p}+\delta$, we have by (4.19) and Hölder’s inequality,

$$\begin{aligned} I_{411}&\lesssim \int |w|^{p-2}|\partial _xw|\rho _{p\eta -\delta }+\int |w|^{p-1}\rho _{p\eta +1-\delta } \\ {}&\lesssim \left( {{\mathbb {A}}}^w_p\right) ^{1/2}\left( \int |w|^{p-2}\rho _{p\eta -2\delta }\right) ^{1/2}+\int |w|^{p-1}\rho _{p\eta +1-\delta } \\ {}&\lesssim \left( {{\mathbb {A}}}^w_p\right) ^{p/(p+2)}+1. \end{aligned}$$

For $I_{412}$, due to $\eta >\frac{1}{p}+\delta$, by (4.23), (4.19), Hölder’s inequality and Young’s inequality, we have

$$\begin{aligned} I_{412}&\lesssim \varepsilon ^\beta \int |w|^{p}|\partial _xw|\rho _{p\eta +\delta }+\int \varepsilon ^\beta |w|^{p+1}\rho _{p\eta +1+\delta } \\ {}&\lesssim \varepsilon ^\beta \int (|w|^{p-2}|\partial _xw|^2\rho _{p\eta }+|w|^{p+2}\rho _{p\eta +2\delta })+\int |w|^{p+1}\rho _{p\eta +1+\delta } \\ {}&\lesssim \varepsilon ^\beta {{\mathbb {A}}}^w_p+\left( {{\mathbb {A}}}^w_p\right) ^{(p+1)/(p+2)}+1. \end{aligned}$$

For $I_{42}$, noting that by the chain rule,

$$\begin{aligned}&H_\varepsilon (v,w)\partial _x w|w|^{p-2}=\partial _x\Big (\int _0^{w}H_\varepsilon (v,r)|r|^{p-2}{\mathord {\mathrm{d}}}r\Big ) \\ {}&\qquad -\int _0^w(\partial _xH_\varepsilon (v,r)+\partial _vH_\varepsilon (v,r)w)|r|^{p-2}{\mathord {\mathrm{d}}}r, \end{aligned}$$

by the integration by parts, we have

$$\begin{aligned} I_{42}&\lesssim \int \left( \int _0^{w}|H_\varepsilon (v,r)| |r|^{p-2}{\mathord {\mathrm{d}}}r\right) |\partial _x\rho _{p\eta }| \\ {}&\quad +\int \left( \int _0^w |\partial _xH_\varepsilon (v,r)| |r|^{p-2}{\mathord {\mathrm{d}}}r\right) \rho _{p\eta } \\ {}&\quad +\int \left( \int _0^w|\partial _vH_\varepsilon (v,r)w| |r|^{p-2}{\mathord {\mathrm{d}}}r\right) \rho _{p\eta } \\ {}&=:I_{421}+I_{422}+I_{423}. \end{aligned}$$

For $I_{421}$, by (4.5) and (4.19) we have

$$\begin{aligned} I_{421}&\lesssim \int \left( \int _0^{w}(\rho ^{-1}_{\delta }+|r|^2) |r|^{p-2}{\mathord {\mathrm{d}}}r\right) \rho _{p\eta +1} \\ {}&\lesssim \int (\rho ^{-1}_{\delta }|w|^{p-1}+|w|^{p+1})\rho _{p\eta +1} \\ {}&\lesssim ({{\mathbb {A}}}_p^w)^{\frac{p+1}{p+2}}+1. \end{aligned}$$

For $I_{422}$, noting that

$$\begin{aligned} |\partial _xH_\varepsilon (x,v,w)|\lesssim \varepsilon ^{-1}\rho ^{-1}_{\delta /\beta }(x)(\langle x\rangle ^\delta +w^{2}), \end{aligned}$$

and

$$\begin{aligned} \eta >\Big [2\tfrac{p+1}{p}+\tfrac{p+2}{\beta p}\Big ]\delta +\tfrac{1}{p}, \end{aligned}$$

by (4.19) with $q=p+1$, $\gamma =-\delta /\beta$ and $q=p-1$, $\gamma =-\delta -\delta /\beta$, we have

$$\begin{aligned} I_{422}&\lesssim \varepsilon ^{-1}\int (\rho ^{-1}_{\delta +\delta /\beta }|w|^{p-1}+\rho ^{-1}_{\delta /\beta }|w|^{p+1})\rho _{p\eta }\lesssim ({{\mathbb {A}}}_p^w)^{\frac{p+1}{p+2}}+1. \end{aligned}$$

For $I_{423}$, by (4.5), (4.10) and (4.19) with $q=p$, $\gamma =-2\delta$, we have

$$\begin{aligned} I_{423}&\lesssim \int (|w|^{p+1}+\rho ^{-2}_{\delta }|w|^{p})\rho _{p\eta }\lesssim 1+({{\mathbb {A}}}_p^w)^{\frac{p+1}{p+2}}. \end{aligned}$$

Finally, for $I_{43}$, by (4.5) and (4.19), we similarly have

$$\begin{aligned} I_{43}\lesssim \int (|w|^{p-1}\rho ^{-1}_{\delta }+|w|^{p+1})\rho _{p\eta +1}\lesssim ({{\mathbb {A}}}_p^w)^{\frac{p+1}{p+2}}+1. \end{aligned}$$

Combining the above calculations, choosing $\varepsilon$ small enough and by Young’s inequality, we obtain

$$\begin{aligned} \frac{1}{2}\partial _t\Vert w\rho _\eta \Vert ^p_{L^p}\lesssim -\frac{c_0}{8}{{\mathbb {A}}}^w_p+1. \end{aligned}$$

Integrating both sides from 0 to T, we obtain the desired estimate. $\square$

Now we can give the proof of Theorem 4.7.

Proof of Theorem 4.7

We follow the proof of Theorem 4.6. Fix $p>1/(\eta -\delta )$. By the $L^p$-theory of PDEs (cf. [38]), we have

$$\begin{aligned} \Vert \partial _t(v\rho _\eta )\Vert _{{{\mathbb {L}}}^{p}_T}+\Vert v\rho _\eta \Vert _{{{\mathbb {H}}}^{2,p}_T}\lesssim _C \Vert H(v,\nabla v)\rho _\eta +(B\cdot \nabla v)\rho _\eta -\Gamma _\rho \Vert _{{{\mathbb {L}}}^p_T}+\Vert v_0\rho _\eta \Vert _{H^{2,p}}, \end{aligned}$$

with $\Gamma _\rho$ defined in the proof of Theorem 4.6. Since $p>1/(\eta -\delta )$, by $|H(v,Q)|\lesssim \langle x\rangle ^\delta +|Q|^2$, we have

$$\begin{aligned} \Vert H(v,\nabla v)\rho _{\eta }\Vert _{{{\mathbb {L}}}^p_T}\lesssim \Vert \rho _{\eta -\delta }\Vert _{L^p}+\Vert |\nabla v|^2\rho _\eta \Vert _{{{\mathbb {L}}}^p_T} \lesssim 1+\Vert \nabla v \rho _{\eta /2}\Vert _{{{\mathbb {L}}}^{2p}_T}^2. \end{aligned}$$

We have by Hölder’s inequality and Sobolev’s embedding,

$$\begin{aligned} \Vert \nabla v \rho _{\eta /2}\Vert _{{{\mathbb {L}}}^{2p}_T}\leqslant&\Vert \nabla v\rho _\eta \Vert _{{{\mathbb {L}}}^\infty _T}^{\theta }\Vert \nabla v\rho _{\eta _0}\Vert _{L^\infty _TL^r}^{1-\theta } \\&\lesssim \Vert \nabla (\nabla v\rho _\eta )\Vert _{{{\mathbb {L}}}^p_T}^{\theta }\Vert \nabla v\rho _{\eta _0}\Vert _{L^\infty _TL^r}^{1-\theta } +\Vert \nabla v\rho _\eta \Vert _{{{\mathbb {L}}}^p_T}^{\theta }\Vert \nabla v\rho _{\eta _0}\Vert _{L^\infty _TL^r}^{1-\theta }, \end{aligned}$$

where $\theta \in (0,1/2)$ and

$$\begin{aligned} r=2p(1-\theta ),\ \ \eta _0=\tfrac{1-2\theta }{2(1-\theta )}\eta . \end{aligned}$$

Let $p_0$ be as in Lemma 4.8. Since $\eta >2\Big (\frac{1+2\beta }{\beta }\delta \vee (\delta _1+\delta )\vee \frac{\delta \vee (2\delta -1)}{2-\zeta }\Big )$, one can choose $\theta$ close to zero and p large enough so that

$$\begin{aligned} \eta _0=\tfrac{1-2\theta }{2(1-\theta )}\eta >\tfrac{1+2\beta }{\beta }\delta \vee (\delta _1+\delta ),\ \ r,p\geqslant p_0. \end{aligned}$$

Thus by (4.18), we obtain

$$\begin{aligned} \Vert \nabla v\rho _{\eta _0}\Vert _{L^\infty _TL^r}+\Vert \nabla v\rho _{\eta }\Vert _{{{\mathbb {L}}}^p_T}\leqslant C, \end{aligned}$$

and therefore,

$$\begin{aligned} \Vert H(v,\nabla v)\rho _\eta \Vert _{{{\mathbb {L}}}^p_T}\leqslant \varepsilon \Vert \nabla ^2(v\rho _\eta )\Vert _{{{\mathbb {L}}}^p_T}+C. \end{aligned}$$

Moreover, as in the proof of Theorem 4.6, one has

$$\begin{aligned} \Vert (B\cdot \nabla v)\rho _\eta -\Gamma _\rho \Vert _{{{\mathbb {L}}}^p_T}\leqslant C. \end{aligned}$$

Thus we obtain the desired estimate as in the proof of Theorem 4.6. $\square$

4.4 Proof of Theorem 4.2

The existence proof follows by the previous a priori estimates and standard compact method. For the uniqueness part we use a probabilistic method.

(Existence). Let $T>0$. For fixed $m\in {{\mathbb {N}}}$, let $\chi ^m_n(x):=\chi ^m(x/n), n\in {{\mathbb {N}}}$ be the cutoff function in ${{\mathbb {R}}}^m$, and $\varrho ^m_n(x):=n^m\varrho ^m(nx), n\in {{\mathbb {N}}}$ be the mollifiers in ${{\mathbb {R}}}^m$, where $\chi ^m\in C^\infty _c({{\mathbb {R}}}^m)$ with $\chi ^m=1$ for $|x|\leqslant 1$ and $\chi ^m=0$ for $|x|>2$, and $\varrho ^m\in C^\infty _c({{\mathbb {R}}}^m)$ is a density function. Define

$$\begin{aligned} B_n(t,x):=B(t,x){{\mathbf {1}}}_{|x|\leqslant n},\ \ \varphi _n(x):=v_0(x)\chi ^d_n(x). \end{aligned}$$

For nonlinear term H, we construct the approximation $H_n$ as follows:

$$\begin{aligned} H_n(t,x,v,Q):=((H(t,x,\cdot ,\cdot )\chi ^{d+1}_n)*\varrho ^{d+1}_n)(v,Q)\chi _n^d(x). \end{aligned}$$

(4.24)

We consider the following approximation equation:

$$\begin{aligned} \partial _t v_n=\mathrm {tr}(a\cdot \nabla ^2 v_n)+B_n\cdot \nabla v_n+H_n(v_n,\nabla v_n),\ \ v_n(0)=\varphi _n. \end{aligned}$$

(4.25)

Note that by the assumptions of Theorem 4.2,

$$\begin{aligned} B_n\in \cap _{p\in [1,\infty ]}{{\mathbb {L}}}^p_T,\ \ \varphi _n\in \cap _{p\in [1,\infty ]}H^{2,p}, \end{aligned}$$

and

$$\begin{aligned} \Vert H_n\Vert _{{{\mathbb {L}}}_T^\infty }+\Vert \partial _v H_n\Vert _{{{\mathbb {L}}}_T^\infty }+\Vert \partial _Q H_n\Vert _{{{\mathbb {L}}}_T^\infty }<\infty . \end{aligned}$$

It is well known that the approximation Eq. (4.25) admits a unique strong solution $v_n\in \cap _{p\geqslant 2} {{\mathbb {H}}}^{2,p}_T$ (cf. [38]). Moreover, by definition, we have the following uniform estimates:

$$\begin{aligned} \Vert B_n\rho _{\delta _1}\Vert _{{{\mathbb {L}}}_T^\infty }\leqslant \Vert B\rho _{\delta _1}\Vert _{{{\mathbb {L}}}_T^\infty }, \end{aligned}$$

and for some C independent of n, in the subcritical case,

$$\begin{aligned} |H_n(v,Q)|\lesssim _C \langle x\rangle ^\delta +|Q|^\zeta , \end{aligned}$$

and in the critical case $d=1$,

$$\begin{aligned}&|H_n(t,x,v,Q)|\lesssim _C \langle x\rangle ^\delta +|Q|^2,\ \ |\partial _vH_n(t,x,v,Q)|\lesssim _C\langle x\rangle ^\delta +|v|^2+|Q|, \\&|H_n(t,x,v,Q)-H_n(t,y,v,Q)|\lesssim _C |x-y|^\beta (\langle x\rangle ^\delta +\langle y\rangle ^\delta +|v|^2+|Q|^2), \end{aligned}$$

Thus by Theorems 4.4, 4.6 and 4.7, we have the following uniform estimates: for $\eta$ being as in (4.7) and p large enough,

$$\begin{aligned} \Vert v_n\rho _{\delta }\Vert _{{{\mathbb {L}}}^\infty _T}+\Vert \partial _t(v_n\rho _\eta )\Vert _{{{\mathbb {L}}}^{p}_T}+\Vert v_n\rho _\eta \Vert _{{{\mathbb {H}}}^{2,p}_T}\leqslant C, \end{aligned}$$

where C is independent of n. By Sobolev’s embedding (cf. [12, Lemma 2.3]), for any $\beta '\in (0,2-\frac{2}{p})$ and $\gamma =1-\frac{\beta '}{2}-\frac{1}{p}$,

$$\begin{aligned} \Vert v_n\rho _\eta \Vert _{C^\gamma _T{{\mathbf {C}}}^{\beta '-d/p}}&\lesssim \Vert v_n\rho _\eta \Vert _{C^\gamma _TH^{\beta ',p}}\\&\lesssim \Vert \partial _t(v_n\rho _\eta )\Vert _{{{\mathbb {L}}}^{p}_T}+\Vert v_n\rho _\eta \Vert _{{{\mathbb {H}}}^{2,p}_T}+\Vert v_0\rho _\eta \Vert _{H^{\beta ',p}}\leqslant C. \end{aligned}$$

Thus by Ascolli-Arzela’s lemma, there are subsequence $n_k$ and $v\in {{\mathbb {L}}}^\infty _T(\rho _{\delta })\cap {{\mathbb {H}}}^{2,p}_T(\rho _\eta )$ such that for all t, x,

$$\begin{aligned} \nabla ^j v_{n_k}(t,x)\rightarrow \nabla ^jv(t,x),\ \ j=0,1, \end{aligned}$$

(4.26)

and for any $R>0$,

$$\begin{aligned} \nabla ^2v_n\rightarrow \nabla ^2v\,\text { weakly in }\,L^2([0,T]\times B_R). \end{aligned}$$

(4.27)

By taking limits for (4.25), one finds that v is a strong solution to (4.1) in the sense of Definition 4.1. Indeed, for any $\psi \in C^\infty _c({{\mathbb {R}}}^d)$, by (4.27) we have

$$\begin{aligned} \lim _{n\rightarrow \infty }\int ^t_0 \langle \mathrm {tr}(a\cdot \nabla ^2 v_n),\psi \rangle {\mathord {\mathrm{d}}}s=\int ^t_0 \langle \mathrm {tr}(a\cdot \nabla ^2 v),\psi \rangle {\mathord {\mathrm{d}}}s \end{aligned}$$

and by (4.26) and the dominated convergence theorem,

$$\begin{aligned} \lim _{n\rightarrow \infty }\int ^t_0 \langle B_n\cdot \nabla v_n,\psi \rangle {\mathord {\mathrm{d}}}s=\int ^t_0 \langle B\cdot \nabla v,\psi \rangle {\mathord {\mathrm{d}}}s \end{aligned}$$

Moreover, since for each $(t,x)\in [0,T]\times {{\mathbb {R}}}^d$ and $R>0$,

$$\begin{aligned} \lim _{n\rightarrow \infty }\sup _{|(v,Q)|\leqslant R}|H_n(t,x,v,Q)-H(t,x,v,Q)|=0, \end{aligned}$$

by (4.26) and the dominated convergence theorem, we also have

$$\begin{aligned} \lim _{n\rightarrow \infty }\int ^t_0 \langle H_n(s,\cdot ,v_n,\nabla v_n),\psi \rangle {\mathord {\mathrm{d}}}s=\int ^t_0 \langle H(s,\cdot ,v,\nabla v),\psi \rangle {\mathord {\mathrm{d}}}s. \end{aligned}$$

Thus we obtain the existence of a strong solution.

(Uniqueness). We prove the uniqueness on the time interval [0, 1] by a probabilisitic method. Let $v_1, v_2$ be two strong solutions of HJB Eq. (4.1) with the same initial value $v_0$. By (4.8), we have

$$\begin{aligned} v_1, v_2\in {{\mathbb {L}}}^\infty _1(\rho _{\delta })\cap L^\infty _1{{\mathscr {C}}}^1(\rho _\eta ). \end{aligned}$$

(4.28)

Let $V:=v_1-v_2$. Then V is a strong solution of the following linear PDE:

$$\begin{aligned} \partial _t V=\mathrm {tr}(a\cdot \nabla ^2 V)+B\cdot \nabla V+G\cdot \nabla V+K\cdot V,\ V(0)=0, \end{aligned}$$

where

$$\begin{aligned} G:=\int ^1_0\partial _Q H(v_1, \nabla v_1+\theta \nabla (v_2-v_1)){\mathord {\mathrm{d}}}\theta , \end{aligned}$$

and

$$\begin{aligned} K:=\int ^1_0\partial _v H(v_1+\theta (v_2-v_1), \nabla v_2){\mathord {\mathrm{d}}}\theta . \end{aligned}$$

By (4.28) and (4.9), there is a constant $C_0>0$ such that for all $(t,x)\in [0,1]\times {{\mathbb {R}}}^d$,

$$\begin{aligned} |G(t,x)|\lesssim _{C_0}\langle x\rangle , \ \ |K(t,x)|\lesssim _{C_0}\langle x\rangle ^2. \end{aligned}$$

(4.29)

Let $T\in (0,1]$ be fixed and determined below. For a space-time function F, let

$$\begin{aligned} F^T(t,x):=F(T-t,x). \end{aligned}$$

Thus under ($\mathbf{H} ^\alpha _1$) and $B\in {{\mathbb {L}}}^\infty _1(\rho _{\delta _1})$, for each $(t,x)\in [0,T]\times {{\mathbb {R}}}^d$, the following SDE admits a unique weak solution starting from x at time t (see [37]):

$$\begin{aligned} X^{t,x}_s=x+\int ^s_t\sqrt{2a^T}(r,X^{t,x}_r){\mathord {\mathrm{d}}}W_r+\int ^s_t(B^T+G^T)(r,X^{t,x}_r){\mathord {\mathrm{d}}}r,\ \ \forall s\in [t,T]. \end{aligned}$$

As in the proof of Theorem 4.4, by Itô’s formula, we have

$$\begin{aligned} \mathrm {e}^{\int ^{t'}_{t} K^T(s, X^{t,x}_s){\mathord {\mathrm{d}}}s} V^T(t',X^{t,x}_{t'})=V^T(t,x)+M_{t'},\ t'\in [t,T], \end{aligned}$$

where $M_{t'}$ is a continuous local martingale. Note that by (4.29) and [54, Lemma 2.2], for $T=T(C_0,d,c_0,\Vert B\Vert _{{{\mathbb {L}}}^\infty _1(\rho _{\delta _1})})$ small enough,

$$\begin{aligned} {{\mathbb {E}}}\mathrm {e}^{2\int ^T_{t} K^T(s, X^{t,x}_s){\mathord {\mathrm{d}}}s}\leqslant {{\mathbb {E}}}\mathrm {e}^{2C_0\sup _{s\in [t,T]}|X^{t,x}_s|^2}<\infty . \end{aligned}$$

By using stopping time technique as in the proof of Theorem 4.4 and taking expectations, we find that for T being small enough, $0\leqslant t\leqslant T$

$$\begin{aligned} V^T(t,x)={{\mathbb {E}}}\mathrm {e}^{\int ^T_{t} K^T(s, X^{t,x}_s){\mathord {\mathrm{d}}}s} V(0,X^{t,x}_T)\equiv 0. \end{aligned}$$

Thus we obtain the uniqueness on small time interval [0, T]. We can proceed to consider [T, 2T] and so on. The proof is complete.

5 HJB equations with distribution-valued coefficients

In this section we focus on Eq. (1.5). Our strategy is summarized as follows: we first decompose Eq. (1.5) into two equations: the linear one with singular f and the nonlinear one without f. For the linear equation, we can obtain the desired estimate by Theorem 3.7. For the nonlinear equation, we introduce Zvonkin’s transformation to kill the singular part so that we can use the results in Sect. 4 to deduce a priori estimates for solutions to the nonlinear equation. Finally we employ the standard compactness argument to construct a solution to (1.5).

Now we fix $\alpha \in (\frac{1}{2},\frac{2}{3})$ and $\kappa \in (0,1)$ being small enough so that

$$\begin{aligned} {\bar{\alpha }}:=\alpha +{\widetilde{\kappa }}\in (\tfrac{1}{2},\tfrac{2}{3}),\ \ {{\widetilde{\kappa }}}:=\kappa ^{1/4},\ \ \delta :=2(\tfrac{9}{2-3\alpha }+1)\kappa <1. \end{aligned}$$

(5.1)

We consider the following singular HJB equation:

$$\begin{aligned} {{\mathscr {L}}}u=\left( \partial _t - \Delta \right) u = b \cdot \nabla u + H(u,\nabla u) + f, \quad u (0) =\varphi , \end{aligned}$$

(5.2)

where $(b,f)\in \cap _{T>0}{{\mathbb {B}}}^\alpha _T({\rho _\kappa })$ and

$$\begin{aligned} H(t,x,u,Q):{{\mathbb {R}}}^+\times {{\mathbb {R}}}^d\times {{\mathbb {R}}}\times {{\mathbb {R}}}^d\rightarrow {{\mathbb {R}}}\end{aligned}$$

satisfies ($\mathbf{H} ^{\delta ,\zeta }_{\mathrm{sub}}$) or ($\mathbf{H} ^{\delta ,\beta }_\mathrm{crit}$) with $\zeta \in [0,2)$, $\beta \in (0,1]$ and for some $C>0$,

$$\begin{aligned} |\partial _u H(t,x,u,Q)|+|\partial _Q H(t,x,u,Q)|\lesssim _C \langle x\rangle ^\delta +|u|+|Q|. \end{aligned}$$

(5.3)

To understand HJB Eq. (5.2), we use the paracontrolled calculus:

$$\begin{aligned} u=\nabla u\prec \!\!\!\prec {{\mathscr {I}}}b+{{\mathscr {I}}}f+u^\sharp +P_t\varphi , \end{aligned}$$

(5.4)

where $u^\sharp$ solves the following equation

$$\begin{aligned} \left\{ \begin{aligned} {{\mathscr {L}}}u^\sharp&=\nabla u\prec b-\nabla u\prec \!\!\!\prec b+\nabla u\succ b+b\circ \nabla u \\ {}&\quad +H(u,\nabla u)-[{{\mathscr {L}}},\nabla u\prec \!\!\!\prec ]{{\mathscr {I}}}b, \\u^\sharp (0)&=0, \end{aligned} \right. \end{aligned}$$

(5.5)

with $b\circ \nabla u$ being defined by (3.5) for $\lambda =0$.

Our aim of this section is to prove the following result.

Theorem 5.1

Let $T>0$, $\beta \in (0,1-{\bar{\alpha }}]$, $\zeta \in [0,2)$ and $\alpha ,{\bar{\alpha }},\kappa ,\delta$ be as in (5.1). Suppose that $(b,f)\in {{\mathbb {B}}}^\alpha _T({\rho _\kappa })$ and ($\mathbf{H} ^{\delta ,\zeta }_{\mathrm{sub}}$) or ($\mathbf{H} ^{\delta ,\beta }_\mathrm{crit}$) as well as (5.3) hold. Let

$$\begin{aligned} \left\{ \begin{aligned}&\eta>\tfrac{2\zeta \delta }{2-\zeta }\vee [2{{\widetilde{\kappa }}}+2\delta ],\&\mathrm{under}\, {(\mathbf{H} ^{\delta ,\zeta }_{\mathrm{sub}})};\\&\eta >2\left[ \tfrac{2(1+2\beta )\delta }{\beta }\vee ({{\widetilde{\kappa }}}+2\delta )\vee \tfrac{(2\delta )\vee (4\delta -1)}{2-\zeta }\right] ,&\mathrm{under}\, {(\mathbf{H} ^{\delta ,\beta }_{\mathrm{crit}})}. \end{aligned} \right. \end{aligned}$$

(5.6)

Fix $\varepsilon \in (0,1)$. For any initial value $\varphi \in {{\mathbf {C}}}^{1+\alpha +\varepsilon }(\rho _{\varepsilon \delta })$, there is a paracontrolled solution $(u, u^\sharp )$ solving (5.4) and (5.5) with regularity

$$\begin{aligned} u\in {\mathbb {S}}_T^{2-{\bar{\alpha }}}(\rho _\eta )\cap {{\mathbb {L}}}^\infty _T(\rho _{2\delta }), \quad u^\sharp \in {\mathbb {S}}_T^{3-2{\bar{\alpha }}}(\rho _{2\eta })\cap {{\mathbb {L}}}_T^\infty (\rho _{2\delta +\kappa }). \end{aligned}$$

Moreover, if $\eta <\frac{1-\alpha }{2}$, then the paracontrolled solution $(u,u^\sharp )$ is unique.

Remark 5.2

(i) Typical examples satisfying ($\mathbf{H} ^{\delta ,\beta }_\mathrm{crit}$) as well as (5.3) are given by

$$\begin{aligned} H(x,u)=g_1(x)|\nabla u|^2+(g_2(x)+F(u))\nabla u+g_3(u)+g_4(x), \end{aligned}$$

where $g_1\in {{\mathbf {C}}}^\beta$, $g_2\in L^\infty (\rho _{\delta _0}), \delta _0<\delta , g_4\in L^\infty (\rho _\delta )$, and $g_3, F\in {{\mathscr {C}}}^1$.

(ii) By (5.6), one sees that $\eta$ can be arbitrarily small as long as $\kappa$ is small.

To show the existence of a paracontrolled solution, we use the approximation method. More precisely, since $(b,f)\in {{\mathbb {B}}}^\alpha _T(\rho _\kappa )$, by the very definition, there is a sequence of $(b_n,f_n)\in L^\infty _T{{\mathscr {C}}}^\infty (\rho _\kappa )$ with

$$\begin{aligned} \sup _n\Big (\ell _T^{b_n}(\rho _\kappa )+{{\mathbb {A}}}^{b_n,f_n}_{T,\infty }(\rho _\kappa )\Big )\leqslant c_0, \end{aligned}$$

and such that for $\lambda \geqslant 0$,

$$\begin{aligned} \left\{ \begin{aligned}&\lim _{n\rightarrow \infty }\Big (\Vert b_n-b\Vert _{L_T^\infty {{\mathbf {C}}}^{-\alpha }(\rho _\kappa )}+\Vert f_n-f\Vert _{L_T^\infty {{\mathbf {C}}}^{-\alpha }(\rho _\kappa )}\Big )=0,\\&\lim _{n\rightarrow \infty }\Vert b_n\circ \nabla {{\mathscr {I}}}_\lambda b_n-b\circ \nabla {{\mathscr {I}}}_\lambda b\Vert _{L_T^\infty {{\mathbf {C}}}^{1-2\alpha }(\rho _\kappa )}=0,\\&\lim _{n\rightarrow \infty }\Vert b_n\circ \nabla {{\mathscr {I}}}_\lambda f_n-b\circ \nabla {{\mathscr {I}}}_\lambda f\Vert _{L_T^\infty {{\mathbf {C}}}^{1-2\alpha }(\rho _\kappa )}=0. \end{aligned} \right. \end{aligned}$$

(5.7)

Moreover, let $\varphi _n$ be the the convolution of $\varphi$ with smooth mollifier so that

$$\begin{aligned} \sup _n\Vert \varphi _n\Vert _{{{\mathbf {C}}}^{1+\alpha +\varepsilon }(\rho _{\varepsilon \delta })}\lesssim \Vert \varphi \Vert _{{{\mathbf {C}}}^{1+\alpha +\varepsilon }(\rho _{\varepsilon \delta })}. \end{aligned}$$

We consider the following approximation equation:

$$\begin{aligned} {{\mathscr {L}}}u_n = b_n \cdot \nabla u_n + H(u_n,\nabla u_n) + f_n, \quad u_n(0) = \varphi _n. \end{aligned}$$

(5.8)

By Theorem 4.2, it is well known that approximation equation (5.8) admits a unique strong solution $u_n$ with

$$\begin{aligned} \Vert u_n\Vert _{{{\mathbb {L}}}^\infty _T(\rho _{\delta })}+\Vert \partial _tu_n\Vert _{{{\mathbb {L}}}^{p}_T(\rho _\eta )}+\Vert u_n\Vert _{{{\mathbb {H}}}^{2,p}_T(\rho _\eta )}\leqslant C_n. \end{aligned}$$

Our aim is of course to establish the following uniform estimate:

$$\begin{aligned} \sup _n\Big (\Vert u_n\Vert _{{\mathbb {S}}_T^{2-{\bar{\alpha }}}(\rho _\eta )}+\Vert u_n\Vert _{{{\mathbb {L}}}^\infty _T(\rho _{2\delta })} +\Vert u^\sharp _n\Vert _{{{\mathbb {S}}}_T^{3-2{\bar{\alpha }}}(\rho _{2\eta })}+\Vert u^\sharp _n\Vert _{{{\mathbb {L}}}_T^\infty (\rho _{2\delta +\kappa })}\Big )\leqslant C, \end{aligned}$$

(5.9)

where $u^\sharp _n$ is defined by (5.4) with (b, f) being replaced by $(b_n,f_n)$.

To show the uniform estimate (5.9), our approach is to transform (5.8) into HJB equation studied in Sect. 4. In the following, for simplicity, we drop the subscript n and use the convention that all the constants appearing below only depend on the parameter set

$$\begin{aligned} \Theta :=(T, d,\alpha ,\beta ,\eta ,\zeta ,\kappa ,c_0,\varepsilon ,\Vert \varphi \Vert _{{{\mathbf {C}}}^{1+\alpha +\varepsilon }(\rho _{\varepsilon \delta })}). \end{aligned}$$

First of all, by Lemma 2.13, one can make the following decomposition for the initial value $\varphi \in {{\mathbf {C}}}^{1+\alpha +\varepsilon }(\rho _{\varepsilon \delta })$: for $\varepsilon _0\in (0,\frac{\varepsilon \alpha }{1-\varepsilon })$,

$$\begin{aligned} \varphi =\varphi _1+\varphi _2,\ \ \varphi _1\in {{\mathbf {C}}}^{1+\alpha +\varepsilon _0},\ \varphi _2\in {{\mathscr {C}}}^2(\rho _\delta ). \end{aligned}$$

Next we make the following decomposition for u:

$$\begin{aligned} u=u_1+u_2, \end{aligned}$$

where $u_1$ solves the following linear equation with non-homogeneous term f

$$\begin{aligned} {{\mathscr {L}}}u_1 = b \cdot \nabla u_1 + f, \quad u_1(0)=\varphi _1, \end{aligned}$$

(5.10)

while $u_2$ solves the following HJB equation

$$\begin{aligned} {{\mathscr {L}}}u_2 = b \cdot \nabla u_2 + H(u_1+u_2,\nabla u_1+\nabla u_2), \quad u_2(0)=\varphi _2. \end{aligned}$$

(5.11)

Clearly, the linear Eq. (5.10) can be uniquely solved by Theorem 3.7 with the solution $u_1\in {{\mathbb {S}}}_T^{2-\alpha }(\rho _\delta )$. Thus it remains to solve (5.11). However, since b is a distribution, we cannot directly apply Theorem 4.2. We use (2.23) and Zvonkin’s transformation to kill the singular part of b.

5.1 Zvonkin’s transformation for HJB equations

In this section we introduce a transformation of phase space to kill the distributional part in the drift of the HJB equation (5.11) so that we can apply the result in Sect. 4. Such a transformation was firstly used by Zvonkin in [56] to study the SDEs with singular drifts. In the literature, it is also called Zvonkin’s transformation. Below we always assume

$$\begin{aligned} b\in L^\infty _T({{\mathscr {C}}}^\infty (\rho _\kappa )),\ \ell ^b_T(\rho _\kappa )\leqslant c_0. \end{aligned}$$

(5.12)

The key step for Zvonkin’s transform is to construct a $C^1$-diffeomorphism such that the solutions to Eq. (5.11) composed with this diffeomorphism satisfy a new equation without the singular part of the drift b. However, a diffeomorphism does not allow polynomial growth for $C^1$-norm as $|x|\rightarrow \infty$. To this end, we decompose b into two parts by Lemma 2.13. By Lemma 2.13 we make the following decomposition:

$$\begin{aligned} b = b_{>} + b_{\leqslant }:={\mathscr {V}}_>b+{\mathscr {V}}_\leqslant b, \end{aligned}$$

We are goint to construct a $C^1$-differmophism to kill the $b_>$ part. Furthermore, we define

$$\begin{aligned} \bar{b}:=b_{>}\circ \nabla {{\mathscr {I}}}_\lambda b_>,\ \ \bar{b}_>&:={\mathscr {V}}_>\bar{b},\quad \bar{b}_\leqslant :={\mathscr {V}}_\leqslant \bar{b}. \end{aligned}$$

(5.13)

Lemma 5.3

For any $m\in {{\mathbb {N}}}$ and $\varepsilon >0$, it holds that

$$\begin{aligned} b_>\in L^\infty _T{{\mathscr {C}}}^m,\ \ {\bar{b}}_\leqslant \in L^\infty _T{{\mathscr {C}}}^m(\rho _{2\kappa +\varepsilon }). \end{aligned}$$

(5.14)

For some $C=C(d,\alpha ,\kappa )>0$, it holds that

$$\begin{aligned} \Vert b_{>}\Vert _{L^\infty _T {{\mathbf {C}}}^{-\alpha -{{\widetilde{\kappa }}}}}+\Vert b_{\leqslant } \Vert _{{{\mathbb {L}}}^\infty _T (\rho _{{{\widetilde{\kappa }}}})}\lesssim _C \sqrt{\ell _T^b(\rho _{\kappa })}, \end{aligned}$$

(5.15)

where ${\tilde{\kappa }}=\kappa ^{1/4}$, and

$$\begin{aligned} \Vert \bar{b}\Vert _{L^\infty _T{{\mathbf {C}}}^{1-2\alpha }(\rho _{{{\widetilde{\kappa }}}})}+ \Vert \bar{b}_>\Vert _{L^\infty _T{{\mathbf {C}}}^{1-2\alpha -{{\widetilde{\kappa }}}}} +\Vert \bar{b}_\leqslant \Vert _{{{\mathbb {L}}}^\infty _T(\rho _{{{\widetilde{\kappa }}}})}\lesssim _C \ell _T^b(\rho _\kappa ). \end{aligned}$$

(5.16)

Proof

(i) Since $b\in L^\infty _T{{\mathscr {C}}}^\infty (\rho _\kappa )$, by Lemma 2.13 one sees that (5.14) holds.

(ii) We use Lemma 2.13 with weight $\rho _{\kappa ^{1/2}}$ to conclude

$$\begin{aligned} \Vert b_{>}\Vert _{L^\infty _T {{\mathbf {C}}}^{-\alpha -{{\widetilde{\kappa }}}}} \lesssim \Vert b_{>}\Vert _{L^\infty _T {{\mathbf {C}}}^{-\alpha -\kappa ^{1/2}}} \lesssim \Vert b \Vert _{L^\infty _T{{\mathbf {C}}}^{-\alpha } (\rho _\kappa )}\leqslant \sqrt{\ell _T^b(\rho _{\kappa })}. \end{aligned}$$

Since $\alpha <1$, we can choose $\varepsilon >0$ being small enough so that

$$\begin{aligned} {\bar{\kappa }}:=\kappa +\kappa ^{1/2}(\alpha +\varepsilon )\leqslant \kappa ^{1/2}-\kappa <\tfrac{2}{3}{\widetilde{\kappa }}-\kappa . \end{aligned}$$

Noting that

$$\begin{aligned} \rho _{{\bar{\kappa }}}(x)=\langle x\rangle ^{-\kappa ^{1/2}(\kappa ^{1/2}+\alpha +\varepsilon )}=\rho ^{\kappa ^{1/2}+\alpha +\varepsilon }_{\kappa ^{1/2}}(x), \end{aligned}$$

by Lemma 2.13 again, we have

$$\begin{aligned} \Vert b_{\leqslant } \Vert _{{{\mathbb {L}}}^\infty _T (\rho _{{{\widetilde{\kappa }}}})}\leqslant \Vert b_{\leqslant } \Vert _{{{\mathbb {L}}}^\infty _T (\rho _{{\bar{\kappa }}})}&=\Vert b_{\leqslant } \Vert _{{{\mathbb {L}}}^\infty _T (\rho ^{\kappa ^{1/2}+\alpha +\varepsilon }_{\kappa ^{1/2}})} \\ {}&\lesssim \Vert b \Vert _{L^\infty _T{{\mathbf {C}}}^{-\alpha } (\rho ^{\kappa ^{1/2}}_{\kappa ^{1/2}})}=\Vert b \Vert _{L^\infty _T{{\mathbf {C}}}^{-\alpha } (\rho _\kappa )}. \end{aligned}$$

(iii) Note that by definition (5.13),

$$\begin{aligned} {\bar{b}}=b\circ \nabla {{\mathscr {I}}}_\lambda b- b\circ \nabla {{\mathscr {I}}}_\lambda (b_{\leqslant }) - b_{\leqslant }\circ \nabla {{\mathscr {I}}}_\lambda b_> \end{aligned}$$

and

$$\begin{aligned} \Vert b\circ \nabla {{\mathscr {I}}}_\lambda b\Vert _{L^\infty _T{{\mathbf {C}}}^{1-2\alpha }(\rho _{2\kappa })}\leqslant \ell _T^b(\rho _\kappa ). \end{aligned}$$

By (2.16), (2.12) and (5.15), we have for $\varepsilon \in (0,1-\alpha )$,

$$\begin{aligned} \Vert b\circ \nabla {{\mathscr {I}}}_\lambda (b_{\leqslant })\Vert _{L^\infty _T{{\mathbf {C}}}^0(\rho _{\kappa +{\bar{\kappa }}})}&\lesssim \Vert b\Vert _{L^\infty _T{{\mathbf {C}}}^{-\alpha }(\rho _\kappa )}\Vert b_{\leqslant }\Vert _{L_T^\infty {{\mathbf {C}}}^{\alpha +\varepsilon -1}(\rho _{{\bar{\kappa }}})} \lesssim \ell _T^b(\rho _\kappa ), \end{aligned}$$

and

$$\begin{aligned} \Vert b_\leqslant \circ \nabla {{\mathscr {I}}}_\lambda (b_{>}) \Vert _{L^\infty _T{{\mathbf {C}}}^{1-\alpha -{{\widetilde{\kappa }}}}(\rho _{{\bar{\kappa }}})}&\lesssim \Vert b_\leqslant \Vert _{{{\mathbb {L}}}_T^\infty (\rho _{{\bar{\kappa }}})}\Vert b_>\Vert _{L^\infty _T{{\mathbf {C}}}^{-\alpha -{{\widetilde{\kappa }}}}} \lesssim \ell _T^b(\rho _\kappa ). \end{aligned}$$

Combining the above estimate we get

$$\begin{aligned} \Vert \bar{b}\Vert _{L^\infty _T{{\mathbf {C}}}^{1-2\alpha }(\rho _{{{\widetilde{\kappa }}}})}\lesssim \Vert \bar{b}\Vert _{L^\infty _T{{\mathbf {C}}}^{1-2\alpha }(\rho _{\kappa +{\bar{\kappa }}})}\lesssim \ell _T^b(\rho _\kappa ). \end{aligned}$$

(iii) As for the other two estimates in (5.16), we use Lemma 2.13 with the weight $\rho _{{{\widetilde{\kappa }}}}$ to have

$$\begin{aligned} \Vert {\bar{b}}_>\Vert _{L_T^\infty {{\mathbf {C}}}^{1-2\alpha -{{\widetilde{\kappa }}}}}\leqslant \Vert \bar{b}_>\Vert _{L_T^\infty {{\mathbf {C}}}^{1-2\alpha -\frac{{\bar{\kappa }}+\kappa }{{{\widetilde{\kappa }}}}}}\lesssim \Vert \bar{b}\Vert _{L^\infty _T{{\mathbf {C}}}^{1-2\alpha }(\rho _{\kappa +{\bar{\kappa }}})}\lesssim \ell _T^b(\rho _\kappa ), \end{aligned}$$

and for $\varepsilon >0$ small enough

$$\begin{aligned} \Vert {\bar{b}}_\leqslant \Vert _{{{\mathbb {L}}}_T^\infty (\rho _{{{\widetilde{\kappa }}}})}\leqslant \Vert \bar{b}_\leqslant \Vert _{{{\mathbb {L}}}_T^\infty (\rho _{{{\bar{\kappa }}+\kappa +{{\widetilde{\kappa }}}(2\alpha -1+\varepsilon )}})}\lesssim \Vert \bar{b}\Vert _{L^\infty _T{{\mathbf {C}}}^{1-2\alpha }(\rho _{\kappa +{\bar{\kappa }}})}\lesssim \ell _T^b(\rho _\kappa ). \end{aligned}$$

The proof is complete. $\square$

To construct a $C^1$-diffeomorphism for killing the singular $b_>$, we consider the following vector-valued parabolic equation:

$$\begin{aligned} {{\mathscr {L}}}_\lambda {\mathbf {u}}= (b_{>} -\bar{b}_\leqslant ) \cdot (\nabla {\mathbf {u}}+{{\mathbb {I}}}),\quad {\mathbf {u}}(0)=\mathbf{0}\in {{\mathbb {R}}}^d. \end{aligned}$$

(5.17)

Remark 5.4

The reason for considering $b_{>} -\bar{b}_\leqslant$ rather than $b_>$ is the following: in order to use (5.17) to construct a $C^1$-diffeomorphism, the solution ${\mathbf {u}}$ to (5.17) must be in unweighted spaces, which requires $\ell ^{b_>}_T(1)<\infty$. However, by (5.16), ${\bar{b}}=b_{>}\circ \nabla {{\mathscr {I}}}_\lambda b_>$ stays in a weighted space. Hence, we shall use $\bar{b}_\leqslant$ to cancel the weight contained in the decomposition of renormalizing $b_>\circ \nabla {\mathbf {u}}$. It should also be also noticed that since $b_{>} -\bar{b}_\leqslant$ still stays in some weighted space, one cannot directly use Lemma 3.4 to construct a $C^1$-diffeomorphism. Fortunately, one still has the following result.

Lemma 5.5

Let $\alpha \in (\frac{1}{2},\frac{2}{3})$ and $\kappa \in (0,(\frac{2}{3}-\alpha )^4)$. Under (5.12), for ${\bar{\alpha }}=\alpha +{{\widetilde{\kappa }}}$, there exist $\lambda =\lambda (\Theta )$ large enough and a constant $C=C(\Theta )>0$ such that

$$\begin{aligned} \Vert {\mathbf {u}}\Vert _{{{\mathscr {C}}}^1}\leqslant 1/2,\ \ \Vert {\mathbf {u}}\Vert _{{\mathbb {S}}^{2-{\bar{\alpha }}}_T}\leqslant C. \end{aligned}$$

(5.18)

Proof

We use the paracontrolled ansatz as in (3.3) and write

$$\begin{aligned} {\mathbf {u}}=\nabla {\mathbf {u}}\prec \!\!\!\prec {{\mathscr {I}}}_\lambda b_>+{{\mathscr {I}}}_\lambda b_>+{\mathbf {u}}^\sharp , \end{aligned}$$

where

$$\begin{aligned} {\mathbf {u}}^\sharp :={{\mathscr {I}}}_\lambda \big (\nabla {\mathbf {u}}\prec b_>-\nabla {\mathbf {u}}\prec \!\!\!\prec b_>+\nabla {\mathbf {u}}\succ b_> +\Gamma ^b_{\mathbf {u}}-[{{\mathscr {L}}}_\lambda ,\nabla {\mathbf {u}}\prec \!\!\!\prec ]{{\mathscr {I}}}_\lambda b_>\big ) \end{aligned}$$

with

$$\begin{aligned} \Gamma ^b_{\mathbf {u}}:=b_>\circ \nabla {\mathbf {u}}-\bar{b}_\leqslant \cdot (\nabla {\mathbf {u}}+{{\mathbb {I}}}). \end{aligned}$$

Recalling that ${\bar{b}}=b_>\circ \nabla {{\mathscr {I}}}_\lambda b_>$, as in (3.5), we have

$$\begin{aligned} \Gamma ^b_{\mathbf {u}}&{=b_>\circ (\nabla ^2 {\mathbf {u}}\prec {{\mathscr {I}}}_\lambda b_>)+b_>\circ (\nabla {\mathbf {u}}\prec \nabla {{\mathscr {I}}}_\lambda b_>)+b_>\circ \nabla {{\mathscr {I}}}_\lambda b_>} \\ {}&\qquad {+\mathrm {com}_1+b_>\circ \nabla {\mathbf {u}}^\sharp -\bar{b}_\leqslant \cdot (\nabla {\mathbf {u}}+{{\mathbb {I}}})} \\ {}&=b_>\circ (\nabla ^2 {\mathbf {u}}\prec {{\mathscr {I}}}_\lambda b_>)+\mathrm {com}(\nabla {\mathbf {u}},\nabla {{\mathscr {I}}}_\lambda b_>,b_>) \\ {}&\quad +\mathrm {com}_1+b_>\circ \nabla {\mathbf {u}}^\sharp +\bar{b}_>\cdot (\nabla {\mathbf {u}}+{{\mathbb {I}}}), \end{aligned}$$

where

$$\begin{aligned} \mathrm {com}_1:=b_>\circ \nabla [\nabla {\mathbf {u}}\prec \!\!\!\prec {{\mathscr {I}}}_\lambda b_>-\nabla {\mathbf {u}}\prec {{\mathscr {I}}}_\lambda b_>]. \end{aligned}$$

Let

$$\begin{aligned} \gamma ,\beta \in ({\bar{\alpha }},2-2{\bar{\alpha }}]. \end{aligned}$$

Except for the last term $\bar{b}_>\cdot (\nabla {\mathbf {u}}+{{\mathbb {I}}})$, we estimate each term of $\Gamma ^b_{\mathbf {u}}$ as in Lemma 3.3 and obtain

$$\begin{aligned} \Vert \Gamma ^b_{\mathbf {u}}\Vert _{L_T^\infty {{\mathbf {C}}}^{1-2{\bar{\alpha }}}}&\lesssim \Vert b_>\Vert ^2_{L^\infty _T{{\mathbf {C}}}^{-{\bar{\alpha }}}}\Vert {\mathbf {u}}\Vert _{{{\mathbb {S}}}_T^{{\bar{\alpha }}+\gamma }} +\Vert b_>\Vert _{L^\infty _T{{\mathbf {C}}}^{-{\bar{\alpha }}}}\Vert \nabla {\mathbf {u}}^\sharp \Vert _{L^\infty _T{{\mathbf {C}}}^\beta } \\ {}&\quad +\Vert \bar{b}_>\cdot (\nabla {\mathbf {u}}+{{\mathbb {I}}})\Vert _{L_T^\infty {{\mathbf {C}}}^{1-2{\bar{\alpha }}}} \\ {}&\lesssim \ell _T^b(\rho _\kappa )\Big (\Vert {\mathbf {u}}\Vert _{{\mathbb {S}}_T^{{\bar{\alpha }}+\gamma }}+1\Big ) +\sqrt{\ell _T^b(\rho _\kappa )}\Vert {\mathbf {u}}^\sharp \Vert _{L_T^\infty {{\mathbf {C}}}^{\beta +1}}, \end{aligned}$$

where we have used (5.15), (5.16) and (2.17). As in Lemma 3.4, for any $\theta \in (1+\frac{3{\bar{\alpha }}}{2}, 2)$, there is a constant $C>0$ independent of $\lambda$ such that for all $\lambda \geqslant 1$,

$$\begin{aligned} \lambda ^{1-\frac{\theta }{2}}(\Vert {\mathbf {u}}\Vert _{{\mathbb {S}}_T^{\theta -{\bar{\alpha }}}}+\Vert {\mathbf {u}}^\sharp \Vert _{{\mathbb {S}}_T^{2\theta -2{\bar{\alpha }}-1}})\leqslant c\ell _T^b(\rho _\kappa )( \Vert {\mathbf {u}}\Vert _{{\mathbb {S}}_T^{\theta -{\bar{\alpha }}}}+\Vert {\mathbf {u}}^\sharp \Vert _{{\mathbb {S}}_T^{2\theta -2{\bar{\alpha }}-1}}+1). \end{aligned}$$

Taking $\lambda$ large enough, we get the first estimate in (5.18). The second estimate follows from the same argument as in Lemma 3.4. $\square$

Now, let us define

$$\begin{aligned} \Phi (t,x):=x+{\mathbf {u}}(t,x). \end{aligned}$$

By Lemma 5.5, it is easy to see that for each $t\in [0,T]$ and $x,y\in {{\mathbb {R}}}^d$,

$$\begin{aligned} \tfrac{1}{2}|x-y|\leqslant |\Phi (t,x)-\Phi (t,y)|\leqslant \tfrac{3}{2}|x-y| \end{aligned}$$

(5.19)

and

$$\begin{aligned} \partial _t\Phi =\Delta \Phi -\lambda {\mathbf {u}}+ (b_{>} -\bar{b}_\leqslant )\cdot \nabla \Phi . \end{aligned}$$

(5.20)

In particular,

$$\begin{aligned} x\mapsto \Phi (t,x)\,\text { is a }\,C^1\text {-diffeomorphism.} \end{aligned}$$

Let $\Phi ^{-1}(t,x)$ be the inverse of $x\mapsto \Phi (t,x)$ and define

$$\begin{aligned} v(t,x):=u_2(t,\Phi ^{-1}(t,x))\Rightarrow v(t,\Phi (t,x))=u_2(t,x), \end{aligned}$$

where $u_2$ solves HJB Eq. (5.11).

In the rest of this subsection, if there is no confusion, we also use $\circ$ to denote the composition of two functions. By the chain rule, we have

$$\begin{aligned} \partial _t v\circ \Phi +\partial _t\Phi \cdot (\nabla v\circ \Phi )=\partial _t u_2,\ \ \nabla u_2=\nabla \Phi \cdot (\nabla v\circ \Phi ) \end{aligned}$$

and

$$\begin{aligned} \Delta u_2=\Delta \Phi \cdot (\nabla v\circ \Phi )+\mathrm {tr}(\widetilde{a}\cdot \nabla ^2 v\circ \Phi ), \end{aligned}$$

where ${\widetilde{a}}_{ij}:=\sum _{k=1}^d(\partial _k\Phi ^i\partial _k\Phi ^j)$, which implies by (5.11) and (5.20) that

$$\begin{aligned} (\partial _t v)\circ \Phi&=\mathrm {tr}({\widetilde{a}}\cdot \nabla ^2 v\circ \Phi )+H(u_1+u_2,\nabla u_1+\nabla u_2) \\ {}&\quad +((b_\leqslant +{\bar{b}}_\leqslant )\cdot \nabla \Phi +\lambda {\mathbf {u}})\cdot (\nabla v\circ \Phi ). \end{aligned}$$

Thus we obtain the following key lemma for solving HJB equation (5.11).

Lemma 5.6

The v defined above solves the following HJB equation:

$$\begin{aligned} \begin{aligned} \partial _t v=\mathrm {tr}\left( a\cdot \nabla ^2v\right) +B\cdot \nabla v+{\widetilde{H}}(v,\nabla v),\quad v(0)=\varphi _2, \end{aligned}\end{aligned}$$

(5.21)

where $a_{ij}:=\sum _{k=1}^d(\partial _k\Phi ^i\partial _k\Phi ^j)\circ \Phi ^{-1}$ and

$$\begin{aligned} B:=((b_\leqslant +\bar{b}_\leqslant )\cdot \nabla \Phi +\lambda {\mathbf {u}})\circ \Phi ^{-1}, \end{aligned}$$

and for $(t,x,v,Q)\in [0,T]\times {{\mathbb {R}}}^d\times {{\mathbb {R}}}\times {{\mathbb {R}}}^d$,

$$\begin{aligned} {\widetilde{H}}(t,x,v,Q):=H\big (t,\cdot , u_1(t,\cdot )+v, \nabla u_1(t,\cdot )+\nabla \Phi (t,\cdot )\cdot Q\big )\circ \Phi ^{-1}(t,x). \end{aligned}$$

Moreover, a satisfies ($\mathbf{H} ^{1-{\bar{\alpha }}}_a$), $B\in {{\mathbb {L}}}^\infty _T(\rho _{{{\widetilde{\kappa }}}})$, and under ($\mathbf{H} ^{\delta ,\zeta }_{\mathrm{sub}}$) or ($\mathbf{H} ^{\delta ,\beta }_\mathrm{crit}$) for $\beta \leqslant 1-{\bar{\alpha }}$, ${\widetilde{H}}$ still satisfies ($\mathbf{H} ^{2\delta ,\zeta }_{\mathrm{sub}}$) or ($\mathbf{H} ^{2\delta ,\beta }_{\mathrm{crit}}$) .

Proof

(i) By (5.19) and (5.18), we have $\frac{1}{2}{{\mathbb {I}}}\leqslant {\widetilde{a}}\leqslant 2{{\mathbb {I}}}$ and

$$\begin{aligned} | a(t,x)- a(t,y)|&\lesssim |\nabla {\mathbf {u}}(t,\Phi ^{-1}(t,x))-\nabla {\mathbf {u}}(t,\Phi ^{-1}(t,y))| \\ {}&\lesssim |\Phi ^{-1}(t,x)-\Phi ^{-1}(t,y)|^{1-{\bar{\alpha }}}\lesssim |x-y|^{1-{\bar{\alpha }}}. \end{aligned}$$

(ii) Note that for some $C\geqslant 1$,

$$\begin{aligned} C^{-1}\langle x\rangle \leqslant \langle \Phi (t,x)\rangle \leqslant C\langle x\rangle ,\ \forall t\in [0,T]. \end{aligned}$$

(5.22)

The assertion $B\in {{\mathbb {L}}}^\infty _T(\rho _{{{\widetilde{\kappa }}}})$ follows by (5.18) and Lemma 5.3.

(iii) We only check that under ($\mathbf{H} ^{\delta ,\beta }_\mathrm{crit}$), ${\widetilde{H}}$ satisfies ($\mathbf{H} ^{2\delta ,\beta }_\mathrm{crit}$). For simplicity, we drop the time variable and we only consider $H_c$ part. By (4.5), we have

$$\begin{aligned}&|H_c\big (x, u_1(x)+v, \nabla u_1(x)+\nabla \Phi (x)\cdot Q\big )| \\&\quad \leqslant c_2\langle x\rangle ^\delta +c'_3(|Q|^2+|\nabla u_1(x)|^2) \leqslant c'_2\langle x\rangle ^{2\delta }+c'_3|Q|^2, \end{aligned}$$

where we used $u_1\in {{\mathbb {S}}}^{2-\alpha }_T(\rho _\delta )$. By (4.6) and (5.3), we have for $|x-y|\leqslant 1$, $\beta \leqslant 1-{\bar{\alpha }}$

$$\begin{aligned}&|H_c\big (x, u_1(x)+v, \nabla u_1(x)+\nabla \Phi (x)\cdot Q\big )-H_c\big (y, u_1(y)+v, \nabla u_1(y)+\nabla \Phi (y)\cdot Q\big )| \\ {}&\quad \lesssim |x-y|^\beta \Big (\langle x\rangle ^\delta +\langle y\rangle ^\delta +|u_1(x)+v|^2+|\nabla u_1(x)+\nabla \Phi (x)\cdot Q|^2\Big ) \\ {}&\qquad +|u_1(x)-u_1(y)|\Big (\langle y\rangle ^\delta +|v|+|u_1(x)|+|u_1(y)|+|\nabla u_1(x)|+|Q|\Big ) \\ {}&\quad \qquad +(|\nabla u_1(x)-\nabla u_1(y)|+|\nabla \Phi (x)-\nabla \Phi (y)||Q|) \\ {}&\quad \qquad \times (\langle y\rangle ^\delta +|u_1(y)|+|v|+|\nabla u_1(x)|+|\nabla u_1(y)|+|Q|) \\ {}&\quad \lesssim |x-y|^{\beta }(\langle x\rangle ^{2\delta }+\langle y\rangle ^{2\delta }+|v|^2+|Q|^2). \end{aligned}$$

Furthermore, we have

$$\begin{aligned}&|\partial _v H_c\big (x, u_1(x)+v, \nabla u_1(x)+\nabla \Phi (x)\cdot Q\big )| \\ {}&\quad \lesssim \langle x\rangle ^\delta + |u_1(x)|+|v|+|\nabla u_1(x)|+|Q|\lesssim \langle x\rangle ^\delta +|v|+|Q|. \end{aligned}$$

Therefore, ${\widetilde{H}}$ satisfies ($\mathbf{H} ^{2\delta ,\beta }_\mathrm{crit}$) by definition and (5.19), (5.22). $\square$

5.2 Proof of Theorem 5.1

By Lemma 5.6 and Theorem 4.2 we can derive the following a priori estimate for the solution to (5.2).

Lemma 5.7

Under (5.12), there is a constant $C=C(\Theta )>0$ such that

$$\begin{aligned} \Vert u\Vert _{{{\mathbb {L}}}^\infty _T(\rho _{2\delta })}+\Vert u\Vert _{{\mathbb {S}}_T^{2-{\bar{\alpha }}}(\rho _\eta )}\leqslant C. \end{aligned}$$

(5.23)

Proof

Recall $u=u_1+u_2$, where $u_1$ solves Eq. (5.10) and $u_2$ solves Eq. (5.11). By Theorem 3.7, one has

$$\begin{aligned} \Vert u_1\Vert _{{\mathbb {S}}_T^{2-{\bar{\alpha }}}(\rho _\delta )}\lesssim 1. \end{aligned}$$

Hence, to prove (5.23), due to $\eta \geqslant 2\delta$, it suffices to prove that

$$\begin{aligned} \Vert u_2\Vert _{{{\mathbb {L}}}^\infty _T(\rho _{2\delta })}+\Vert u_2\Vert _{{\mathbb {S}}_T^{2-{\bar{\alpha }}}(\rho _\eta )}\lesssim 1. \end{aligned}$$

(5.24)

Note that by Lemma 5.6, $v=u_2(\Phi )$ solves (5.21). In particular, by Lemma 5.6 and Theorem 4.2, for p large enough and $\eta$ satisfying (5.6),

$$\begin{aligned} \Vert v\Vert _{{{\mathbb {L}}}_T^\infty (\rho _{2\delta })}+ \Vert \partial _tv\Vert _{{{\mathbb {L}}}_T^p(\rho _\eta )}+\Vert v\Vert _{{{\mathbb {H}}}^{2,p}_T(\rho _\eta )}\lesssim 1, \end{aligned}$$

(5.25)

which implies by [12, Lemma 2.3],

$$\begin{aligned} \Vert v\Vert _{C_T^{(2-{\bar{\alpha }})/2}L^\infty (\rho _\eta )}\lesssim 1. \end{aligned}$$

(5.26)

By (5.22), we have

$$\begin{aligned} \Vert u_2\Vert _{{{\mathbb {L}}}^\infty _T(\rho _{2\delta })}=\Vert v(\Phi )\rho _{2\delta }\Vert _{{{\mathbb {L}}}^\infty _T} \asymp \Vert v(\Phi )\rho _{2\delta }(\Phi )\Vert _{{{\mathbb {L}}}^\infty _T}=\Vert v\Vert _{{{\mathbb {L}}}^\infty _T(\rho _{2\delta })}, \end{aligned}$$

and by (2.17), (5.25) and (5.18),

$$\begin{aligned}&\Vert \nabla u_2\Vert _{L^\infty _T{{\mathbf {C}}}^{1-{\bar{\alpha }}}(\rho _\eta )} =\Vert \nabla v\circ \Phi \cdot \nabla \Phi \Vert _{L^\infty _T{{\mathbf {C}}}^{1-{\bar{\alpha }}}(\rho _\eta )} \\ {}&\qquad \lesssim \Vert \nabla v(\Phi )\Vert _{L^\infty _T{{\mathbf {C}}}^{1-{\bar{\alpha }}}(\rho _\eta )} \Vert \nabla \Phi \Vert _{L^\infty _T{{\mathbf {C}}}^{1-{\bar{\alpha }}}} \\ {}&\qquad \lesssim \Vert \nabla v\Vert _{L^\infty _T{{\mathbf {C}}}^{1-{\bar{\alpha }}}(\rho _\eta )} (\Vert {\mathbf {u}}\Vert _{L^\infty _T{{\mathbf {C}}}^{2-{\bar{\alpha }}}}+1)\lesssim 1, \end{aligned}$$

where in the second inequality, we have used that for $|x-y|\leqslant 1$,

$$\begin{aligned} \rho _\eta (x)|\nabla v(\Phi (x))-\nabla v(\Phi (y))|&{\mathop {\lesssim }\limits ^{(5.22)}}\rho _\eta (\Phi (x))|\nabla v(\Phi (x))-\nabla v(\Phi (y))| \\ {}&{\mathop {\lesssim }\limits ^{(2.1), (5.19)}} |\Phi (x)-\Phi (y)|^{1-{\bar{\alpha }}}\Vert \nabla v\Vert _{L^\infty _T{{\mathbf {C}}}^{1-{\bar{\alpha }}}(\rho _\eta )}. \end{aligned}$$

Moreover, by (5.22), we also have

$$\begin{aligned} \Vert u_2(t)-u_2(s)\Vert _{L^\infty (\rho _\eta )}&\lesssim \Vert v(t,\Phi (t))-v(t,\Phi (s))\Vert _{L^\infty (\rho _\eta )}+\Vert v(t)-v(s)\Vert _{L^\infty (\rho _\eta )}\\&\leqslant \Vert \Phi (t)-\Phi (s)\Vert _{L^\infty }\int ^1_0\Vert \nabla v(t,\Gamma ^{t,s}_r)\Vert _{L^\infty (\rho _\eta )}{\mathord {\mathrm{d}}}r\\&\quad +\Vert v(t)-v(s)\Vert _{L^\infty (\rho _\eta )}, \end{aligned}$$

where $\Gamma ^{t,s}_r(x):=r\Phi (t,x)+(1-r)\Phi (s,x)$. Since for any $r\in [0,1]$ and $t,s\in [0,T]$,

$$\begin{aligned} \Gamma ^{t,s}_r(x)=x+r{\mathbf {u}}(t,x)+(1-r){\mathbf {u}}(s,x), \end{aligned}$$

by (5.18), we have

$$\begin{aligned} \rho _\eta (\Gamma ^{t,s}_r(x))\asymp \rho _\eta (x). \end{aligned}$$

Hence, by (5.18) and (5.26),

$$\begin{aligned} \frac{\Vert u_2(t)-u_2(s)\Vert _{L^\infty (\rho _\eta )}}{|t-s|^{(2-{\bar{\alpha }})/2}}\lesssim 1. \end{aligned}$$

Combining the above estimates, we obtain (5.24). The proof is complete. $\square$

Next we apply (5.23), (5.4) and (5.5) to derive the following a priori estimate for $u^\sharp$ as done in Lemma 3.3.

Lemma 5.8

Under (5.12), there is a constant $C=C(\Theta )>0$ such that

$$\begin{aligned} \Vert u^\sharp \Vert _{{{\mathbb {L}}}_T^\infty (\rho _{2\delta +\kappa })}+\Vert u^\sharp \Vert _{{\mathbb {S}}_T^{3-2{\bar{\alpha }}}(\rho _{2\eta })}\leqslant C. \end{aligned}$$

(5.27)

Proof

First of all, by (5.4) and (5.23), we have

$$\begin{aligned} \Vert u^\sharp \Vert _{{{\mathbb {L}}}_T^\infty (\rho _{2\delta +\kappa })} +\Vert u^\sharp \Vert _{L^\infty _T{{\mathbf {C}}}^{2-{\bar{\alpha }}}(\rho _{\eta +\kappa })}\lesssim 1. \end{aligned}$$

(5.28)

Next we estimate each term on the right hand side of (5.5) by using Lemma 2.10.

• By (2.21), (2.4), and ${\bar{\alpha }}=\alpha +{{\widetilde{\kappa }}}$, we have

  $$\begin{aligned} \Vert \nabla u\prec b-\nabla u\prec \!\!\!\prec b\Vert _{L^\infty _T{{\mathbf {C}}}^{1-2{\bar{\alpha }}}(\rho _{\eta +\kappa })}\lesssim \Vert u\Vert _{{{\mathbb {S}}}_T^{2-{\bar{\alpha }}}(\rho _{\eta })}\Vert b\Vert _{L^\infty _T{{\mathbf {C}}}^{-\alpha }(\rho _{\kappa })}\lesssim 1. \end{aligned}$$

• By (2.15) we have

  $$\begin{aligned} \Vert \nabla u\succ b\Vert _{L_T^\infty {{\mathbf {C}}}^{1-2{\bar{\alpha }}}(\rho _{\eta +\kappa })}\lesssim \Vert u\Vert _{L_T^\infty {{\mathbf {C}}}^{2-{\bar{\alpha }}}(\rho _{\eta })}\Vert b\Vert _{L_T^\infty {{\mathbf {C}}}^{-\alpha }(\rho _\kappa )}\lesssim 1. \end{aligned}$$

• By (2.20) and (2.12) we have

  $$\begin{aligned} \Vert [{{\mathscr {L}}},\nabla u\prec \!\!\!\prec ]{{\mathscr {I}}}b\Vert _{L_T^\infty {{\mathbf {C}}}^{1-2{\bar{\alpha }}}(\rho _{\eta +\kappa })}\lesssim \Vert u\Vert _{{{\mathbb {S}}}_T^{2-{\bar{\alpha }}}(\rho _{\eta })}\Vert b\Vert _{L_T^\infty {{\mathbf {C}}}^{-\alpha }(\rho _\kappa )}\lesssim 1. \end{aligned}$$

• By the growth of H and (5.23), we have

  $$\begin{aligned} \Vert H(u,\nabla u)\Vert _{{{\mathbb {L}}}^\infty _T(\rho _{2\eta })} \lesssim 1+\Vert \nabla u\Vert _{{{\mathbb {L}}}^\infty _T(\rho _{\eta })}^2\lesssim 1. \end{aligned}$$

• By Lemma 3.3 with $\gamma =2-2{\bar{\alpha }}$, $\beta \in ({\bar{\alpha }},2-2{\bar{\alpha }})$, we have

  $$\begin{aligned} \Vert b\circ \nabla u\Vert _{L_T^\infty {{\mathbf {C}}}^{1-2{\bar{\alpha }}}(\rho _{2\eta })}\lesssim \Vert u\Vert _{{{\mathbb {S}}}_T^{2-{\bar{\alpha }}}(\rho _{2\eta -2\kappa })} +\Vert u^\sharp \Vert _{L_T^\infty {{\mathbf {C}}}^{\beta +1}(\rho _{2\eta -\kappa })}+1, \end{aligned}$$

  and by interpolation inequality (2.5) with $\theta =\frac{\eta -2\kappa }{\eta -\kappa }$, (5.28) and Young’s inequality,

  $$\begin{aligned} \Vert u^\sharp \Vert _{L_T^\infty {{\mathbf {C}}}^{\beta +1}(\rho _{2\eta -\kappa })}&\lesssim \Vert u^\sharp \Vert _{L_T^\infty {{\mathbf {C}}}^{3-2{\bar{\alpha }}}(\rho _{2\eta })}^{\theta } \Vert u^\sharp \Vert _{L^\infty _T{{\mathbf {C}}}^{2-{\bar{\alpha }}}(\rho _{\eta +\kappa })}^{1-\theta } \\ {}&\lesssim \varepsilon \Vert u^\sharp \Vert _{L_T^\infty {{\mathbf {C}}}^{3-2{\bar{\alpha }}}(\rho _{2\eta })}+1, \end{aligned}$$

  where we choose $\beta$ such that $\beta \leqslant (1-{\bar{\alpha }})(\theta +1)$ since $\kappa$ is small enough.

Combining the above calculations and by (2.11) with $\theta =2$ and $q=\infty$, we obtain

$$\begin{aligned} \Vert u^\sharp \Vert _{{\mathbb {S}}_T^{3-2{\bar{\alpha }}}(\rho _{2\eta })}\lesssim \varepsilon \Vert u^\sharp \Vert _{L_T^\infty {{\mathbf {C}}}^{3-2{\bar{\alpha }}}(\rho _{2\eta })}+1, \end{aligned}$$

which in turn implies the desired estimate. $\square$

Now we are in a position to give the proof of Theorem 5.1.

Proof of Theorem 5.1

(Existence) By (5.23) and (5.27), we obtain the uniform estimate (5.9). Now by Ascoli-Arzelà’s lemma, there are a subsequence still denoted by n and

$$\begin{aligned} (u, u^\sharp )\in {\mathbb {S}}_T^{2-{\bar{\alpha }}}(\rho _{\eta }) \times {\mathbb {S}}_T^{3-2{\bar{\alpha }}}(\rho _{2\eta }) \end{aligned}$$

such that for each $\gamma >0$,

$$\begin{aligned} (u_n, u^\sharp _n)\rightarrow (u, u^\sharp ) \, \mathrm{in }\,{\mathbb {S}}_T^{2-{\bar{\alpha }}-\gamma }(\rho _{\eta +\gamma }) \times {\mathbb {S}}_T^{3-2{\bar{\alpha }}-\gamma }(\rho _{2\eta +\gamma }). \end{aligned}$$

By (5.7) and taking weak limits for approximation equation (5.4) and (5.5) with (b, f) being replaced by $(b_n,f_n)$, one sees that $(u, u^\sharp )$ solves (5.4) and (5.5) (see [18] for more details).

(Uniqueness) Let $u, {\bar{u}}$ be two paracontrolled solutions to (5.2) in the sense of Theorem 5.1 starting from the same initial value. Let $U:=u-{\bar{u}}$. It is easy to see that U is a paracontrolled solution to the following linear equation

$$\begin{aligned} \partial _tU=\Delta U+(b+R)\cdot \nabla U+K\cdot U,\quad U(0)=0, \end{aligned}$$

(5.29)

where

$$\begin{aligned}&R:=\int _0^1\nabla _QH(u,\nabla u+s\nabla ({\bar{u}}-u)){\mathord {\mathrm{d}}}s, \\&K:=\int _0^1\partial _uH(u+s({\bar{u}}-u),\nabla {\bar{u}}){\mathord {\mathrm{d}}}s. \end{aligned}$$

Note that by (5.3) and $u,{\bar{u}}\in {\mathbb {S}}_T^{2-{\bar{\alpha }}}(\rho _\eta )$,

$$\begin{aligned} |R|+|K|\lesssim \rho ^{-1}_\delta +|u|+|{\bar{u}}|+|\nabla {\bar{u}}|+|\nabla u|\lesssim \rho ^{-1}_\eta . \end{aligned}$$

Then uniqueness follows from Theorem A.2. $\square$

6 Applications

In this section we apply the main results in Sect. 5 to the KPZ type Eqs. (1.3) and (1.4).

6.1 KPZ type equations

Consider the following KPZ type equation:

$$\begin{aligned} {{\mathscr {L}}}h=(\partial _x h)^{\diamond 2}+g(h)+\xi ,\quad h(0)=h_0 \end{aligned}$$

(6.1)

where $g\in {{\mathscr {C}}}^1$ and $\xi$ is a space-time white noise on ${\mathbb {R}}^+\times {\mathbb {R}}$ on some stochastic basis $(\Omega ,{{\mathscr {F}}},({{\mathscr {F}}}_t)_{t\geqslant 0}, {{{\mathbb {P}}}})$. Here the nonlinear term $(\partial _x h)^{\diamond 2}=``(\partial _x h)^{2}-\infty ''$ with for and the approximation $\xi _n$ being defined below.

For $g\equiv 0$ Eq. (6.1) is the classical KPZ equation. The motivation for adding the nonlinear term g comes from geometric stochastic heat equations with values in a Riemannian manifold M studied in [4] via regularity structure theory and in [11, 49] by Dirichlet form, which, in local coordinates, can be written as

$$\begin{aligned} \partial _tu^\alpha =\partial _x^2 u^\alpha +\Gamma ^\alpha _{\beta \gamma }(u)\partial _xu^\beta \partial _xu^\gamma +h^\alpha (u)+\sigma _i^\alpha (u)\xi _i, \end{aligned}$$

(6.2)

where $\Gamma$ denotes the Christoffel symbols for the Levi-Civita connection and $\xi _i$ are i.i.d. space-time white noises and $\sigma _i$ are a collection of vector fields on the manifold. We use Einstein’s convention of summation over repeated indices. The first three terms in (6.2) correspond to Eells-Sampson’s harmoic map flow [14] and $h^\alpha$ corresponds to the $\frac{1}{8}\nabla R$ with R the scalar curvature of M. For more background on (6.2) we refer to [4] and [11, 49]. As (6.2) is driven by multiplicative noise, there are more than forty terms required for renormalization and regularity structure theory is required to derive local well-posedness of (6.2). It is also interesting to study (6.2) on the whole line (see [11] for different long time behavior compared to the finite volume case). As directly obtaining global well-posedness to Eq. (6.2) by PDE argument is out of reach by the techniques so far, we study (6.1) and apply our main result.

We define the 2n periodization of $\xi$ by

$$\begin{aligned} \tilde{\xi }_n(\psi )=\xi (\psi _n) \text { where } \psi _n(t,x)={\mathbf {1}}_{ [-n,n)}(x)\sum _{y\in 2n{\mathbb {Z}}}\psi (t,x+y). \end{aligned}$$

Let $\varphi \in C^\infty _c({\mathbb {R}})$ be even and such that $\varphi (0)=1$ and define the spatial regularization of $\tilde{\xi }_n$

$$\begin{aligned} \xi _n=\varphi (n^{-1}\partial _x)\tilde{\xi }_n={\mathcal {F}}^{-1}(\varphi (n^{-1}\cdot ){\mathcal {F}}\tilde{\xi }_n). \end{aligned}$$

The regularity of the space-time white noise $\xi$ is more rough than the coefficient f given in (1.5). To apply Theorem 5.1 we need to introduce the following random fields and use Da Prato-Debussche trick (cf. [13]) to decompose (6.1) into (1.5) and the following equations, which is the usual way for the KPZ equation (cf. [23, 26, 45]):

(6.3)

all with zero initial conditions except $Y(0)(x)=Cx+B(x)$ and $Y_n(0)$ defined similarly as $\xi _n$ with $\xi$ replaced by $Cx+B(x)$, where B is a two sided Brownian motion, which is independent of space-time white noise $\xi$, and $C\in {\mathbb {R}}$. The choice of the initial condition is due to our interest in the KPZ equation starting from its invariant measure (cf. [48, Section 1.4] and [16]). Here and are renormalization constants. For simplicity of notation we also set

$$\begin{aligned} X_n=\partial _xY_n, \quad X=\partial _x Y, \quad X^{(\cdot )}=\partial _xY^{(\cdot )}, \end{aligned}$$

where $(\cdot )$ stands for the above trees. In the following we draw a table for the regularity of each $Y^{(\cdot )}$. For $\gamma >0$ the homogeneities $\alpha _\tau \in {\mathbb {R}}$ are given by

Lemma 6.1

With the above notations, there exist random distributions

and divergence constants , such that for every $\tau \in {{\mathscr {Y}}}$,

$$\begin{aligned} \tau \in \cap _{\kappa >0}{\mathbb {S}}_T^{\alpha _\tau }(\rho _\kappa ), \end{aligned}$$

for $\alpha _\tau$ given in the above table. Moreover, for $\tau _n$ defined in (6.3) $\tau _n\rightarrow \tau$ in $L^p(\Omega ,{\mathbb {S}}_T^{\alpha _\tau }(\rho _\kappa ))$ for every $p\in [1,\infty )$ and every $\kappa >0$. Furthermore, $Y_n\rightarrow Y$ in $L^p(\Omega ,{\mathbb {S}}_T^{\frac{1}{2}-\gamma }(\rho _{1+\kappa })$ for every $p\in [1,\infty )$. Finally, there exist random distribution $\nabla {{\mathscr {I}}}^{t}_s(X)\circ X$ such that

$$\begin{aligned} \sup _{0\leqslant s\leqslant t\leqslant T}\Vert \nabla {{\mathscr {I}}}^{t}_s(X_n)\circ X_n(t)-\nabla {{\mathscr {I}}}^{t}_s(X)\circ X(t)\Vert _{{{\mathbf {C}}}^{-\gamma }(\rho _\kappa )}\rightarrow 0 \text { in }L^p(\Omega ). \end{aligned}$$

Proof

Most terms except in (6.3) have been considered in [45, Theorem 3.6]. These two terms can also been obtained by similar calculation as in [23, Theorem 9.3] (see also [55, Section 3.3.1, Section A.2]). The last convergence result for $\nabla {{\mathscr {I}}}^{t}_s(X)\circ X(t)$ can be obtained similarly as in [45, Lemma C.1]. For reader’s convenience we spell out more details for completeness and we follow the notation of [23, Section 9].

Let W be the space-time white noise in Fourier space. We write $\nabla {{\mathscr {I}}}^{t}_s(X_n)\circ X_n(t)$ as

$$\begin{aligned} \nabla {{\mathscr {I}}}^{t}_s(X_n)\circ X_n(t) =\int&e^{i k_{[12]}x}\psi _0(k_1,k_2) H_{t-s_1}^n(k_1)\\&\times \int _s^t {\mathord {\mathrm{d}}}\sigma H_{t-\sigma }(k_2)H_{\sigma -s_2}^n(k_2)W({\mathord {\mathrm{d}}}\eta _1)W({\mathord {\mathrm{d}}}\eta _2), \end{aligned}$$

with $H_t(k)=i k\mathrm {e}^{-k^2t}{\mathbf {1}}_{t\geqslant 0}, H_t^n(k)=H_t(k)\varphi (n^{-1}k)$, $\eta _i=(s_i,k_i)$, $s_{-i}=s_i, k_{-i}=k_i$, ${\mathord {\mathrm{d}}}\eta _i={\mathord {\mathrm{d}}}s_i{\mathord {\mathrm{d}}}k_i$, $k_{[12]}=k_1+k_2$, $\psi _0(k_1,k_2)=\sum _{|i-j|\leqslant 1}\theta _i(k_1)\theta _j(k_2)$ for $\theta _i$ being the dyadic partition of unity. By Wiener chaos decomposition the term in the zeroth order chaos is given by

$$\begin{aligned} \int H_{t-s_1}^n(k_1)\int _s^t {\mathord {\mathrm{d}}}\sigma H_{t-\sigma }(-k_1)H_{\sigma -s_1}^n(-k_1){\mathord {\mathrm{d}}}\eta _1, \end{aligned}$$

which is zero by using the fact that the integrand is antisymmetric under the change of variables $k_1\rightarrow -k_1$. For the second order chaos we calculate for $0\leqslant s\leqslant r\leqslant t$

$$\begin{aligned}&{\mathbb {E}}|\Delta _q(\nabla {{\mathscr {I}}}^{t}_s(X_n)\circ X_n-\nabla {{\mathscr {I}}}^{t}_r(X_n)\circ X_n)|^2\\&\lesssim \int |\theta _q(k_{[12]})|^2 \psi _0(k_1,k_2)^2|H_{t-s_1}(k_1)|^2\Big |\int _s^r {\mathord {\mathrm{d}}}\sigma H_{t-\sigma }(k_2)H_{\sigma -s_2}^n(k_2)\Big |^2{\mathord {\mathrm{d}}}\eta _1{\mathord {\mathrm{d}}}\eta _2\\&\lesssim |r-s|^\varepsilon \int _{E^2} |\theta _q(k_{[12]})|^2 \psi _0(k_1,k_2)^2(|k_2|+1)^{-2+2\varepsilon }{\mathord {\mathrm{d}}}k_1{\mathord {\mathrm{d}}}k_2\\&\lesssim |r-s|^\varepsilon 2^{2q\varepsilon }, \end{aligned}$$

where the implicit constant is independent of n. The rest of the proof follows by standard arguments as in [23]. $\square$

We make the following decomposition

where ${\widetilde{h}}$ satisfies the following equation

(6.4)

Here we use (6.3).

Using Lemma 6.1, we obtain the following lemma.

Lemma 6.2

There exists a measurable set $\Omega _0$ with ${{{\mathbb {P}}}}(\Omega _0)=1$ such that for every $\kappa >0$, $\gamma \in (0,\tfrac{1}{4})$ and $\omega \in \Omega _0$

Proof

By Lemma 2.10 and (2.11) we have that

and

to have

Other terms follows directly from Lemma 6.1. $\square$

As a result ${\widetilde{h}}$ satisfies (1.5) with b, f given above. We say that h is a paracontrolled solution to (6.1) if ${\widetilde{h}}$ is a paracontrolled solution to (6.4) in the sense of (5.4) and (5.5).

Since $\gamma$ can be arbitrary small, we apply Theorem 5.1 to obtain the following result.

Theorem 6.3

Suppose $g\in {{\mathscr {C}}}^1$. For every initial condition ${\widetilde{h}}(0)\in {{\mathbf {C}}}^{\frac{3}{2}+\varepsilon +\gamma }(\rho _{\varepsilon \delta })$ where $0<\varepsilon<1, \gamma \in (0,\tfrac{1}{4}), 0<\delta :=40\kappa <1$, there exists a unique paracontrolled solution

$$\begin{aligned} ({\widetilde{h}},{\widetilde{h}}^{\sharp })\in ({\mathbb {S}}^{\frac{3}{2}-\kappa ^{1/4}-\gamma }_T(\rho _\eta )\cap {{\mathbb {L}}}^\infty _T(\rho _{2\delta }),{{\mathbb {S}}}_T^{2-2\kappa ^{1/4}-2\gamma }(\rho _{2\eta })\cap {{\mathbb {L}}}^\infty _T(\rho _{2\delta +\kappa })) \end{aligned}$$

to (6.4), where

$$\begin{aligned} 2 ({\kappa ^{1/4}}+80\kappa )<\eta <\tfrac{1}{4}. \end{aligned}$$

Proof

In the following we check other conditions of Theorem 5.1. The condition for $H=H_c+H_s$ is satisfied easily by Lemma 6.1 where $H_c=Q^2$, . In the following we prove $(b,f)\in {\mathbb {B}}^\alpha _T(\rho _\kappa )$. The approximation $\{(b_n,f_n)\}_n$ for (b, f) is given as in Lemma 6.2 with the corresponding tree $\tau$ replaced by $\tau _n$ in Lemma 6.1. In the following we prove that for every $\kappa >0$

$$\begin{aligned} \sup _n(\ell _T^{b_n}(\rho _\kappa )+{{\mathbb {A}}}_{T,\infty }^{b_n,f_n}(\rho _\kappa ))<\infty , \end{aligned}$$

(6.5)

with $\ell _T^{b_n}(\rho _\kappa )$ and ${{\mathbb {A}}}_{T,\infty }^{b_n,f_n}(\rho _\kappa ))$ defined in (2.25) and (2.24), respectively. In the following we omit the subscript n for simplicity and all the following bounds are uniform in n and $\lambda$. Note that

By the last result in Lemma 6.1 and Lemma 2.16 we deduce the first term

$$\begin{aligned} \Vert \nabla {{\mathscr {I}}}_\lambda X\circ X\Vert _{ L_T^\infty {{\mathbf {C}}}^{-\gamma }(\rho _{\kappa })}\lesssim 1. \end{aligned}$$

Other terms on the right hand side can be calculated by Lemma 2.10 and (2.11) to have

and

On the other hand, note that

with . By Lemma 2.10 and (2.11) we know

$$\begin{aligned}\Vert \nabla {{\mathscr {I}}}_\lambda f_1\circ b\Vert _{L_T^\infty {{\mathbf {C}}}^{-\gamma }(\rho _{2\kappa })}\lesssim \Vert f_1\Vert _{L_T^\infty {{\mathbf {C}}}^{-2\gamma }(\rho _{\kappa })} \Vert b\Vert _{L_T^\infty {{\mathbf {C}}}^{-\frac{1}{2}-\gamma }(\rho _{\kappa })}\lesssim 1 , \end{aligned}$$

and

It remains to consider the term and we use the commutator introduced in Lemma 2.11 and Lemma 2.12 to have

By Lemmas 2.12, 2.11 and Lemma 6.1 we have

where we used time regularity of , which follows from (2.4). Combining all the above estimates, we deduce (6.5) follows. Furthermore, we know that the convergence in Definition 2.14 also holds by Lemma 6.1 and Lemma 2.16, which gives that $(b,f)\in {{\mathbb {B}}}_T^\alpha (\rho _\kappa )$. Then the result follows from Theorem 5.1. $\square$

Remark 6.4

• 1. The exponent $\eta$ of the weight could be arbitrarily small since $\kappa$ is arbitrarily small. This result improves the weight for the solution to the KPZ equation obtained in [45].

• 2. In the finite volume case, the initial value $\varphi$ could be more rough, and the fixed point argument allows $\varphi \in \cup _{\varepsilon >0}{{\mathbf {C}}}^\varepsilon$ (see [26]). In the infinite volume case, the singularity near $t=0$ seems to break the energy estimate in Lemma 4.8. We shall study this problem in the future.

• 3. We may also consider the following more general singular SPDEs

  $$\begin{aligned} \partial _th=\partial _x^2 h +|\partial _xh|^2+g(h)+K(h)\partial _xh+\xi , \end{aligned}$$

  (6.6)

  for $g, K\in {{\mathscr {C}}}^1$. We have the decomposition

  

  with ${\widetilde{h}}$ satisfying

  

  (6.7)

  for b, f given in Lemma 6.2. Since requires further renormalization and in this paper we mainly concentrate on the singular renormalized terms from $|\partial _xh|^2$, we consider the following simplified equation

  

  (6.8)

  where $K, g\in {{\mathscr {C}}}^1$ and the most singular terms coming from $(\partial _xh)^2$ in (6.6) have been included. We can apply Theorem 5.1 to obtain the same global well-posedness for Eq. (6.8).

• 4. A challenging question is whether PDE arguments can be used to deduce global well-posedness of vector-valued generalized KPZ equations since it is not clear whether the maximum principle can be extended to cover such a situation. We leave this for our future work.

6.2 Modified KPZ equations

In this subsection we consider the following modified KPZ equation:

$$\begin{aligned} {{\mathscr {L}}}h=g(x)(\partial _x h)^{\diamond 2}+K(x)\partial _xh+\xi ,\quad h(0)=h_0 \end{aligned}$$

(6.9)

where $g,K\in {{\mathscr {C}}}^1$ and $\xi$ is a space-time white noise on ${\mathbb {R}}^+\times {\mathbb {R}}$ on some stochastic basis $(\Omega ,{{\mathscr {F}}},({{\mathscr {F}}}_t)_{t\geqslant 0}, {{{\mathbb {P}}}})$. This model can be derived similarly like KPZ equation as surface growth model where the growth rate also depending on position x (c.f. [36]). We emphasize that for this model we cannot use Cole–Hopf’s transformation to directly obtain the well-posedness since for $w=e^{gh}$ there exists some new nonlinear terms in the equation of w which cannot be cancelled.

Here the nonlinear term requires renormalization and we define the spatial regularization of $\xi$ as in Sect. 6.1. To apply Theorem 5.1 we also introduce the following random fields as in Sect. 6.1 and use Da Prato-Debussche trick to decompose (6.9) into (1.5) and the following equations:

(6.10)

all with zero initial conditions except $Y(0)(x)=B(x)+Cx$, $C\in {{\mathbb {R}}}$, and $Y_n(0)$ defined similarly as $\xi _n$ with $\xi$ replaced by $B(x)+Cx$, where B is a two sided Brownian motion, which is independent of space-time white noise $\xi$. Here and are renormalization constants as in (6.3). We also set

$$\begin{aligned} X_n=\partial _xY_n, \quad X=\partial _x Y, \quad \bar{X}^{(\cdot )}=\partial _x\bar{Y}^{(\cdot )}, \end{aligned}$$

where $(\cdot )$ stands for the above tree. The regularity and the homogeneities of each $\bar{Y}^{(\cdot )}$ are the same as the corresponding ${Y}^{(\cdot )}$ if the trees in the superscript are the same.

Lemma 6.5

With the above notations, there exist random distributions

and divergence constants , such that for every $\tau \in {{\mathscr {Y}}}$,

$$\begin{aligned} \tau \in \cap _{\kappa >0}{\mathbb {S}}_T^{\alpha _\tau }(\rho _\kappa ), \end{aligned}$$

for $\alpha _\tau$ given in the above table. Moreover, for $\tau _n$ defined in (6.10) $\tau _n\rightarrow \tau$ in $L^p(\Omega ,{\mathbb {S}}_T^{\alpha _\tau }(\rho _\kappa ))$ for every $p\in [1,\infty )$ and every $\kappa >0$.

Proof

If terms in the bracket of (6.10) converge in the corresponding space as $n\rightarrow \infty$, we can obtain results easily by Schauder estimate. However, does not converge in spatial distribution space and we have to do probabilistic calculation again. We follow the method and notation in [15]. Let $K_{j,x}(y)=2^jK(2^j(x-y))$ be the kernel associated with the j-th Littlewood-Paley block $\Delta _j$ on ${{\mathbb {R}}}$. For a function f we write $\Delta _jf(x)=\int K_{j,x}(y)f(y){\mathord {\mathrm{d}}}y$. We also use P to denote the heat kernel on ${{\mathbb {R}}}\times {{\mathbb {R}}}$, i.e. $P(t,x)=(4\pi t)^{-\tfrac{1}{2}}\mathrm {e}^{-\tfrac{|x|^2}{4t}}1_{t\geqslant 0}$. For fixed $\bar{\zeta }=(t,\bar{x})$ and for $j\geqslant -1$ define the measure

$$\begin{aligned} \mu _{j}({\mathord {\mathrm{d}}}\zeta ):=\Big [\int K_{j,{\bar{x}}}(x)P(t-s,x-y){\mathord {\mathrm{d}}}x\Big ]1_{s\geqslant 0}{\mathord {\mathrm{d}}}\zeta , \end{aligned}$$

with $\zeta =(s,y)$. For $\zeta _i=(s_i,x_i)$, set $|\zeta _1-\zeta _2|:=|s_1-s_2|^{\frac{1}{2}}+|x_1-x_2|$. Then by similar calculation as in [27, Section 10] and using [27, Lemma 10.14] we know

(6.11)

By [15, (87)] we find

$$\begin{aligned} \Big |\int K_{j,{\bar{x}}}(x)P(t-s_1,x-x_1){\mathord {\mathrm{d}}}x\Big |\lesssim \frac{2^{-(1-\varepsilon /2) j}}{(|\bar{x}-x_1|+|t-s_1|^{1/2}+2^{-j})^{2-\varepsilon /2}}, \end{aligned}$$

which combined with (6.11) and [27, Lemma 10.14] implies that could be controlled by $2^{-(2-\varepsilon )j}$ for $\varepsilon >0$ small enough. Then the desired estimate for follows by standard techniques (c.f. [23, Lemma 9.8]).

We also give more details for the most complicated term (see also [55, Section 3.3.1] for the calculation of a similar term). For fixed ${\bar{\zeta }}=(t,\bar{x})\in {{\mathbb {R}}}\times {{\mathbb {R}}}^3$ and $q\in {{\mathbb {Z}}}$, $q\geqslant -1$, define the measure

$$\begin{aligned} \mu _{q}({\mathord {\mathrm{d}}}\zeta _1,{\mathord {\mathrm{d}}}\zeta _2):=\Big [\int K_{q,{\bar{x}}}(x)&\sum _{|i-j|\leqslant 1}K_{i,x}(y)K_{j,x}(x_2)\partial _xP(t-s_1,y-x_1){\mathord {\mathrm{d}}}x{\mathord {\mathrm{d}}}y\Big ]\\&\delta (t-s_2)1_{s_1\geqslant 0}{\mathord {\mathrm{d}}}\zeta _1{\mathord {\mathrm{d}}}\zeta _2, \end{aligned}$$

with $\zeta _i=(s_i,x_i)\in {{\mathbb {R}}}\times {{\mathbb {R}}}$ for $i=1,2$. ${\tilde{\mu }}_{q}({\mathord {\mathrm{d}}}\zeta _1,{\mathord {\mathrm{d}}}\zeta _2)$ is defined similarly with $\partial _xP$ replaced $\partial _xP*\partial _xP$.

We decompose with $I_i$ in the space of i-th Wiener chaos. Then by similar calulation as in [27, Section 10] and using [27, Lemma 10.14] we know

$$\begin{aligned} {\mathbb {E}}|\Delta _q I_4|^2\lesssim \int |\zeta _1-\zeta _1'|^{-1-\varepsilon }|\zeta _2-\zeta _2'|^{-1} |\mu _{q}({\mathord {\mathrm{d}}}\zeta _1,{\mathord {\mathrm{d}}}\zeta _2)||\mu _{q}({\mathord {\mathrm{d}}}\zeta _1',{\mathord {\mathrm{d}}}\zeta _2')|, \end{aligned}$$

which by [15, Lemma A.19]³ can be contolled by $2^{q\varepsilon }$ for $\varepsilon >0$ small. For $I_2$ we have the decomposition

where we refer to [27, Section 10], [23, Section 9] for the meaning of the graph. We use [27, Lemmas 10.14, 10.16] to have

$$\begin{aligned} {\mathbb {E}}|\Delta _q I_{21}|^2\lesssim \int |\zeta _1-\zeta _1'|^{-1-\varepsilon }|\zeta _2-\zeta _2'|^{-1} |\mu _{q}({\mathord {\mathrm{d}}}\zeta _1,{\mathord {\mathrm{d}}}\zeta _2)||\mu _{q}({\mathord {\mathrm{d}}}\zeta _1',{\mathord {\mathrm{d}}}\zeta _2')|, \end{aligned}$$

and use [27, Lemma 10.16] to have

$$\begin{aligned} {\mathbb {E}}|\Delta _q I_{22}|^2\lesssim \int |\zeta _1-\zeta _1'|^{-3\varepsilon }|\zeta _1-\zeta _2|^{-1+\varepsilon }|\zeta _1'-\zeta _2'|^{-1+\varepsilon } |\mu _{q}({\mathord {\mathrm{d}}}\zeta _1,{\mathord {\mathrm{d}}}\zeta _2)||\mu _{q}({\mathord {\mathrm{d}}}\zeta _1',{\mathord {\mathrm{d}}}\zeta _2')|, \end{aligned}$$

and use [27, Lemma 10.14, (10.37 a)] to have

$$\begin{aligned} {\mathbb {E}}|\Delta _q I_{23}|^2&\lesssim \int |\zeta _1-\zeta _1'|^{-2}|\zeta _1-\zeta _2|^{-\varepsilon }|\zeta _1'-\zeta _2'|^{-\varepsilon } |\mu _{q}({\mathord {\mathrm{d}}}\zeta _1,{\mathord {\mathrm{d}}}\zeta _2)||\mu _{q}({\mathord {\mathrm{d}}}\zeta _1',{\mathord {\mathrm{d}}}\zeta _2')| \\ {}&+ \int |\zeta _1-\zeta _1'|^{-2}|\zeta _1-\zeta _2|^{-1}|\zeta _1'-\zeta _2'|^{-1} |{\tilde{\mu }}_{q}({\mathord {\mathrm{d}}}\zeta _1,{\mathord {\mathrm{d}}}\zeta _2)||{\tilde{\mu }}_{q}({\mathord {\mathrm{d}}}\zeta _1',{\mathord {\mathrm{d}}}\zeta _2'|. \end{aligned}$$

Then by [15, Lemma A.19] ${\mathbb {E}}|\Delta _q I_{2i}|^2$, $i=1, 2, 3,$ can be contolled by $2^{q\varepsilon }$ for $\varepsilon >0$ small.

Different from the classical case in Sect. 6.1, $I_0$ contains g. We use $g\in {{\mathscr {C}}}^2$ to have $|g(x)-g(y)|\lesssim |x-y|$. Then we can shift g to the vertex $x_2$ and use [27, Lemmas 10.14, 10.16] to obtain

By [15, Lemma A.16], we find for $\delta \in (0,1)$

$$\begin{aligned} |\int K_{i,x}(y)\partial _xP(t-s_1,y-x_1){\mathord {\mathrm{d}}}y|\lesssim 2^{-i\delta } (|t-s_1|^{1/2}+|x-x_1|)^{-2-\delta }, \end{aligned}$$

which combined with $|g^2(x_2)-g^2(x)|\lesssim |x-x_1|+|x_2-x_1|$ and [15, Lemma A.16], [27, Lemmas 10.14] implies that .

Then the required regularity of follows by standard argument (c.f. [23]). $\square$

We make the following decomposition

where ${\widetilde{h}}$ satisfies the following equation

(6.12)

Using Lemma 6.5, we obtain the following lemma.

Lemma 6.6

There exists a measurable set $\Omega _0$ with ${{{\mathbb {P}}}}(\Omega _0)=1$ such that for every $\kappa >0$, $\gamma >0$ and $\omega \in \Omega _0$

Proof

The proof follows from the proof of Lemma 6.2, Lemma 6.5 and $g, K\in {{\mathscr {C}}}^1$. $\square$

As a result ${\widetilde{h}}$ satisfies (1.5) with b, f given above. We say that h is a paracontrolled solution to (6.9) if ${\widetilde{h}}$ is a paracontrolled solution to (6.12) in the sense of (5.4) and (5.5).

Since $\gamma$ can be arbitrarily small, we apply Theorem 5.1 to obtain the following result.

Theorem 6.7

Let $g, K\in {{\mathscr {C}}}^1$. For every initial condition ${\widetilde{h}}(0)\in {{\mathbf {C}}}^{\frac{3}{2}+\varepsilon +\gamma }(\rho _{\varepsilon \delta })$ where $0<\varepsilon<1, \gamma \in (0,\tfrac{1}{4}), 0<\delta :=40\kappa <1$, there exists a unique paracontrolled solution

$$\begin{aligned} ({\widetilde{h}},{\widetilde{h}}^{\sharp })\in ({\mathbb {S}}^{\frac{3}{2}-\kappa ^{1/4}-\gamma }_T(\rho _\eta )\cap {{\mathbb {L}}}^\infty _T(\rho _{2\delta }),{{\mathbb {S}}}_T^{2-2\kappa ^{1/4}-2\gamma }(\rho _{2\eta })\cap {{\mathbb {L}}}^\infty _T(\rho _{2\delta +\kappa })) \end{aligned}$$

to (6.12), where

$$\begin{aligned} 2 ({\kappa ^{1/4}}+80\kappa )<\eta <\tfrac{1}{4}. \end{aligned}$$

Proof

In the following we check other conditions of Theorem 5.1. The condition for H is satisfied easily. In the following we prove $(b,f)\in {\mathbb {B}}^\alpha _T(\rho _\kappa )$. The approximation $\{(b_n,f_n)\}_n$ for (b, f) is given as in Lemma 6.6 with the corresponding tree $\tau$ replaced by $\tau _n$ in Lemma 6.5. In the following we prove that for every $\kappa >0$

$$\begin{aligned} \sup _n(\ell _T^{b_n}(\rho _\kappa )+{{\mathbb {A}}}_{T,\infty }^{b_n,f_n}(\rho _\kappa ))<\infty , \end{aligned}$$

(6.13)

with $\ell _T^{b_n}(\rho _\kappa )$ and ${{\mathbb {A}}}_{T,\infty }^{b_n,f_n}(\rho _\kappa ))$ defined in (2.25) and (2.24), respectively. In the following we omit the subscript n for simplicity and all the following bounds are uniform in n and $\lambda$. We first consider

(6.14)

For the first term $\nabla {{\mathscr {I}}}_\lambda (gX)\circ [gX]$ we use Lemma 2.17 and Lemma 6.1 to have

$$\begin{aligned} \Vert \nabla {{\mathscr {I}}}_\lambda (gX)\circ (gX)\Vert _{ L_T^\infty {{\mathbf {C}}}^{-\gamma }(\rho _{2\kappa })}\lesssim 1. \end{aligned}$$

Other terms on the right hand side of (6.14) can be calculated by Lemma 2.10 and (2.11):

and

On the other hand, we know

with . By Lemma 2.10 and (2.11) we know

$$\begin{aligned}\Vert \nabla {{\mathscr {I}}}_\lambda f_1\circ b\Vert _{L_T^\infty {{\mathbf {C}}}^{-\gamma }(\rho _{2\kappa })}\lesssim \Vert f_1\Vert _{L_T^\infty {{\mathbf {C}}}^{-2\gamma }(\rho _{\kappa })} \Vert b\Vert _{L_T^\infty {{\mathbf {C}}}^{-\frac{1}{2}-\gamma }(\rho _{\kappa })}\lesssim 1 , \end{aligned}$$

and

We use Lemma 2.17 and Lemma 6.1 to have

$$\begin{aligned} \Vert \nabla {{\mathscr {I}}}_\lambda (KX)\circ (gX)\Vert _{L_T^\infty {{\mathbf {C}}}^{-\gamma }(\rho _{2\kappa })}\lesssim 1. \end{aligned}$$

It remains to consider the term and we use the commutator introduced in Lemma 2.11 and Lemma 2.12 to have

We have further decomposition for the last term

By Lemmas 2.12, 2.11 and Lemma 6.1

where we used time regularity of , which follows from (2.4). Combining all the above estimates, we deduce that (6.13) follows. Furthermore, we know that the convergence in Definition 2.14 also holds by using Lemma 6.1 and Lemma 2.16, which gives that $(b,f)\in {{\mathbb {B}}}_T^\alpha (\rho _\kappa )$. Then the result follows from Theorem 5.1. $\square$

Appendices

Appendix A: Uniqueness of paracontrolled solutions

In this subsection we use Hairer and Labbé’s argument [30] to show the uniqueness of paracontrolled solutions. For this aim, we use the following time-dependent exponential weight: for $\ell \in (0,1)$,

$$\begin{aligned} \mathbf{e}^\ell _t(x):=\exp (-(1+t)\langle x\rangle ^\ell ),\ t\geqslant 0,\ x\in {{\mathbb {R}}}^d. \end{aligned}$$

We can similarly define the Hölder space with weight $\mathbf{e}^\ell$ (see [45]). For instance,

$$\begin{aligned} \Vert f\Vert _{L^\infty _T{{\mathbf {C}}}^\alpha (\mathbf{e}^\ell )}:=\sup _{t\in [0,T]}\Vert f(t,\cdot )\Vert _{{{\mathbf {C}}}^\alpha (\mathbf{e}^\ell _t)}, \end{aligned}$$

and for $\alpha \in (0,1)$,

$$\begin{aligned} \Vert f\Vert _{C^\alpha _TL^\infty (\mathbf{e}^\ell )}&:= \sup _{0\leqslant t\leqslant T} \Vert f(t)\mathbf{e}^\ell _t\Vert _{L^\infty } +\sup _{0\leqslant s\ne t\leqslant T} \frac{\Vert f(t)-f(s)\Vert _{L^\infty (\mathbf{e}^\ell _{t\vee s})}}{|t-s|^{\alpha }}. \end{aligned}$$

In particular, for $\alpha \in (0,2)$, we also set

$$\begin{aligned} {{\mathbb {S}}}^\alpha _T(\mathbf{e}^\ell ):=\Vert f\Vert _{L^\infty _T{{\mathbf {C}}}^\alpha (\mathbf{e}^\ell )}+\Vert f\Vert _{C^{\alpha /2}_TL^\infty (\mathbf{e}^\ell )}. \end{aligned}$$

By [43, Lemma 2.10], for any $T>0$, there is a $C=C(T,\ell ,d)>0$ such that for all $s,t\in [0,T]$ and $j\geqslant -1$,

$$\begin{aligned} \Vert \Delta _jP_tf\Vert _{L^\infty (\mathbf{e}^\ell _s)}\lesssim \mathrm {e}^{-2^{2j}t}\Vert \Delta _jf\Vert _{L^\infty (\mathbf{e}^\ell _s)}. \end{aligned}$$

(A.1)

Moreover, Lemmas 2.8, 2.10, 2.11 and 2.12 still hold for exponential weight $\mathbf{e}^\ell _t$ (see [45]). The following result corresponds to Lemma 2.9.

Lemma A.1

Let $\alpha ,\ell \in (0,1)$, $\kappa \in (0,(1-\frac{\alpha }{2})\ell )$. For any $q\in (\frac{1}{1-\alpha /2-\kappa /\ell },\infty ]$ and $T>0$, there is a constant $C=C(T, d,\alpha ,\ell ,\theta ,\kappa ,q)>0$ such that

$$\begin{aligned} \Vert {{\mathscr {I}}}f\Vert _{{{\mathbb {S}}}_T^{2-\frac{2}{q}-\frac{2\kappa }{\ell }-\alpha }(\mathbf{e}^\ell )}\lesssim _C \Vert f\Vert _{L^q_T{{\mathbf {C}}}^{-\alpha }(\rho _{\kappa }\mathbf{e}^\ell )}. \end{aligned}$$

Proof

First of all we have the following simple observation:

$$\begin{aligned} \mathbf{e}^\ell _t(x)\lesssim \langle x\rangle ^{-\kappa } \mathbf{e}^\ell _s(x)/|t-s|^{\kappa /\ell },\ \ 0\leqslant s<t<\infty . \end{aligned}$$

(A.2)

Let $\frac{1}{p}+\frac{1}{q}=1$ and $t\in (0,T]$. By (A.1) and Hölder’s inequality, we have for $j\geqslant -1$,

$$\begin{aligned} \Vert \Delta _j{{\mathscr {I}}}f(t)\Vert _{L^\infty (\mathbf{e}^\ell _t)}&\lesssim \int ^t_0\mathrm {e}^{-2^{2j}(t-s)}\Vert \Delta _jf(s)\Vert _{L^\infty (\mathbf{e}^\ell _t)}{\mathord {\mathrm{d}}}s \\ {}&\lesssim \int ^t_0\frac{\mathrm {e}^{-2^{2j}(t-s)}}{|t-s|^{\kappa /\ell }}\Vert \Delta _jf(s)\Vert _{L^\infty (\rho _\kappa \mathbf{e}^\ell _s)}{\mathord {\mathrm{d}}}s \\ {}&\lesssim 2^{\alpha j}\left( \int ^t_0\frac{\mathrm {e}^{-p2^{2j}(t-s)}}{|t-s|^{p\kappa /\ell }}{\mathord {\mathrm{d}}}s\right) ^{1/p} \Vert f\Vert _{L^q_t{{\mathbf {C}}}^{-\alpha }(\rho _\kappa \mathbf{e}^\ell )}\\&\lesssim 2^{-(\frac{2}{p}-\frac{2\kappa }{\ell }-\alpha ) j}\Vert f\Vert _{L^q_t{{\mathbf {C}}}^{-\alpha }(\rho _\kappa \mathbf{e}^\ell )}, \end{aligned}$$

which in turn gives that

$$\begin{aligned} \Vert {{\mathscr {I}}}f\Vert _{L^\infty _T{{\mathbf {C}}}^{2-\frac{2}{q}-\frac{2\kappa }{\ell }-\alpha }(\mathbf{e}^\ell )} \lesssim \Vert f\Vert _{L^q_T{{\mathbf {C}}}^{-\alpha }(\rho _{\kappa }\mathbf{e}^\ell )}. \end{aligned}$$

(A.3)

On the other hand, for $0\leqslant t_1<t_2\leqslant T$, we have

$$\begin{aligned} \Vert {{\mathscr {I}}}f(t_2)- {{\mathscr {I}}}f(t_1)\Vert _{L^\infty (\mathbf{e}^\ell _{t_2})}&\leqslant \Vert (P_{t_2-t_1}-I){{\mathscr {I}}}f(t_1)\Vert _{L^\infty (\mathbf{e}^\ell _{t_2})}\\&\quad +\left\| \int _{t_1}^{t_2}P_{t_2-s}f(s) {\mathord {\mathrm{d}}}s\right\| _{L^\infty (\mathbf{e}^\ell _{t_2})}=:I_1+I_2. \end{aligned}$$

For $I_1$, by (2.10) and (A.3) we have

$$\begin{aligned} I_1&\lesssim (t_2-t_1)^{1-\frac{\alpha }{2}-\frac{1}{q}- \frac{\kappa }{\ell }} \Vert {{\mathscr {I}}}f(t_1)\Vert _{{{\mathbf {C}}}^{2-\alpha -\frac{2}{q}- \frac{2\kappa }{\ell }}(\mathbf{e}^\ell _{t_2})} \\ {}&\lesssim (t_2-t_1)^{1-\frac{\alpha }{2}-\frac{1}{q}- \frac{\kappa }{\ell }}\Vert f\Vert _{L^q_T{{\mathbf {C}}}^{-\alpha }(\rho _{\kappa }{} \mathbf{e}^\ell )}. \end{aligned}$$

For $I_2$, by (2.8), (A.2) and Hölder’s inequality, we have

$$\begin{aligned} I_2&\lesssim \int _{t_1}^{t_2}(t_2-s)^{-\frac{\alpha }{2}}\Vert f(s)\Vert _{{{\mathbf {C}}}^{-\alpha }(\mathbf{e}^\ell _{t_2})}{\mathord {\mathrm{d}}}s \\ {}&\lesssim \int _{t_1}^{t_2}(t_2-s)^{-\frac{\alpha }{2}-\frac{\kappa }{\ell }}\Vert f(s)\Vert _{{{\mathbf {C}}}^{-\alpha }(\rho _\kappa \mathbf{e}^\ell _s)}{\mathord {\mathrm{d}}}s \\ {}&\lesssim (t_2-t_1)^{1-\frac{\alpha }{2}-\frac{1}{q}- \frac{\kappa }{\ell }}\Vert f\Vert _{L^q_T{{\mathbf {C}}}^{-\alpha }(\rho _{\kappa }{} \mathbf{e}^\ell )}. \end{aligned}$$

Combining the above estimates, we obtain the desired estimate. $\square$

Now we consider the following linear equation:

$$\begin{aligned} {{\mathscr {L}}}u=(b+{\bar{b}})\cdot \nabla u+hu,\ \ u(0)\equiv 0, \end{aligned}$$

(A.4)

where $b\in \cap _{T>0}{{\mathbb {B}}}^\alpha _T(\rho _\kappa )$ and ${\bar{b}},h\in \cap _{T>0}L^\infty _T(\rho _\eta )$. Let

$$\begin{aligned} (u,u^\sharp )\in \cap _{T>0}{{\mathbb {S}}}^{2-\alpha }_T(\rho _\eta )\times {{\mathbb {S}}}^{3-2\alpha }_T(\rho _{2\eta }) \end{aligned}$$

be the paracontrolled solution of PDE (A.4). That is,

$$\begin{aligned} u=\nabla u\prec \!\!\!\prec {{\mathscr {I}}}b+u^\sharp , \end{aligned}$$

(A.5)

with $u^\sharp$ solving the following PDE in weak sense

$$\begin{aligned} {{\mathscr {L}}}u^\sharp&=\nabla u\prec b-\nabla u\prec \!\!\!\prec b+\nabla u\succ b+b\circ \nabla u\nonumber \\ {}&\quad +{\bar{b}}\cdot \nabla u+hu-[{{\mathscr {L}}}, \nabla u\prec \!\!\!\prec ]{{\mathscr {I}}}b, \end{aligned}$$

(A.6)

where

$$\begin{aligned} b\circ \nabla u&=b\circ (\nabla ^2 u\prec {{\mathscr {I}}}b)+(b\circ \nabla {{\mathscr {I}}}b)\cdot \nabla u+\text {com}\nonumber \\&\quad +\text {com}_1+b\circ \nabla u^\sharp , \end{aligned}$$

(A.7)

and

$$\begin{aligned} \text {com}_1:=b\circ \nabla [\nabla u\prec \!\!\!\prec {{\mathscr {I}}}b-\nabla u\prec {{\mathscr {I}}}b] \end{aligned}$$

and

$$\begin{aligned} \text {com}:=\mathrm {com}(\nabla u, \nabla {{\mathscr {I}}}b,b). \end{aligned}$$

Theorem A.2

Let $\ell \in (0,1)$ and $\kappa \in (0,\tfrac{(2-3\alpha )\ell }{6})$, $\eta \in (0,\frac{(1-\alpha )\ell }{2})$. Suppose that

$$\begin{aligned}&b\in \cap _{T>0}{{\mathbb {B}}}^\alpha _T(\rho _\kappa ),\ \ {\bar{b}},h\in \cap _{T>0}{{\mathbb {L}}}^\infty _T(\rho _\eta ),\\&\beta \in (\alpha ,(2-2\alpha -\tfrac{6\kappa }{\ell })\wedge (1-\tfrac{2\eta }{\ell })),\ \ \gamma \in (\alpha ,2-2\alpha -\tfrac{4\kappa }{\ell }). \end{aligned}$$

The unique paracontrolled solution to PDE (A.4) in the sense of Definition 3.1 with

$$\begin{aligned} (u,u^\sharp )\in {{\mathbb {S}}}_T^{\gamma +\alpha }(\mathbf{e}^\ell )\times L_T^\infty {{\mathbf {C}}}^{\beta +1}(\mathbf{e}^\ell ) \end{aligned}$$

is zero.

Proof

Let $T>0$. Choose q large enough such that

$$\begin{aligned}\alpha<\gamma \leqslant 2-2\alpha -\tfrac{2}{q}-\tfrac{4\kappa }{\ell },\ \ \alpha <\beta \leqslant (2-2\alpha -\tfrac{2}{q}-\tfrac{6\kappa }{\ell })\wedge (1-\tfrac{2\eta }{\ell }). \end{aligned}$$

First of all, by Lemmas A.1 and 2.10, we have

$$\begin{aligned} \begin{aligned}&\Vert u\Vert _{{{\mathbb {S}}}_T^{2-\alpha -\frac{2}{q}-\frac{4\kappa }{\ell }}(\mathbf{e}^\ell )} \\ {}&\lesssim \Vert b\prec \nabla u+b\succ \nabla u+b\circ \nabla u\Vert _{L^q_T{{\mathbf {C}}}^{-\alpha }(\rho _{2\kappa }{} \mathbf{e}^\ell )}+\Vert {\bar{b}}\cdot \nabla u+hu\Vert _{L_T^qL^\infty (\rho _{\eta }{} \mathbf{e}^\ell )} \\ {}&\lesssim \Vert b\Vert _{L^\infty _T{{\mathbf {C}}}^{-\alpha }(\rho _\kappa )}\Vert \nabla u\Vert _{L^q_TL^\infty (\mathbf{e}^\ell )} +\Vert b\circ \nabla u\Vert _{L^q_T{{\mathbf {C}}}^{-\alpha }(\rho _{2\kappa }{} \mathbf{e}^\ell )} \\ {}&+\Vert {\bar{b}}\Vert _{{{\mathbb {L}}}_T^\infty (\rho _\eta )}\Vert \nabla u\Vert _{L_T^qL^\infty (\mathbf{e}^\ell )}+\Vert h\Vert _{{{\mathbb {L}}}_T^\infty (\rho _\eta )}\Vert u\Vert _{L_T^qL^\infty (\mathbf{e}^\ell )}, \end{aligned} \end{aligned}$$

and by the corresponding version of Lemma 2.12 for exponential weight $\mathbf{e}^\ell$ (see [45, Lemma 2.10]),

$$\begin{aligned} \begin{aligned} \Vert u^\sharp \Vert _{L_T^\infty {{\mathbf {C}}}^{\beta +1}(\mathbf{e}^\ell )}&\lesssim \Vert \nabla u\prec b-\nabla u\prec \!\!\!\prec b+\nabla u\succ b-[{{\mathscr {L}}}, \nabla u\prec \!\!\!\prec ]{{\mathscr {I}}}b\Vert _{L^\infty _T{{\mathbf {C}}}^{1-2\alpha -\frac{2}{q}-\frac{4\kappa }{\ell }}(\rho _\kappa \mathbf{e}^\ell )}\\&\quad +\Vert b\circ \nabla u\Vert _{L^q_T{{\mathbf {C}}}^{1-2\alpha }(\rho _{2\kappa }\mathbf{e}^\ell )}+\Vert {\bar{b}}\cdot \nabla u+hu\Vert _{{{\mathbb {L}}}_T^\infty (\rho _{\eta }\mathbf{e}^\ell )} \\&\lesssim \Vert b\circ \nabla u\Vert _{L^q_T{{\mathbf {C}}}^{1-2\alpha }(\rho _{2\kappa }\mathbf{e}^\ell )}+\Vert u\Vert _{{{\mathbb {S}}}_T^{2-\alpha -\frac{2}{q}-\frac{4\kappa }{\ell }}(\mathbf{e}^\ell )} \\ {}&\quad +\Vert {\bar{b}}\Vert _{{{\mathbb {L}}}_T^\infty (\rho _\eta )}\Vert \nabla u\Vert _{{{\mathbb {L}}}_T^\infty (\mathbf{e}^\ell )}+\Vert h\Vert _{{{\mathbb {L}}}_T^\infty (\rho _\eta )}\Vert u\Vert _{{{\mathbb {L}}}_T^\infty (\mathbf{e}^\ell )} \\ {}&\lesssim \Vert u\Vert _{{{\mathbb {S}}}_T^{2-\alpha -\frac{2}{q}-\frac{4\kappa }{\ell }}(\mathbf{e}^\ell )}+\Vert b\circ \nabla u\Vert _{L^q_T{{\mathbf {C}}}^{1-2\alpha }(\rho _{2\kappa }{} \mathbf{e}^\ell )}. \end{aligned}\end{aligned}$$

Moreover, by Lemma 3.3 with $(\rho ,{\bar{\rho }})=(\rho _\kappa ,\mathbf{e}^\ell _t)$,

$$\begin{aligned} \Vert (b\circ \nabla u)(t)\Vert _{{{\mathbf {C}}}^{1-2\alpha }(\rho _{2\kappa }\mathbf{e}^\ell _t)} \lesssim \Vert u\Vert _{{{\mathbb {S}}}^{\gamma +\alpha }_t(\mathbf{e}^\ell )}+\Vert u^\sharp (t)\Vert _{{{\mathbf {C}}}^{\beta +1}(\rho _\kappa \mathbf{e}^\ell _t)}. \end{aligned}$$

Combining the above three estimates, we obtain

$$\begin{aligned}&\Vert u\Vert _{{{\mathbb {S}}}_T^{\gamma +\alpha }(\mathbf{e}^\ell )}+\Vert u^\sharp \Vert _{L_T^\infty {{\mathbf {C}}}^{\beta +1}(\mathbf{e}^\ell )} \\ {}&\lesssim \Vert \nabla u\Vert _{L^q_TL^\infty (\mathbf{e}^\ell )} +\Vert u\Vert _{L_T^qL^\infty (\mathbf{e}^\ell )}+\Vert b\circ \nabla u\Vert _{L^q_T{{\mathbf {C}}}^{1-2\alpha }(\rho _{2\kappa }{} \mathbf{e}^\ell )}\\&\lesssim \left( \int ^T_0\Big (\Vert u\Vert ^q_{{{\mathbb {S}}}^{\gamma +\alpha }_t(\mathbf{e}^\ell )}+\Vert u^\sharp (t)\Vert ^q_{{{\mathbf {C}}}^{\beta +1}(\rho _\kappa \mathbf{e}^\ell _t)}\Big ){\mathord {\mathrm{d}}}t\right) ^{1/q}, \end{aligned}$$

which implies $u\equiv 0$ by Gronwall’s inequality. $\square$

Appendix B: Exponential moment estimates for SDEs

In this section we consider the following SDE:

$$\begin{aligned} {\mathord {\mathrm{d}}}X_t=b(t,X_t){\mathord {\mathrm{d}}}t+\sigma (t,X_t){\mathord {\mathrm{d}}}W_t, \ X_0=x. \end{aligned}$$

We have the following exponential moment estimates for $X_t$.

Lemma B.1

Suppose that $\sigma$ is bounded and b is linear growth. Then for any $\alpha \in [0,2)$ and $T, \gamma >0$, there is a constant $C>0$ such that for all $x\in {{\mathbb {R}}}^d$,

$$\begin{aligned} {{\mathbb {E}}}\mathrm {e}^{\gamma \sup _{t\in [0,T]} \langle X_t\rangle ^\alpha }\leqslant C\mathrm {e}^{\langle x\rangle ^\alpha }. \end{aligned}$$

Proof

Let $\beta \in (\alpha ,2)$. Recall $\langle x\rangle ^\beta =(1+|x|^2)^{\beta /2}$. By Itô’s formula, we have

$$\begin{aligned} M_t:=\mathrm {e}^{-\lambda t}\langle X_t\rangle ^\beta&=\langle x\rangle ^\beta +\int ^t_0 \eta _s{\mathord {\mathrm{d}}}s+\int ^t_0\xi _s{\mathord {\mathrm{d}}}W_s, \end{aligned}$$

where

$$\begin{aligned} \eta _s&:=\mathrm {e}^{-\lambda s}\beta \Big [X_s\cdot b(s,X_s)+ \mathrm {tr}(\sigma \sigma ^*)(s,X_s)/2\Big ]\langle X_s\rangle ^{\beta -2}\\&\quad +\beta (\tfrac{\beta }{2}-1)\mathrm {e}^{-\lambda s}|\sigma ^*(s,X_s) X_s|^2 \langle X_s\rangle ^{\beta -4}-\lambda \mathrm {e}^{-\lambda s} \langle X_s\rangle ^\beta , \end{aligned}$$

and

$$\begin{aligned} \xi _s:=\beta \mathrm {e}^{-\lambda s}\sigma ^*(s,X_s)X_s\langle X_s\rangle ^{\beta -2}. \end{aligned}$$

By the linear growth of b and the boundedness of $\sigma$, there is a $\lambda$ large enough so that

$$\begin{aligned} \eta _s\leqslant 0 \end{aligned}$$

and

$$\begin{aligned} |\xi _s|^2\leqslant C\mathrm {e}^{-\lambda s}\langle X_s\rangle ^{2(\beta -1)}\leqslant C M^{2-\frac{2}{\beta }}_s. \end{aligned}$$

Now by [33, Theorem 1.1], we obtain the desired estimate. $\square$