The W ^(p,q)_1,2 -solvability for a class of fully nonlinear parabolic equations

Abstract

The solvability in the Sobolev-Lorentz spaces $W^{1,2}_{(p,q)}(\Omega _{T})$ with $p>d+1$ and $q>0$ is proved for a class of fully nonlinear parabolic equations with small BMO nonlinearities in (x, t)-variables over a bounded parabolic domain with $C^{1,1}$-smooth lateral boundary. Here, we make use of the unified approach based on the Fefferman-Stein theorem in accordance with almost all pointwise estimate of the sharp functions to establish the estimates of $D^2u$ and $D_tu$ in the framework of Lorentz spaces.

1 Introduction

Nonlinear Calderón-Zygmund theory for elliptic and parabolic partial differential equations have been extensively studied in recent decades. In particular, an interior $W^{2,p}$-estimate with $p>d$ for a class of fully nonlinear elliptic equations, was first obtained by Caffarelli in [6] via the Aleksandrov-Bakel’man-Pucci estimate, the covering argument and the Harnack inequality by using an improvement of Krylov-Safonov’s technique. A similar interior estimate was extended by Wang in [26] to fully nonlinear parabolic equations by adapting the Aleksandrov-Bakel’man-Pucci-Krylov-Tso maximum principle and the compactness method. Later, Escauriaza in [11] further sharpened Caffarelli’s result in [6] to the range $p>d-\varepsilon$ with a small $\varepsilon >0$ depending on the ellipticity constants and d by considering the reverse Hölder inequality. Meanwhile, we also notice that a solvability in $W^{2,1}_{p,loc}(\Omega _T) \cap C(\overline{\Omega }_T)$ for the initial-boundary problem of fully nonlinear parabolic equations was obtained by Crandall, Kocan and Świech in [7]. Winter in [27] further made use of the Alexandroff maximum principle and the weak Harnack inequality on the boundary setting to establish the boundary estimate as well as the $W^{2,p}$-solvability for the Dirichlet boundary problem associated nonlinear elliptic equations. Recently, an interesting work from Krylov [14] provided a unified approach to show the $L^p$-solvability for nondivergence elliptic and parabolic equations under the regular assumption of $VMO_{x}$ principle coefficients. This approach mainly relies on the pointwise estimates of sharp functions for the spatial derivatives of solutions due to the Fefferman-Stein theorem. Furthermore, such an approach was developed by Dong, Kim and Krylov to attain the $W^{2,p}$-solvability for fully nonlinear elliptic and parabolic equations with the nonlinearities being VMO in independent variables. For examples, Krylov [15] showed the $W^{2,p}$-solvability with $p>d$ to the Bellman’s equations with the VMO nonlinearity by his unified argument. Dong-Krylov-Li [8] demonstrated an interior $W^{2,p}$-solvability for $p>d$ and $W^{2,1}_{p}$-solvability for $p>d+1$, respectively, to fully nonlinear elliptic and parabolic equations. More generally, Krylov [16] studied the existence and uniqueness in $W^{2,p}$ for $p>d$ to the strong solution of fully nonlinear elliptic equations $H(u,Du,D^{2}u,x)=0$. Dong and Krylov in [9] proved the solvability in $W^{2,1}_{p}$ for $p>d+1$ to the strong solutions of fully nonlinear parabolic equations $\partial _{t} u(t,x)+H(u,Du,D^{2}u;t,x)=0$ under some relaxed convexity assumptions instead of requiring H to be convex or concave with respect to $D^{2}u$. Very recently, Dong and Krylov [10] also derived the regularity in the mixed-norm Sobolev spaces to fully nonlinear elliptic and parabolic equations by improving the Fefferman-Stein theorem in the mixed-norm weighted Lebesgue spaces.

This paper is actually a continuation of Dong-Krylov-Li’s work in [8], which extends it in two folds: the Fefferman-Stein theorem in the Lorentz spaces and Lorentz regularity of fully nonlinear parabolic equations. Here, the Fefferman-Stein inequality in the Lorentz spaces is also inspired by Dong and Krylov’s paper [10] involving that in the mixed-norm form. Our main aim is to attain a global estimate in the Sobolev-Lorentz spaces by using a unified approach to fully nonlinear parabolic equations with the nonlinearities being small BMO oscillation in independent variable in a bounded $C^{1,1}$ parabolic domain. To state our problem under consideration more precisely, let us recall some related notations. Let $\Omega$ be a bounded domain of $\mathbb {R}^{d}$ for $d\ge 2$ with $C^{1,1}$-smooth boundary, and $\Omega _{T}:= \Omega \times (0,T)$ for $T>0$ with its usual parabolic boundary $\partial \Omega _{T} := \Big ( \partial \Omega \times (0,T] \Big ) \cup \Big ( \Omega \times \{0\} \Big )$. We consider the following zero Cauchy-Dirichlet problem of a fully nonlinear parabolic equation:

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle { \partial _{t} u(x,t) + H(D^{2}u, Du, u, x,t) = 0 } &{} \quad \text {in} \,\,\, \Omega _{T}, \\ u = 0 &{} \quad \text {on} \,\,\, \partial \Omega _{T}, \end{array} \right. \end{aligned}$$

(1.1)

where $D^{2} u = \Big ( D_{ij} u\Big )_{d\times d}$ denotes the Hessian matrix of u, and $D u=\Big ( D_{i} u \Big )_{d\times 1}$ denotes its gradient.

As usual, we use $\mathring{W}^{1,\,2}_{2}(\Omega _{T})$ to denote the set of all functions $u\in L^2(\Omega _{T})$ that their weak derivatives $Du,D^2u,D_tu$ belong to $L^2$-spaces, and u vanishes on the parabolic boundary $\partial \Omega _{T}$ in the trace sense. Therefore, a strong solution of (1.1) which is treated throughout this paper is a Sobolev function $u \in \mathring{W}^{1,\,2}_{2}(\Omega _{T})$ satisfying (1.1) almost everywhere in $\Omega _{T}$, for more details to see [18, Chapter VII]. In this context, we suppose that the nonlinearity $H(D^{2}u, Du, u, x,t)$ can be decomposed into following two parts:

$$\begin{aligned} H(D^{2}u, Du, u, x,t) = F(D^{2}u, x,t) + G(D^{2}u, Du, u, x,t). \end{aligned}$$

Let ${\mathcal {S}}$ be the set of all symmetric $d\times d$-matrices, and they satisfy the following structural assumptions:

H1. We suppose that $F(u'',x,t)$ is convex and positive homogeneous of degree one with respect to $u'' \in {\mathcal {S}}$, and there exist two constants $0<\lambda \le \Lambda < \infty$ such that

$$\begin{aligned} \lambda \, |\xi |^{2} \le F(u''+ \xi \xi ^{*}, x,t) - F(u'', x,t) \le \Lambda \, |\xi |^{2} \end{aligned}$$

for any $u'' \in {\mathcal {S}}$, $\xi \in \mathbb {R}^{d}$ and each $(x,t)\in \Omega _{T}$.

H2. For any fixed $u''\in {\mathcal {S}}$, $u'\in \mathbb {R}^{d}$ and $(x,t)\in \Omega _T$, we let $G(u'', u', u, x,t)$ be a monotone nonincreasing function in $u\in \mathbb {R}$. Moreover, there exists a positive constant K such that

$$\begin{aligned} \left| G(u'',u',u,x,t) - G(u'',v',v,x,t) \right| \le K \, \Big ( |u'-v'| + |u-v| \Big ) \end{aligned}$$

and

$$\begin{aligned} \left| G(u'',u',u,x,t)\right| \le \psi \left( |u''| \right) \, |u''| + K \, \Big ( |u'| + |u| \Big ) + g(x,t) \end{aligned}$$

for any $u'' \in {\mathcal {S}}$, $u',v'\in \mathbb {R}^{d}$, $u,v\in \mathbb {R}$ and each $(x,t)\in \Omega _{T}$. In the above, the function $\psi (\rho ): \overline{\mathbb {R}^{+}} \rightarrow \overline{\mathbb {R}^{+}}$ is bounded, monotonically decreasing and $\psi (\rho ) \rightarrow 0$ as $\rho \rightarrow 0$, while g(x, t) is a given function specified later.

Before stating our main result, it is necessary to introduce some notations. For any $r>0$ and $(x,t)\in \mathbb {R}^{d+1}$, we denote

$$\begin{aligned} B_{r}(x) = \left\{ y \in \mathbb {R}^{d}: \,\, |x-y|<r \right\} \quad \text {and} \quad Q_{r}(x,t) = B_{r}(x) \times (t, \, t+r^{2}). \end{aligned}$$

For convenience, we write $B_{r}=B_{r}(0)$ and $Q_{r}=Q_{r}(0,0)$ in the following context. An average of f(x, t) over $Q_{r}$ is denoted by

$$\begin{aligned} \bar{f}_{Q_{r}} = \fint _{Q_{r}} f(x,t) \, dxdt = \frac{1}{|Q_{r}|} \int _{Q_{r}} f(x,t) \, dxdt, \end{aligned}$$

where $|Q_{r}|$ is $(d+1)$-dimensional Lebesgue measure of $Q_{r}$. To our aim, we need to impose the following $(\delta , R_{0})$-vanishing assumption on the leading term in (1.1) for some $\delta$ and $R_{0}>0$.

Assumption 1.1

For any $u''\in S$ with $|u''|=1$, we say that $F(u'',x,t)$ is $(\delta , R_{0})$-vanishing of (x, t). If there exists $R_{0}>0$ and a function $\widetilde{F}(u'')$ (independent of $z=(x,t)$) satisfying (H1), such that for any $r\in (0, R_{0}]$ and $(s,y) \in \overline{\Omega }_{T}$, we have

$$\begin{aligned} \sup _{u''\in S, \,\,|u''|=1} \fint _{Q_{r}(s,y)} \left| F(u'', x,t) - \widetilde{F}(u'') \right| \, dxdt \le \delta , \end{aligned}$$

where $F(u'', x,t)$ is zero extended from $Q_{r}(s,y) \cap \Omega _{T}$ to $Q_{r}(s,y)$ for (x, t) close to the parabolic boundary, and the parameter $\delta >0$ will be specified later.

We refer to [8] for the same assumptions as H1, H2 and Assumption 1.1, where Dong, Krylov and Li established the $L^{p}$ estimate with $p>d+1$ for the zero Cauchy-Dirichlet problems (1.1) in a bounded $C^{1,1}$-smooth domain. In the article, our aim is to extend it to a regularity in the scale of Lorentz space which is extremely useful when dealing with borderline existence as a refined version of Lebesgue spaces. So, it is necessary to recall the Lorentz space and Sobolev-Lorentz space since our Sobolev-Lorentz regularity for solutions is taken into consideration. The Lorentz space is a two-parameter scale of spaces which make the classical Lebesgue spaces refined in some sense, and the Sobolev-Lorentz space is similarly done from Sobolev space. More precisely, we have

Definition 1.2

• (i)  Let U be an open subset in $\mathbb {R}^{d+1}$. For $p\in [1, \infty )$ and $q\in (0, + \infty )$, the Lorentz space $L^{(p,q)}(U)$ is the set of all measurable functions $f: U \rightarrow \mathbb {R}^{d+1}$ such that

  $$\begin{aligned} \Vert f\Vert _{L^{(p,q)}(U)} := \Bigg ( \, p \int _{0}^{\infty }\Big ( \, \kappa ^{p}|\{(x,t)\in U: |f(x,t)|>\kappa \}| \, \Big )^{\frac{q}{p}}\frac{d\kappa }{\kappa } \, \Bigg )^{\frac{1}{q}} < + \infty . \end{aligned}$$

  (1.2)

  For $q=\infty$, the space $L^{(p,\infty )}(U)$ is set to be the usual Marcinkiewicz space with quasinorm

  $$\begin{aligned} \Vert f\Vert _{L^{(p,\infty )}(U)} := \sup _{\kappa>0} \Big ( \, \kappa ^{p} |\{(x,t)\in U: |f(x,t)|>\kappa \}| \, \Big )^{\frac{1}{p}} < + \infty . \end{aligned}$$

  (1.3)

• (ii)  The Sobolev-Lorentz space $W^{1, \, 2}_{(p,q)}(U)$ is defined by

  $$\begin{aligned} W^{1, \, 2}_{(p,q)}(U) = \left\{ \, f : \,\, f, \, \partial _{t}f, \, Df, \, D^{2}f \, \in \, L^{(p,q)}(U) \, \right\} , \end{aligned}$$

  where the quasinorm of f is

  $$\begin{aligned} \left\| f\right\| _{W^{1, \, 2}_{(p,q)}(U)} := \left\| f\right\| _{L^{(p,q)}(U)} + \left\| \partial _{t}f \right\| _{L^{(p,q)}(U)} + \left\| Df\right\| _{L^{(p,q)}(U)} + \left\| D^{2}f\right\| _{L^{(p,q)}(U)}. \end{aligned}$$

The regularity in the scale of Lorentz spaces concerning partial differential equations was originated from Talenti’s work [23] based on the symmetrization argument. In 2010, Mingione [21] developed some sharp estimates in the Lorentz spaces for nonlinear elliptic equations in divergence form by using the duality exponent. Using the geometric argument first introduced by Byun and Wang [4], Mengesha and Phuc [20] recently obtained a global weighted Lorentz estimate to quasilinear elliptic equations with small BMO coefficients in Reifenberg flat domains. In addition, Baroni [2] showed a local Lorentz estimate for parabolic systems of p-Laplacian type by improving the classic Hardy inequality and the reverse Hölder inequality, and a similar result was also obtained in [3] for parabolic obstacle problems of p-Laplacian type by using the same technique. A global extension in the weighted Lorentz spaces was established by Tian and Zheng in [24] to the zero Cauchy-Dirichlet problems of nonlinear parabolic equations with small BMO nonlinearities in Reifenberg domains. Recently, Bui and Duong [5] got a global weighted Lorentz estimate to the weak solutions of divergence nonlinear parabolic equations with nonstandard growth over Reifenberg domains. Very recently, Tian and Zheng [25] also proved a global estimate of weak solutions in variable Lorentz spaces to parabolic obstacle problems with nonstandard growth over a quasiconvex domain based on the large-M-inequality principle originated from Acerbi and Mingione’s work [1]. Zhang and Zheng in [28] also showed local Lorentz estimates for a class of fully nonlinear parabolic and elliptic equations by using the large-M-inequality principle.

We now summarize a global Sobolev-Lorentz estimate for fully nonlinear parabolic equation (1.1) as follows.

Theorem 1.3

Let $\Omega$ be a bounded $C^{1,1}$-smooth domain and $T>0$. For the Cauchy-Dirichlet problems (1.1) with assumptions H1 and H2, then there exists a small constant $\delta =\delta (d,\lambda ,\Lambda ,p,q,\partial \Omega ) >0$ such that if the $F(u'',x,t)$ is $(\delta , R_{0})$-vanishing shown as Assumption 1.1for some $R_{0}>0$, then the following is true:

• (i)   For $u\in W^{1,2}_{(p,q)}(\Omega _{T})$ satisfying (1.1) with $g \in L^{(p,q)}(\Omega _{T})$ for $p\in (d+1, \, +\infty )$ and $q\in (0, \, +\infty ]$, we have

  $$\begin{aligned} \left\| u \right\| _{W^{1,2}_{(p,q)}(\Omega _{T})} \le N \left( \left\| g \right\| _{L^{(p,q)}(\Omega _{T})} + \bar{\rho } \left\| \psi \right\| _{L^{\infty }} \right) \end{aligned}$$

  (1.4)

  for some $\bar{\rho }>0$, where N is a positive constant depending only on $d,p,q,\lambda ,\Lambda ,\delta ,R_{0},K,T,|\Omega |$ and $\Vert \partial \Omega \Vert _{C^{1,1}}$ while $q<\infty$, and the constant is independent of q while $q=\infty$.

• (ii)   For any $g \in L^{(p,q)}(\Omega _{T})$ with $p\in (d+1, \, +\infty )$ and $q\in (0, \, +\infty ]$, there is a unique solution $u\in W^{1,2}_{(p,q)}(\Omega _{T})$ of (1.1). In particular, if $\psi (0)=0$ in (H2), then there is a unique solution $u\in W^{1,2}_{(p,q)}(\Omega _{T})$ of (1.1) with the estimate

  $$\begin{aligned} \left\| u \right\| _{W^{1,2}_{(p,q)}(\Omega _{T})} \le N \left\| g \right\| _{L^{(p,q)}(\Omega _{T})}. \end{aligned}$$

  (1.5)

It is also worth noting that a solvability theorem in the space $W^{1,2}_{p,loc}(\Omega _T) \cap C(\overline{\Omega }_T)$ can be found in [7] for the initial boundary value problem for a class of fully nonlinear parabolic equations. The solvability in the Sobolev spaces $W^{1,2}_p(\overline{\Omega }_T)$ for $p > d +1$ of the terminal boundary value problem is proved by Dong, Krylov and Li in [8] for such fully nonlinear parabolic equations (1.1), in bounded cylindrical domains, in the case of VMO “coefficients”. Here, we further prove the solvability theorem 1.3 in the scale of Sobolev-Lorentz spaces under the assumptions of H1 and H2 along this line from Dong-Krylov’s argument. A key ingredient of our main proof is that we need to improve the Fefferman-Stein type theorem of sharp functions in the Lorentz spaces. We note that Krylov in [15] gave a modified Fefferman-Stein theorem in Lebesgue spaces by a filtration of partitions, which leads to that the $L^p$-solvability was obtained for Bellman’s equations with VMO coefficients in the whole space. Furthermore, Dong and Krylov [10] improved the Fefferman-Stein theorem in the mixed-norm weighted Lebesgue spaces, which yields a mixed-norm regularity for fully nonlinear elliptic and parabolic equations. Inspired by their works, we are going to establish a variant of the Fefferman-Stein theorem in the Lorentz spaces on the basis of the Calderón-Zygmund decomposition in the parabolic setting. Another important point is that the partitions of unity do not work for global estimates of the solutions in the Lorentz-Sobolev spaces, since the commuting transformation of integral does not hold in the Lorentz spaces. To this end, we make use of the idea of local $L^{p}$-estimate for nondivergence parabolic equations to handle the mean oscillation estimates of Hessian introduced by Krylov’s paper [15], which originated from Lin’s paper [19] for nondivergence elliptic equations.

Let us now consider the following Dirichlet problem of fully nonlinear elliptic equations:

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle { H(D^{2}u, Du, u, x) = 0 } &{} \quad \text {in} \,\,\, \Omega , \\ u = 0 &{} \quad \text {on} \,\,\, \partial \Omega . \end{array} \right. \end{aligned}$$

(1.6)

As a direct consequence of main Theorem , we immediately have the following solvability regarding the elliptic setting.

Corollary 1.4

Let $\Omega$ be a bounded $C^{1,1}$-smooth domain. For the Dirichlet problem (1.6) with assumptions H1 and H2, then there exists a small constant $\delta =\delta (d,\lambda ,\Lambda ,p,q,\partial \Omega ) >0$ such that if the $F(u'',x)$ is $(\delta , R_{0})$-vanishing for some $R_{0}>0$, then the following is true:

• (i)   For $u\in W^{2}_{(p,q)}(\Omega )$ satisfying (1.6) with $g \in L^{(p,q)}(\Omega )$ with $p\in (d, \, +\infty )$ and $q\in (0, \, +\infty ]$, we have

  $$\begin{aligned} \left\| u \right\| _{W^{2}_{(p,q)}(\Omega )} \le N \left( \left\| g \right\| _{L^{(p,q)}(\Omega )} + \bar{\rho } \left\| \psi \right\| _{L^{\infty }} \right) \end{aligned}$$

  (1.7)

  for some $\bar{\rho }>0$, where N is a positive constant depending only on $d,p,q,\lambda ,\Lambda ,\delta ,R_{0},K,|\Omega |$ and $\Vert \partial \Omega \Vert _{C^{1,1}}$ if $q<\infty$, while the constant is not dependent of q if $q=\infty$. Specifically, if $\psi (0)=0$ in (H2), then we have

  $$\begin{aligned} \left\| u \right\| _{W^{2}_{(p,q)}(\Omega )} \le N \left\| g \right\| _{L^{(p,q)}(\Omega )}. \end{aligned}$$

  (1.8)

• (ii)    For any $g \in L^{(p,q)}(\Omega )$ with $p\in (d, \, +\infty )$ and $q\in (0, \, +\infty ]$, there is a unique solution $u\in W^{2}_{(p,q)}(\Omega )$ of (1.6).

The remainder of this paper is organized as follows. In Sect. 2, we present the Fefferman-Stein type theorem in the Lorentz spaces, and some necessary preliminary results concerning the Gagliardo-Nirenberg interpolation inequality and Alexandrov’s estimate in the Lorentz spaces. In Sect. 3 we complete the proof of Theorem 1.3. Moreover, as a corollary of main Theorem 1.3, a global Sobolev-Lorentz estimate of the solution is proved for parabolic Bellman’s equations with small BMO coefficients in a bounded $C^{1,1}$-smooth domain.

2 Preliminary results

This section is devoted to some related notations and basic facts which will be useful in the main proof. Throughout this paper, we denote $N(d,p,q,\lambda ,\Lambda ,\cdots )$ or $N_{i}(d,p,q,\lambda ,\Lambda ,\cdots )$ for $i=0, 1,2,\cdots$, a universal constant depending only on prescribed quantities and possibly varying from line to line. First, we give a generalized version of the Fefferman-Stein theorem of sharp functions in the Lorentz spaces. Let $\mathfrak {C}$ be a set of all ${\mathcal {C}}_{(R^{2}, \, R)}(x,t)$ for any $R>0$ with the fixed point $(x,t) \in \mathbb {R}^{d+1}$. When f is locally integrable on $\mathbb {R}^{d+1}$, we define the parabolic maximal function ${\mathcal {M}}f$ by

$$\begin{aligned} {\mathcal {M}}f(x,t) = \sup _{{\mathcal {C}} \in \mathfrak {C}} \fint _{\mathcal {C}} |f(s,y)| \, dyds. \end{aligned}$$

(2.1)

For $\gamma \in (0,1]$, we set

$$\begin{aligned} f^{\sharp }_{\gamma }(x,t) = \sup _{\mathcal {C} \in \mathfrak {C}} \left( \fint _{\mathcal {C}} \fint _{\mathcal {C}} |f(s,y) - f(\tau ,z)|^{\gamma } \,\, dyds dzd\tau \right) ^{\frac{1}{\gamma }}. \end{aligned}$$

(2.2)

Remark 2.1

While $\gamma =1$, we observe that $f\in BMO$ is equivalent to $f^{\sharp }_{1} \in L^{\infty }$. We also note that $BMO\cong \mathcal {L}^{p,d}$(the Campanato spaces) for any $1\le p< \infty$ (cf. [22, Chapter IV]), which leads to this fact that the norms $\Vert f^{\sharp }_{\gamma }\Vert _{L^{\infty }}$ and $\Vert f^{\sharp }_{1}\Vert _{L^{\infty }}$ are equivalent for any $\gamma \in (0,1]$.

The Hardy-Littlewood maximal function theorem states that if $f\in L^{p}$ for $1<p\le \infty$, then $\mathcal {M}f \in L^{p}$ and $\Vert \mathcal {M}f \Vert _{L^{p}} \le N(p,d) \, \Vert f \Vert _{L^{p}}$, for instance, see [22, Chapter I]. By interpolating argument, Mengesha and Phuc in [20, Lemma 3.11] extended the above boundedness of the Hardy-Littlewood maximal function from the Lebesgue spaces to the weighted Lorentz spaces. Similarly, if $f^{\sharp }_{\gamma } \in L^{p}$ for $1<p<\infty$, then $\mathcal {M}f \in L^{p}$ and $\Vert \mathcal {M}f \Vert _{L^{p}} \le N(d,p,\gamma ) \, \Vert f^{\sharp }_{\gamma }\Vert _{L^{p}}$ as given by Krylov in [15, Theorem 5.3]. Therefore, the interpolation method is also applied to the Fefferman-Stein sharp function in the Lorentz spaces just as what Mengesha and Phuc did. Indeed, for any fixed $1<p<\infty$, we take $p_1 = \frac{p+1}{2}$ to obtain

$$\begin{aligned} \big \Vert \mathcal {M}f \big \Vert _{L^{p_1}(\mathbb {R}^{d+1})} \le N_1 \, \big \Vert f^{\sharp }_{\gamma } \big \Vert _{L^{p_1}(\mathbb {R}^{d+1})} \end{aligned}$$

for any $f^{\sharp }_{\gamma } \in L^{p_1}(\mathbb {R}^{d+1})$; while we take $p_2 = p+1$ to show

$$\begin{aligned} \big \Vert \mathcal {M}f \big \Vert _{L^{p_2}(\mathbb {R}^{d+1})} \le N_2 \, \big \Vert f^{\sharp }_{\gamma } \big \Vert _{L^{p_2}(\mathbb {R}^{d+1})} \end{aligned}$$

for any $f^{\sharp }_{\gamma } \in L^{p_2}(\mathbb {R}^{d+1})$. Note that for any $1<p<\infty$, $1<p_1< p<p_2<+\infty$. By taking $\vartheta =1-\frac{1}{p}$ such that $\frac{1}{p}=\frac{1-\vartheta }{p_1}+\frac{\vartheta }{p_2}$, we use the interpolation theorem for Lorentz spaces (cf. [12, Theorem 1.4.19]) to obtain the following conclusion

$$\begin{aligned} \big \Vert \, \mathcal {M}f \, \big \Vert _{L^{(p,q)}(\mathbb {R}^{d+1})} \le N_0N_1^{1-\vartheta }N^{\vartheta }_2 \, \big \Vert \, f^{\sharp }_{\gamma } \, \big \Vert _{L^{(p,q)}(\mathbb {R}^{d+1})}. \end{aligned}$$

In summary, we have the following conclusion.

Lemma 2.2

Let $1<p<\infty$, $0<q\le \infty$ and $\gamma \in (0,1]$. Then, for any $f^{\sharp }_{\gamma } \in L^{(p,q)}(\mathbb {R}^{d+1})$ there exists a constant N depending only on d, p, q and $\gamma$ such that

$$\begin{aligned} \big \Vert \, \mathcal {M}f \, \big \Vert _{L^{(p,q)}(\mathbb {R}^{d+1})} \le N \, \big \Vert \, f^{\sharp }_{\gamma } \, \big \Vert _{L^{(p,q)}(\mathbb {R}^{d+1})}. \end{aligned}$$

Furthermore, we can also show the improved Fefferman-Stein Theorem (Lemma 2.2) in a half space via the odd or even extensions, see Lemma 5 in [8]. To this end, we write

$$\begin{aligned} \mathbb {R}^{d+1}_{+} = \left\{ \, (x,t) = (x,t^{1}, \cdots ,x^{d}) \in \mathbb {R}^{d+1}: \,\, x^{1}>0 \, \right\} . \end{aligned}$$

Lemma 2.3

Let $1<p<\infty$, $0<q\le \infty$ and $\gamma \in (0,1]$. Then, for any $f^{\sharp }_{\gamma } \in L^{(p,q)}(\mathbb {R}^{d+1}_{+})$ there exists a constant N depending only on d, p, q and $\gamma$ such that

$$\begin{aligned} \big \Vert \, \mathcal {M}f \, \big \Vert _{L^{(p,q)}(\mathbb {R}^{d+1}_{+})} \le \, N \, \big \Vert \, f^{\sharp }_{\gamma } \, \big \Vert _{L^{(p,q)}(\mathbb {R}^{d+1}_{+})}. \end{aligned}$$

Next let us extend the Gagliardo–Nirenberg interpolation inequality from the Lebesgue spaces to the Lorentz spaces. We first recall the conclusion by Dong and Krylov in [10, Lemma 6.5 (ii)]. For a fix $\rho >0$ and $f(x,t)\in L^1_{loc}(\mathbb {R}^{d+1})$, we introduce the confined maximal function and the confined sharp function:

$$\begin{aligned} \mathbb {M}_{\rho }f(x,t)= & {} \sup _{\left\{ \begin{array}{c} \mathcal {C} \in \mathfrak {C} \,: \\ R \in [\rho ,\infty ) \end{array}\right\} } \fint _{\mathcal {C}} |f(s,y)| \, dyds, \\ f^{\sharp }_{1, \,\rho }(x,t)= & {} \sup _{\left\{ \begin{array}{c} \mathcal {C} \in \mathfrak {C} \,: \\ R \in (0,\rho ] \end{array}\right\} } \fint _{\mathcal {C}} \fint _{\mathcal {C}} |f(s,y) - f(\tau ,z)| \, dyds dzd\tau . \end{aligned}$$

Lemma 2.4

For any $\rho >0$, $p\in [1,\infty )$, and $f\in W^{1,2}_{p,\,\text {loc}}(\mathbb {R}^{d+1})$, we have

$$\begin{aligned} \mathbb {M}_{\rho } \left( |Df|^{p} \right) \le N \, \mathbb {M}_{\rho }^{\frac{1}{2}} \left( |D^{2}f|^{p} \right) \mathbb {M}_{\rho }^{\frac{1}{2}} \left( |f|^{p} \right) + N \,\rho ^{-p} \,\mathbb {M}_{\rho } \left( |f|^{p} \right) , \end{aligned}$$

where $N=N(d,p)$ is a positive constant.

Indeed, Lemma 2.4 yields the following interpolation inequality in the Lorentz space.

Lemma 2.5

Let $1<p<\infty$ and $0<q\le \infty$. Then for any $f \in W^{1,\,2}_{(p,q),\,\text {loc}}(\mathbb {R}^{d+1})$, we have

$$\begin{aligned} \left\| \,Df\,\right\| _{L^{(p,q)}(\mathbb {R}^{d+1})} \le N \left( \varepsilon \left\| \,D^{2}f\,\right\| _{L^{(p,q)}(\mathbb {R}^{d+1})} + \varepsilon ^{-1} \left\| \,f\,\right\| _{L^{(p,q)}(\mathbb {R}^{d+1})} \right) \end{aligned}$$

(2.3)

for any $\varepsilon >0$, where $N=N(p,q,d)$ is a positive constant.

Proof

Indeed, we note that for any $\rho >0$,

$$\begin{aligned} (Df)^{\sharp }_{1}(x,t) \le (Df)^{\sharp }_{1, \,\rho }(x,t) + 2 \mathbb {M}_{\rho }(|Df|)(x,t) \quad \text {in} \quad \mathbb {R}^{d+1}. \end{aligned}$$

(2.4)

By the Poincaré inequality we get

$$\begin{aligned} (Df)^{\sharp }_{1, \,\rho }(x,t) \le N \rho \mathcal {M}(|D^{2}f|)(x,t). \end{aligned}$$

Following Lemma 2.4 with $p=1$ it yields

$$\begin{aligned} \mathbb {M}_{\rho }(|Df|)(x,t)\le & {} N \rho \mathcal {M}(|D^{2}f|)(x,t) + N \rho ^{-1} \,\mathcal {M}\left( |f| \right) (x,t). \end{aligned}$$

Putting the above two estimates into (2.4) deduces

$$\begin{aligned} (Df)^{\sharp }_{1}(x,t) \le N \rho \mathcal {M}(|D^{2}f|)(x,t) + N \rho ^{-1} \,\mathcal {M}\left( |f| \right) (x,t) \end{aligned}$$

for any $(x,t) \in \mathbb {R}^{d+1}$.

Now, we make use of Lemma 2.2 and the Hardy-Littlewood maximal function theorem in the Lorentz spaces to obtain the required result by taking $\varepsilon =\rho$. This completes the proof. $\square$

In what follows, we consider the Cauchy-Dirichlet problem of linear parabolic equations in nondivergence form:

$$\begin{aligned} \left\{ \begin{array}{ll} \partial _{t} u + Lu := \partial _{t} u + a^{ij}(x,t) D_{ij} u + b^{i}(x,t)D_{i}u - c(x,t)u = \varphi (x,t) &{} \quad \text {in} \quad \Omega _{T}, \\ u=0 &{} \quad \text {on} \quad \partial \Omega _{T}. \end{array} \right. \end{aligned}$$

(2.5)

Here, we suppose that all coefficients $a^{ij}(x,t)$, $b^{i}(x,t)$ and c(x, t) are bounded measurable functions defined on $\Omega _{T}$ and the symmetric coefficient matrix $A(x,t)=\Big (a^{ij}(x,t)\Big )$ satisfies an uniformly ellipticity, which means that there exist positive constants $0<{\lambda }\le {\Lambda }< \infty$ such that

$$\begin{aligned} |b^{i}(x,t)| \le {\Lambda }, \quad i=1, \cdots , d; \quad |c(x,t)|\le {\Lambda } \end{aligned}$$

(2.6)

and

$$\begin{aligned} {\lambda } |\xi |^{2} \le A(x,t) \, \xi \xi ^{*} \le {\Lambda } |\xi |^{2} \end{aligned}$$

(2.7)

for any $\xi \in \mathbb {R}^{d}$ and almost all $(x,t)\in \Omega _{T}$.

In the following we recall the classical Aleksandrov estimate to the Cauchy-Dirichlet problem (2.5), see [13, Lemma 3].

Lemma 2.6

Let $p\ge d+1$. Assume that $u \in \mathring{W}^{1,\, 2}_{p}(\Omega _{T})$ is a solution of (2.5) with (2.6) and (2.7). If $\varphi (x,t) \in L^{p}(\Omega _{T})$ and $c(x,t) \ge 0$ in $\Omega _{T}$, then

$$\begin{aligned} {\text {ess}\sup }_{(x,t) \in \Omega _{T}}|u(x,t)| \le N \Big \Vert |c|^{\frac{d+1-p}{p}} |\det A|^{\frac{1}{p}} \varphi \, \Big \Vert _{L^{p}(\Omega _{T})}, \end{aligned}$$

where $N=N(d,p,{\lambda },{\Lambda },|\Omega |)$.

Combining Lemma 2.6 with the imbedding inequality in the Lorentz spaces, it yields the following $L^{(p,q)}$-estimate to the problem (2.5).

Lemma 2.7

For $d+1<p<\infty$ and $0<q\le \infty$, let $u \in \mathring{W}^{1,\, 2}_{(p,q)}(\Omega _{T})$ be the solution of (2.5) with (2.6) and (2.7). If $\varphi (x,t) \in L^{(p,q)}(\Omega _{T})$ and $c(x,t) \ge 0$ in $\Omega _{T}$, then

$$\begin{aligned} \left\| \,u\,\right\| _{L^{(p,q)}(\Omega _{T})} \le N \left\| \,\varphi \,\right\| _{L^{(p,q)}(\Omega _{T})}, \end{aligned}$$

(2.8)

where $N=N(d,p,q,{\lambda },{\Lambda },T,|\Omega |)$.

Proof

We first use Lemma 2.6 with $p=d+1$, and the assumptions (2.6, 2.7) to obtain

$$\begin{aligned} |u(x,t)| \le N \left\| \varphi \, \right\| _{L^{d+1}(\Omega _{T})} \end{aligned}$$

for almost all $(x,t) \in \Omega _{T}$. Furthermore, by the imbedding inequality in the Lorentz spaces (cf. [12]), for any $p>d+1$ it yields

$$\begin{aligned} \left\| \varphi \right\| _{L^{d+1}(\Omega _{T})} \le N \, |\Omega _{T}|^{\frac{1}{d+1}-\frac{1}{p}} \left\| \varphi \right\| _{L^{(p,q)}(\Omega _{T})}, \end{aligned}$$

where $N=N(p,q,d)$. Putting the above two estimates together leads to the required result. This completes the proof. $\square$

3 Proof of Theorem 1.3

This section is devoted to our main proof. We first consider the following parabolic equation

$$\begin{aligned} \partial _{t} u(x,t) + F(D^{2}u, x,t) =h(x,t) \qquad \text {in} \quad \mathbb {R}^{d+1}. \end{aligned}$$

(3.1)

Hereafter, we suppose that Assumption 1.1 holds with $\mathbb {R}^{d+1}$ instead of $\Omega _{T}$. To obtain the Sobolev-Lorentz estimate of (3.1) in the whole space, for any fixed point $(t_{0}, x_{0})\in \mathbb {R}^{d+1}$, we begin with establishing the following mean oscillation estimate on $Q_{r}(t_{0},x_{0})$ for $r>0$.

Lemma 3.1

For $\kappa \ge 2$, $r>0$, $\beta \in (1, \infty )$ and $\alpha \in (0,1)$, let $u \in W^{1,\, 2}_{\beta (d+1)}(\mathbb {R}^{d+1})$ be the solution of (3.1). Then, there exists a small constant $\delta _0=\delta _0(d,\lambda ,\Lambda ,\alpha ,\beta ,\gamma ,R_0)>0$ with some $R_{0}>0$ such that the following holds: if Assumption1.1holds for any $\delta \in (0,\delta _0)$, then we have

$$\begin{aligned}&\fint _{Q_{r}(t_{0}, x_{0})} \fint _{Q_{r}(t_{0}, x_{0})} \left| D^{2}u(x,t) - D^{2}u(s,y)\right| ^{\gamma } \, dxdtdyds \nonumber \\&\quad \le N \kappa ^{d+2} \left( \fint _{Q_{\kappa r}(t_{0}, x_{0})}|h|^{d+1} \, dxdt \right) ^{\frac{\gamma }{d+1}}\nonumber \\&\qquad + N \kappa ^{d+2} \delta ^{\frac{\gamma }{(d+1)\, \beta '}} \left( \fint _{Q_{\kappa r}(t_{0}, x_{0})} |D^{2}u|^{(d+1) \, \beta } \, dxdt \right) ^{\frac{\gamma }{(d+1)\, \beta }} \nonumber \\&\qquad + N \kappa ^{- \alpha \gamma } \left( \fint _{Q_{\kappa r}(t_{0}, x_{0})} |D^{2}u|^{d+1} \, dxdt \right) ^{\frac{\gamma }{d+1}} \nonumber \\&\qquad + N \kappa ^{d+2} R_{0}^{-2\gamma } \left( \fint _{Q_{\kappa r}(t_{0}, x_{0})} |u|^{d+1} \, dxdt\right) ^{\frac{\gamma }{d+1}} \nonumber \\&\qquad + N \kappa ^{d+2} R_{0}^{-\gamma } \left( \fint _{Q_{\kappa r}(t_{0}, x_{0})} |Du|^{d+1} \, dxdt\right) ^{\frac{\gamma }{d+1}} \end{aligned}$$

(3.2)

for some $\gamma \in (0,1]$, where $\beta ' = {\beta }/{(\beta -1)}$ and $N=N(d,\delta ,\lambda ,\Lambda ,\beta )$.

Proof

The proof is divided into two folds.

Case 1. If $0 < \kappa r \le R_{0}$, in the Assumption 1.1 we take

$$\begin{aligned} \widetilde{F}(D^{2}u) = \bar{F}_{Q_{\kappa r}(t_{0}, x_{0})} (D^{2}u). \end{aligned}$$

Consider

$$\begin{aligned} \partial _{t} u(x,t) + \widetilde{F}(D^{2}u) = \tilde{h}(x,t) \qquad \text {in} \quad \mathbb {R}^{d+1}, \end{aligned}$$

(3.3)

where

$$\begin{aligned} \tilde{h}(x,t) = h(x,t) + \widetilde{F}(D^{2}u) - F(D^{2}u, x,t). \end{aligned}$$

Let us take $\displaystyle {\hat{u}'' = \frac{D^{2}u}{|D^{2}u|}}$, then $\hat{u}''\in S$ and $|\hat{u}''|=1$. With (H1) we get

$$\begin{aligned}&\fint _{Q_{\kappa r}(t_{0}, x_{0})} \left| F(D^{2}u, x,t) - \widetilde{F}(D^{2}u) \right| ^{d+1} \, dxdt \nonumber \\&\quad \le \sup _{u''\in S, \,\, |u''|=1} \fint _{Q_{\kappa r}(t_{0}, x_{0})} \left| F(u'', x,t) - \widetilde{F}(u'') \right| ^{d+1} |D^{2}u|^{d+1} \, dxdt \nonumber \\&\quad \le N \delta ^{\frac{1}{\beta '}} \left( \fint _{Q_{\kappa r}(t_{0}, x_{0})} |D^{2}u|^{(d+1)\beta } \, dxdt \right) ^{\frac{1}{\beta }}, \end{aligned}$$

where $\delta$ is a small positive constant shown as in Assumption 1.1 and $N=N(d, \lambda , \Lambda )$. It combined with Lemma 5.6 in [8] yields

$$\begin{aligned}&\fint _{Q_{r}(t_{0}, x_{0})} \fint _{Q_{r}(t_{0}, x_{0})} \left| D^{2}u(x,t) - D^{2}u(s,y)\right| ^{\gamma } \, dxdtdyds \nonumber \\&\quad \le N \kappa ^{d+2} \left( \fint _{Q_{\kappa r}(t_{0}, x_{0})} |h|^{d+1} \, dxdt \right) ^{\frac{\gamma }{d+1}}\nonumber \\&\qquad + N \kappa ^{d+2} \delta ^{\frac{\gamma }{(d+1)\, \beta '}} \left( \fint _{Q_{\kappa r}(t_{0}, x_{0})} |D^{2}u|^{(d+1) \, \beta } \, dxdt \right) ^{\frac{\gamma }{(d+1)\, \beta }} \nonumber \\&\qquad + N \kappa ^{- \alpha \gamma } \left( \fint _{Q_{\kappa r}(t_{0}, x_{0})} |D^{2}u|^{d+1} \, dxdt \right) ^{\frac{\gamma }{d+1}} \end{aligned}$$

(3.4)

for some $\gamma \in (0,1]$.

Case 2. If $\kappa r > R_{0}$, we remark that

$$\begin{aligned}&\fint _{Q_{r}(t_{0}, x_{0})} \fint _{Q_{r}(t_{0}, x_{0})} \left| D^{2}u(x,t) - D^{2}u(s,y)\right| ^{\gamma } \, dxdtdyds \nonumber \\&\quad \le 2 \fint _{Q_{r}(t_{0}, x_{0})} |D^{2}u(x,t)|^{\gamma } \, dxdt. \end{aligned}$$

(3.5)

It is a well-known fact that condition (H1) implies that there is an operator $L_{0} = a^{ij}(x,t) D_{ij}$ with assumption (2.7) such that $\partial _{t} u(x,t) + F(D^{2}u, x,t) = \partial _{t} u(x,t) + L_{0}u(x,t)$. Therefore, we use Corollary 4.2 in [15] with dilations and the standard cut-off argument to obtain

$$\begin{aligned} \fint _{Q_{\kappa r/2}(t_{0}, x_{0})} |D^{2}u(x,t)|^{\gamma } \, dxdt\le\, &\, {} N \Bigg ( \left( \fint _{Q_{\kappa r}(t_{0}, x_{0})} |h|^{d+1} \, dxdt\right) ^{\frac{\gamma }{d+1}} \nonumber \\&+ \frac{1}{(\kappa r)^{2\gamma }} \left( \fint _{Q_{\kappa r}(t_{0}, x_{0})} |u|^{d+1} \, dxdt\right) ^{\frac{\gamma }{d+1}} \\&+ \frac{1}{(\kappa r)^{\gamma }} \left( \fint _{Q_{\kappa r}(t_{0}, x_{0})} |Du|^{d+1} \, dxdt\right) ^{\frac{\gamma }{d+1}} \Bigg ) \end{aligned}$$

for some $\gamma \in (0,1)$. With (3.5) and $\kappa \ge 2$ it leads to that

$$\begin{aligned}&\fint _{Q_{r}(t_{0}, x_{0})} \fint _{Q_{r}(t_{0}, x_{0})} \left| D^{2}u(x,t) - D^{2}u(s,y)\right| ^{\gamma } \, dxdtdyds \nonumber \\&\quad \le N \kappa ^{d+2} \Bigg ( \left( \fint _{Q_{\kappa r}(t_{0}, x_{0})} |h|^{d+1} \, dxdt\right) ^{\frac{\gamma }{d+1}} \nonumber \\&\qquad + R_{0}^{-2\gamma } \left( \fint _{Q_{\kappa r}(t_{0}, x_{0})} |u|^{d+1} \, dxdt\right) ^{\frac{\gamma }{d+1}} \nonumber \\&\qquad + R_{0}^{-\gamma } \left( \fint _{Q_{\kappa r}(t_{0}, x_{0})} |Du|^{d+1} \, dxdt\right) ^{\frac{\gamma }{d+1}} \Bigg ). \end{aligned}$$

(3.6)

Putting the above two cases together completes the proof of the lemma. $\square$

Using Lemma 3.1 and the modified Fefferman-Stein Lemma 2.2, we get the following theorem.

Theorem 3.2

For $p\in (d+1, \, +\infty )$ and $q\in (0, \, +\infty ]$, let $u \in W^{1,\, 2}_{(p,q)}(\mathbb {R}^{d+1})$ satisfying (3.1) with $h\in L^{(p,q)}(\mathbb {R}^{d+1})$. Then there exists a small constant $\delta _0 = \delta _0(d,\lambda ,\Lambda ,R_{0}) > 0$ with some $R_{0}>0$ such that if Assumption 1.1holds for any $\delta \in (0,\delta _0)$, then we have

$$\begin{aligned}&\left\| D^{2}u \right\| _{L^{(p,q)}(\mathbb {R}^{d+1})} + \left\| \partial _{t} u \right\| _{L^{(p,q)}(\mathbb {R}^{d+1})} \nonumber \\&\quad \le N \left( \left\| h \right\| _{L^{(p,q)}(\mathbb {R}^{d+1})} + \left\| u \right\| _{L^{(p,q)}(\mathbb {R}^{d+1})} \right) , \end{aligned}$$

(3.7)

where $N=N(d,p,q,\lambda ,\Lambda ,\delta ,R_{0})$.

Proof

By Lemma 3.1 with $\alpha \in (0,1)$, we obtain

$$\begin{aligned}&\left( D^{2}u \right) ^{\sharp }_{\gamma }(t_{0}, x_{0}) \\&\quad \le N \kappa ^{\frac{d+2}{\gamma }} \mathcal {M}^{\frac{1}{d+1}}\left( |h|^{d+1} \right) (t_{0}, x_{0})\nonumber \\&\qquad + N \kappa ^{\frac{d+2}{\gamma }} \delta ^{\frac{1}{(d+1)\, \beta '}} \mathcal {M}^{\frac{1}{(d+1)\, \beta }}\left( |D^{2}u|^{(d+1) \, \beta } \right) (t_{0}, x_{0}) \\&\qquad + N \kappa ^{- \alpha } \mathcal {M}^{\frac{1}{d+1}}\left( |D^{2}u|^{d+1} \right) (t_{0}, x_{0})\\&\qquad + N \kappa ^{\frac{d+2}{\gamma }}R_{0}^{-2} \mathcal {M}^{\frac{1}{d+1}}\left( |u|^{d+1} \right) (t_{0}, x_{0}) \\&\qquad + N \kappa ^{\frac{d+2}{\gamma }}R_{0}^{-1} \mathcal {M}^{\frac{1}{d+1}}\left( |Du|^{d+1} \right) (t_{0}, x_{0}) \end{aligned}$$

for some $\gamma \in (0,1]$. Therefore, with the modified Fefferman-Stein Lemma 2.2 we get

$$\begin{aligned} \left\| D^{2}u \right\| _{L^{(p,q)}(\mathbb {R}^{d+1})}\le & {} N \left( \kappa ^{\frac{d+2}{\gamma }} \delta ^{\frac{1}{(d+1)\, \beta '}} + \kappa ^{- \alpha } \right) \left\| D^{2}u \right\| _{L^{(p,q)}(\mathbb {R}^{d+1})} \nonumber \\&+ N \kappa ^{\frac{d+2}{\gamma }} \left( \left\| h \right\| _{L^{(p,q)}(\mathbb {R}^{d+1})} \right. \nonumber \\&+ \left. \left\| u \right\| _{L^{(p,q)}(\mathbb {R}^{d+1})} + \left\| Du \right\| _{L^{(p,q)}(\mathbb {R}^{d+1})} \right) , \end{aligned}$$

(3.8)

where $N=N(d,p,q,\lambda ,\Lambda ,\delta ,R_{0})$. Now, let us first take $\kappa >0$ being big number, and then take small $\delta >0$ such that

$$\begin{aligned} N \left( \kappa ^{\frac{d+2}{\gamma }} \delta ^{\frac{1}{(d+1)\, \beta '}} + \kappa ^{- \alpha } \right) \le \frac{1}{2}. \end{aligned}$$

Therefore, it follows from the Gagliardo-Nirenberg interpolation inequality (2.3) that

$$\begin{aligned} \left\| D^{2}u \right\| _{L^{(p,q)}(\mathbb {R}^{d+1})}\le & {} N \left( \left\| h \right\| _{L^{(p,q)}(\mathbb {R}^{d+1})} \right. \nonumber \\&\left. + \left\| u\right\| _{L^{(p,q)}(\mathbb {R}^{d+1})} \right) . \end{aligned}$$

(3.9)

On the other hand, we notice that

$$\begin{aligned} \partial _{t} u(x,t) = - F(D^{2}u, x,t) + h(x,t) \qquad \text {in} \quad \mathbb {R}^{d+1}. \end{aligned}$$

By (H1) we have

$$\begin{aligned} \left\| \partial _{t} u \right\| _{L^{(p,q)}(\mathbb {R}^{d+1})}\le & {} N \left\| D^{2}u \right\| _{L^{(p,q)}(\mathbb {R}^{d+1})} + \left\| h \right\| _{L^{(p,q)}(\mathbb {R}^{d+1})}\nonumber \\\le & {} N \left( \left\| h \right\| _{L^{(p,q)}(\mathbb {R}^{d+1})} + \left\| u \right\| _{L^{(p,q)}(\mathbb {R}^{d+1})} \right) . \end{aligned}$$

(3.10)

Combining (3.9) with (3.10) yields (3.7). $\square$

In the following we consider the initial-boundary problem of (3.1) over a half space. We let

$$\begin{aligned} \partial \, \mathbb {R}^{d+1}_{+} = \left\{ \, (x,t) = (x,t^{1}, \cdots ,x^{d}) \in \mathbb {R}^{d+1}: \,\, x^{1}=0 \, \right\} \end{aligned}$$

and consider

$$\begin{aligned} \left\{ \begin{array}{ll} \partial _{t} u(x,t) + F(D^{2}u, x,t) = h(x,t) &{} \quad \text {in} \quad \mathbb {R}^{d+1}_{+}, \\ u=0 &{} \quad \text {on} \quad \partial \, \mathbb {R}^{d+1}_{+}. \end{array} \right. \end{aligned}$$

(3.11)

Here, we suppose that there exists a small constant $\delta >0$ such that Assumption 1.1 holds with $\overline{\mathbb {R}^{d+1}_{+}}$ instead of $\Omega _{T}$. For any $r>0$ and $(x,t)\in \overline{\mathbb {R}^{d+1}_{+}}$, we denote

$$\begin{aligned} Q_{r}^{+}(x,t) = Q_{r}(x,t) \cap \mathbb {R}^{d+1}_{+}, \qquad Q_{r}^{+} = Q_{r}^{+}(0,0), \qquad Q_{r}^{+}(x^{1}) = Q_{r}^{+}(0,x^{1},0). \end{aligned}$$

Let us fix $(t_{0}, x_{0})\in \overline{\mathbb {R}^{d+1}_{+}}$. To attain the mean oscillation estimate on $Q^{+}_{r}(t_{0},x_{0})$ for any $r>0$ we mainly make use of Lemma 7.4 in [8] in terms of the odd extension technique.

Lemma 3.3

For $\kappa \ge 2$, $r>0$, $\beta \in (1, \infty )$ and $\alpha \in (0,1)$, let $u \in \mathring{W}^{1,\, 2}_{d+1}(\mathbb {R}_{+}^{d+1})$ be the solution of (3.11) and $(t_{0}, x_{0})\in \overline{\mathbb {R}_{+}^{d+1}}$. Then there exists a small constant $\delta _0=\delta _0(d,\lambda ,\Lambda ,\beta )>0$ such that the following holds: if Assumption 1.1holds for any $\delta \in (0,\delta _0)$, then we have

$$\begin{aligned}&\fint _{Q^{+}_{r}(t_{0}, x_{0})} \fint _{Q^{+}_{r}(t_{0}, x_{0})} \left| D^{2}u(x,t) - D^{2}u(s,y)\right| ^{\gamma } \, dxdtdyds \\&\quad \le N \kappa ^{d+2} \left( \fint _{Q^{+}_{\kappa r}(t_{0}, x_{0})} |h|^{d+1} \, dxdt \right) ^{\frac{\gamma }{d+1}}\\&\qquad + N \kappa ^{d+2} \delta ^{\frac{\gamma }{(d+1)\, \beta '}} \left( \fint _{Q^{+}_{\kappa r}(t_{0}, x_{0})} |D^{2}u|^{(d+1) \, \beta } \, dxdt \right) ^{\frac{\gamma }{(d+1)\, \beta }} \\&\qquad + N \kappa ^{- \alpha \gamma } \left( \fint _{Q^{+}_{\kappa r}(t_{0}, x_{0})} |D^{2}u|^{d+1} \, dxdt \right) ^{\frac{\gamma }{d+1}} \\&\qquad + N \kappa ^{d+2} R_{0}^{-2\gamma } \left( \fint _{Q^{+}_{\kappa r}(t_{0}, x_{0})} |u|^{d+1} \, dxdt \right) ^{\frac{\gamma }{d+1}} \\&\qquad + N \kappa ^{d+2} R_{0}^{-\gamma } \left( \fint _{Q^{+}_{\kappa r}(t_{0}, x_{0})} |Du|^{d+1} \, dxdt \right) ^{\frac{\gamma }{d+1}} \end{aligned}$$

with some $\gamma \in (0,1]$, where $\beta ' = {\beta }/{(\beta -1)}$ and $N=N(d,\delta ,\lambda ,\Lambda ,\beta )$.

If f is measurable function defined in $\mathbb {R}^{d+1}_{+}$, then for any $(x,t) \in \overline{\mathbb {R}^{d+1}_{+}}$ and $r>0$,

$$\begin{aligned} \fint _{Q^{+}_{r}(x,t)} |f(s,y)| \, dyds \le 2 \fint _{Q_{r}(x,t)} |f(s,y) I_{\mathbb {R}^{d+1}_{+}}| \, dyds \le N(d)\, \mathcal {M} \, (f \, I_{\mathbb {R}^{d+1}_{+}} )\,(x,t), \end{aligned}$$

where $I_{\mathbb {R}^{d+1}_{+}}$ is the indicator function of $\mathbb {R}^{d+1}_{+}$. On the other hand, for the smallest $r>0$ such that $\mathcal {C} \subset Q^{+}_{r}$, we have

$$\begin{aligned}&\left( \fint _{\mathcal {C}} \fint _{\mathcal {C}} \left| f(x,t) - f(s,y)\right| ^{\gamma } \, dxdtdyds \right) ^{\frac{1}{\gamma }} \\&\quad \le N(d,\gamma ) \left( \fint _{Q^{+}_{r}} \fint _{Q^{+}_{r}} \left| f(x,t) - f(s,y)\right| ^{\gamma } \, dxdtdyds \right) ^{\frac{1}{\gamma }}. \end{aligned}$$

With Lemma 3.3 we obtain the following result.

Corollary 3.4

Under the same hypotheses as in Lemma 3.3, we have

$$\begin{aligned}&\left( D^{2}u \right) ^{\sharp }_{\gamma }(t_{0}, x_{0}) \\&\quad \le N \kappa ^{\frac{d+2}{\gamma }} \mathcal {M}^{\frac{1}{d+1}}\left( |h|^{d+1} \, I_{\mathbb {R}^{d+1}_{+}} \right) (t_{0}, x_{0})\\&\qquad + N \kappa ^{\frac{d+2}{\gamma }} \delta ^{\frac{1}{(d+1)\, \beta '}} \mathcal {M}^{\frac{1}{(d+1)\, \beta }}\left( |D^{2}u|^{(d+1) \, \beta } \, I_{\mathbb {R}^{d+1}_{+}} \right) (t_{0}, x_{0}) \\&\qquad + N \kappa ^{- \alpha } \mathcal {M}^{\frac{1}{d+1}}\left( |D^{2}u|^{d+1} \, I_{\mathbb {R}^{d+1}_{+}} \right) (t_{0}, x_{0})\nonumber \\&\qquad + N \kappa ^{\frac{d+2}{\gamma }}R_{0}^{-2} \mathcal {M}^{\frac{1}{d+1}}\left( |u|^{d+1} I_{\mathbb {R}^{d+1}_{+}} \right) (t_{0}, x_{0}) \\&\qquad + N \kappa ^{\frac{d+2}{\gamma }}R_{0}^{-1} \mathcal {M}^{\frac{1}{d+1}}\left( |Du|^{d+1} I_{\mathbb {R}^{d+1}_{+}} \right) (t_{0}, x_{0}) \end{aligned}$$

for any $(t_{0}, x_{0})\in \overline{\mathbb {R}_{+}^{d+1}}$, where $N=N(d,\delta ,\lambda ,\Lambda ,\beta )$.

Hence, the following regularity theorem in $\mathring{W}^{1,\, 2}_{(p,q)}(\mathbb {R}^{d+1}_{+})$ can be deduced from Corollary 3.4 and Lemma 2.3 with the same way as the proof of Theorem 3.2.

Theorem 3.5

For $p\in (d+1, \, +\infty )$ and $q\in (0, \, +\infty ]$, let $u \in \mathring{W}^{1,\, 2}_{(p,q)}(\mathbb {R}^{d+1}_{+})$ satisfying (3.11) with $h\in L^{(p,q)}(\mathbb {R}^{d+1}_{+})$. Then there exists a small constant $\delta _0=\delta _0(d,\lambda ,\Lambda ,R_{0})>0$ for some $R_{0}>0$ such that if Assumption 1.1holds for any $\delta \in (0,\delta _0)$ in $\mathbb {R}^{d+1}_{+}$, then we have

$$\begin{aligned}&\left\| D^{2}u \right\| _{L^{(p,q)}(\mathbb {R}^{d+1}_{+})} + \left\| \partial _{t} u \right\| _{L^{(p,q)}(\mathbb {R}^{d+1}_{+})}\nonumber \\&\quad \le N \left( \left\| h \right\| _{L^{(p,q)}(\mathbb {R}^{d+1}_{+})} + \left\| u \right\| _{L^{(p,q)}(\mathbb {R}^{d+1}_{+})} \right) , \end{aligned}$$

(3.12)

where $N=N(d,p,q,\lambda ,\Lambda ,\delta ,R_{0})$.

On the basis of Theorems 3.2 and 3.5, we now prove the main Theorem 1.3 to the Cauchy-Dirichlet problems (1.1) in $\Omega _{T}$. Here $\Omega$ is a bounded $C^{1,1}$ domain and there exists a small constant $\delta >0$ such that Assumption 1.1 holds in $\Omega _{T}$.

Proof of Theorem 1.3

(i)  Let us define

$$\begin{aligned} h(x,t) = - G(D^{2}u, Du, u, x,t). \end{aligned}$$

(3.13)

We employ the technique of flattening the boundary and Theorems 3.2 and 3.5 which is a similar way shown as in [17]. Then, we obtain

$$\begin{aligned} \left\| D^{2}u \right\| _{L^{(p,q)}(\Omega _{T})} + \left\| \partial _{t} u \right\| _{L^{(p,q)}(\Omega _{T})} \le N \left( \left\| h \right\| _{L^{(p,q)}(\Omega _{T})} + \left\| u \right\| _{L^{(p,q)}(\Omega _{T})} \right) , \end{aligned}$$

(3.14)

where $N=N(d,p,q,\lambda ,\Lambda ,\delta ,R_{0},T, C^{1,1} \,\text {norm of}\,\partial \Omega )$. The definition of h together with Assumption (H2) leads to that

$$\begin{aligned} \left\| h \right\| _{L^{(p,q)}(\Omega _{T})}= & {} \left\| G(D^{2}u, Du, u, x,t) \right\| _{L^{(p,q)}(\Omega _{T})} \nonumber \\\le & {} \left\| \psi \left( |D^{2}u| \right) \, |D^{2}u| \right\| _{L^{(p,q)}(\Omega _{T})} + K \, \left( \left\| Du \right\| _{L^{(p,q)}(\Omega _{T})} \right. \nonumber \\&\left. + \left\| u \right\| _{L^{(p,q)}(\Omega _{T})} \right) + \left\| g \right\| _{L^{(p,q)}(\Omega _{T})}. \end{aligned}$$

(3.15)

Recall that the function $\psi (\rho ): \bar{\mathbb {R}}^{+} \rightarrow \bar{\mathbb {R}}^{+}$ is bounded, monotonically decreasing and $\psi (\rho ) \rightarrow 0$ as $\rho \rightarrow 0$. Therefore, for some $\bar{\rho }>0$ we obtain

$$\begin{aligned}&\left\| \psi \left( |D^{2}u| \right) \, |D^{2}u| \right\| _{L^{(p,q)}(\Omega _{T})}\nonumber \\&\quad \le \psi \left( \bar{\rho } \right) \, \left\| D^{2}u \right\| _{L^{(p,q)}(\Omega _{T})} + N(p,q) \, \bar{\rho } \left\| \psi \right\| _{L^{\infty }} |\Omega _{T}|^{\frac{1}{p}}. \end{aligned}$$

(3.16)

This together with (3.15) and (3.14) yield

$$\begin{aligned}&\left\| D^{2}u \right\| _{L^{(p,q)}(\Omega _{T})} + \left\| \partial _{t} u \right\| _{L^{(p,q)}(\Omega _{T})}\\&\quad \le N \, \psi \left( \bar{\rho } \right) \, \left\| D^{2}u \right\| _{L^{(p,q)}(\Omega _{T})} + N \, \bar{\rho } \left\| \psi \right\| _{L^{\infty }} |\Omega _{T}|^{\frac{1}{p}} \\&\qquad + N \, \left( \left\| Du \right\| _{L^{(p,q)}(\Omega _{T})} + \left\| u \right\| _{L^{(p,q)}(\Omega _{T})} + \left\| g \right\| _{L^{(p,q)}(\Omega _{T})} \right) , \end{aligned}$$

where $N=N(d,p,q,\lambda ,\Lambda ,\delta ,R_{0},T, K, C^{1,1} \,\text {norm of}\,\partial \Omega )$. Taking $\bar{\rho }$ large enough that $\displaystyle { N \, \psi \left( \bar{\rho } \right) \le \frac{1}{2}}$ and using again the Gagliardo-Nirenberg interpolation inequality (2.3) in the Lorentz spaces, we obtain

$$\begin{aligned}&\left\| D^{2}u \right\| _{L^{(p,q)}(\Omega _{T})} + \left\| \partial _{t} u \right\| _{L^{(p,q)}(\Omega _{T})}\nonumber \\&\quad \le N \, \left( \left\| u \right\| _{L^{(p,q)}(\Omega _{T})} + \left\| g \right\| _{L^{(p,q)}(\Omega _{T})} + \bar{\rho } \left\| \psi \right\| _{L^{\infty }} \right) , \end{aligned}$$

(3.17)

where $N=N(d,p,q,\lambda ,\Lambda ,\delta ,R_{0},T, K, |\Omega |, C^{1,1} \,\text {norm of}\,\partial \Omega )$. Finally, to estimate the Lorentz norm of u, we rewrite the Cauchy-Dirichlet problem (1.1) as

$$\begin{aligned} \left\{ \begin{array}{ll} \partial _{t} u(x,t) + F(D^{2}u, x,t) + G(D^{2}u, Du, u, x,t) - G(D^{2}u, 0, 0, x,t) = - G(D^{2}u, 0, 0, x,t) &{} \quad \text {in} \quad \Omega _{T}, \\ u=0 &{} \quad \text {on} \quad \partial \Omega _{T}. \end{array} \right. \end{aligned}$$

Conditions (H1) and (H2) imply that there exists an operator $L \, \cdot = \,a^{ij}\, D_{ij} \, \cdot + \,b^{i}\, D_{i} \, \cdot - \, c \, \cdot$ with (2.6) and (2.7), such that

$$\begin{aligned} \left\{ \begin{array}{ll} \partial _{t} u(x,t) + Lu = - G(D^{2}u, 0, 0, x,t) &{} \quad \text {in} \quad \Omega _{T}, \\ u=0 &{} \quad \text {on} \quad \partial \Omega _{T}. \end{array} \right. \end{aligned}$$

Since G is a monotonically nonincreasing function in u, by the Alexandrov estimate in the Lorentz spaces (cf. Lemma 2.7) we obtain

$$\begin{aligned} \left\| \,u\,\right\| _{L^{(p,q)}(\Omega _{T})} \le N \left\| \,G(D^{2}u, 0, 0, x,t)\,\right\| _{L^{(p,q)}(\Omega _{T})}, \end{aligned}$$

(3.18)

where $N=N(d,p,q,\lambda ,\Lambda ,T,|\Omega |)$. This combined with Assumption (H2) yields

$$\begin{aligned} \left\| \,u\,\right\| _{L^{(p,q)}(\Omega _{T})} \le N \, \left( \, \left\| \psi \left( |D^{2}u| \right) \, |D^{2}u| \right\| _{L^{(p,q)}(\Omega _{T})} + \left\| g \right\| _{L^{(p,q)}(\Omega _{T})} \right) . \end{aligned}$$

Therefore, with the above estimate and (3.17), we use the same argument as in the proof of (3.16) to obtain

$$\begin{aligned} \left\| D^{2}u \right\| _{L^{(p,q)}(\Omega _{T})} + \left\| \partial _{t} u \right\| _{L^{(p,q)}(\Omega _{T})} \le N \, \left( \left\| g \right\| _{L^{(p,q)}(\Omega _{T})} + \bar{\rho } \left\| \psi \right\| _{L^{\infty }} \right) \end{aligned}$$

(3.19)

for some $\bar{\rho }>0$, where $N=N(d,p,q,\lambda ,\Lambda ,\delta ,R_{0},T, K, |\Omega |, \Vert \partial \Omega \Vert _{C^{1,1}})$.

(ii)   With the a prior estimate (3.19) in hand, we make use of an approximating argument to the nonlinear function H and the domain $\Omega$, which is just as the same proof of [8, Theorem 1.2], to lead to the existence and uniqueness claims of (1.1) in the Sobolev-Lorentz spaces $W^{1,\, 2}_{(p,q)}(\Omega _{T})$. It completes the proof. $\square$

Notice that parabolic Bellman’s equation is a typical prototype as fully nonlinear parabolic equations. Therefore, as a consequence of Theorem 1.3, it yields the following Sobolev-Lorentz estimate of strong solutions to parabolic Bellman’s equations under the assumption of small BMO coefficients in a bounded $C^{1,1}$ domain. More precisely, we consider the initial-boundary problem of parabolic Bellman’s equation

$$\begin{aligned} \left\{ \begin{array}{ll} \partial _{t} u(x,t) + \underset{\omega \in \mathcal {A}}{\sup } \,\, \Big [ a^{ij}(\omega ,x,t) D_{ij}u(x,t) + b^{i}(\omega ,x,t) D_{i}u(x,t) \\ \qquad \qquad \qquad \quad - c(\omega , x,t) u(x,t) + g(\omega ,x,t) \Big ] = 0 &{} \,\,\text {in} \,\, \Omega _{T}, \\ u(x,t)=0 &{} \,\, \text {on} \,\, \partial \Omega _{T}, \end{array} \right. \end{aligned}$$

(3.20)

where the set $\mathcal {A}$ is a separable metric space, $a(\omega ,x,t) = \left( a^{ij}(\omega ,x,t)\right)$, $b(\omega ,x,t)=\left( b^{i}(\omega ,x,t)\right)$, $c(\omega , x,t)\ge 0$ and $g(\omega ,x,t)$ are given functions. Here, we assume that these functions are measurable in (x, t) for each $\omega$, continuous in $\omega$ for each (x, t) and there exist positive constants $0<{\lambda }\le {\Lambda }< \infty$ such that

$$\begin{aligned} |b(\omega ,x,t)| \le {\Lambda }, \qquad c(\omega , x,t)\le {\Lambda }, \qquad \hat{g}(x,t):= \underset{\omega \in \mathcal {A}}{\sup } \left[ g(\omega ,x,t)\right] < \infty \end{aligned}$$

(3.21)

and

$$\begin{aligned} {\lambda } |\xi |^{2} \le a^{ij}(\omega ,x,t) \, \xi _{i} \xi _{j} \le {\Lambda } |\xi |^{2} \end{aligned}$$

(3.22)

for any $\xi \in \mathbb {R}^{d}$, $\omega \in \mathcal {A}$ and almostly all $(x,t)\in \Omega _{T}$. Moreover, $a^{ij}(\omega ,x,t)$ is $(\delta , R_{0})$-vanishing of (x, t) which means that there exists $R_{0}>0$ such that for any $r\in (0, R_{0}]$ and $(s,y) \in \overline{\Omega }_{T}$,

$$\begin{aligned} \sup _{i,j} \fint _{Q_{r}(s,y)} \underset{\omega \in \mathcal {A}}{\sup } \, \left| a^{ij} (\omega ,x,t) - \bar{a}^{ij}_{Q_{r}(s,y)}(\omega ) \right| \, dxdt \le \delta , \end{aligned}$$

(3.23)

where the parameter $\delta >0$ will be specified later.

One can check that H1,H2 and Assumptions 1.1 hold in Theorem 1.3 by taking

$$\begin{aligned} F= & {} \underset{\omega \in \mathcal {A}}{\sup } \,\, a^{ij}(\omega ,x,t) D_{ij}u(x,t) \quad \text {and} \\ G= & {} \underset{\omega \in \mathcal {A}}{\sup } \,\, \Big [ b^{i}(\omega ,x,t) D_{i}u(x,t) - c(\omega , x,t) u(x,t) + g(\omega ,x,t) \Big ], \end{aligned}$$

which immediately leads to the following conclusion.

Corollary 3.6

For $p\in (d+1, \, +\infty )$, $q\in (0, \, +\infty ]$, $\Omega \in C^{1,1}$, let $u \in \mathring{W}^{1,\,2}_{(p,q)}(\Omega _{T})$ with $T>0$ be the solution of the Cauchy-Dirichlet problems (3.20) with assumptions (3.21, 3.22) and $\hat{g} \in L^{(p,q)}(\Omega _{T})$. Then, there exists a small constant $\delta =\delta (d,p,q,\lambda ,\Lambda ,R_{0})>0$ for some $R_{0}>0$ such that if the condition (3.23) holds, then we have

$$\begin{aligned} \left\| u \right\| _{W^{1,2}_{(p,q)}(\Omega _{T})} \le N \left\| \hat{g} \right\| _{L^{(p,q)}(\Omega _{T})}. \end{aligned}$$

(3.24)

Here, N is a positive constant depending only on $d,p,q,\lambda ,\Lambda ,\delta ,R_{0},T,|\Omega |$ and $\Vert \partial \Omega \Vert _{C^{1,1}}$.