Maximal regularity for non-autonomous evolution equations

Abstract

We consider the maximal regularity problem for non-autonomous evolution equations

(1)

Each operator $A(t)$ is associated with a sesquilinear form ${\mathop {{\varvec{\mathfrak {a}}}}(t; \cdot , \cdot )}$ on a Hilbert space $H$. We assume that these forms all have the same domain and satisfy some regularity assumption with respect to $t$ (e.g., piecewise $\alpha$-Hölder continuous for some ). We prove maximal $L_p$-regularity for all $u_0$ in the real-interpolation space . The particular case where $p = 2$ improves previously known results and gives a positive answer to a question of Lions (Équations différentielles opérationnelles et problèmes aux limites, Springer, Berlin, 1961) on the set of allowed initial data $u_0$.

1 Introduction and main results

Let $H$ be a real or complex Hilbert space and let $V$ be another Hilbert space with dense and continuous embedding $V \hookrightarrow H$. We denote by $V'$ the (anti-)dual of $V$ and by $\left[ \cdot \,|\,\cdot \right] _H$ the scalar product of $H$ and $\left\langle \cdot , \cdot \right\rangle$ the duality pairing $V'\times V$. The latter satisfies (as usual) $\left\langle v, h \right\rangle = \left[ v\,|\,h\right] _H$ whenever $v \in H$ and $h \in V$. By the standard identification of $H$ with $H'$ we then obtain continuous and dense embeddings $V \hookrightarrow H \eqsim H' \hookrightarrow V'$. We denote by $\Vert . \Vert _V$ and $\Vert . \Vert _H$ the norms of $V$ and $H$, respectively.

We are concerned with the non-autonomous evolution equation

$$\begin{aligned} \left\{ \begin{array}{l} u'(t) + A(t)\,u(t) = f(t), \quad t \in (0, \tau ] \\ u(0)=u_0, \end{array}\right. \end{aligned}$$

(P)

where each operator $A(t)$, $t \in [0, \tau ]$, is associated with a sesquilinear form ${\mathop {{\varvec{\mathfrak {a}}}}(t; \cdot , \cdot )}$. Throughout this article we will assume that

• H1 (Constant form domain) ${\mathcal {D}(\mathop {{\varvec{\mathfrak {a}}}}(t; \cdot , \cdot )) = V}$.

• H2 (Uniform boundedness) there exists $M>0$ such that for all $t \in [0, \tau ]$ and $u, v \in V$, we have ${|\mathop {{\varvec{\mathfrak {a}}}}(t; u, v)|\le M \Vert u \Vert _V \Vert v \Vert _V}$.

• H3 (Uniform quasi-coercivity) there exist $\alpha >0$, $\delta \in \mathbb {R}$ such that for all $t \in [0, \tau ]$ and all $u, v \in V$ we have ${\alpha \Vert u \Vert _V^2 \le \text {Re}\mathop {{\varvec{\mathfrak {a}}}}(t; u, u) + \delta \Vert u \Vert _H^2}$.

Recall that $u \in H$ is in the domain $\mathcal {D}(A(t))$ if there exists $h\in H$ such that for all $v \in V$: ${\mathop {{\varvec{\mathfrak {a}}}}(t; u, v) = \left[ h\,|\,v\right] _H}$. We then set $A(t)u := h$. We mention that equality of the form domains, i.e., ${\mathcal {D}(\mathop {{\varvec{\mathfrak {a}}}}(t; \cdot , \cdot )) = V}$ for $t \in [0, \tau ]$ does not imply equality of the domains $\mathcal {D}(A(t))$ of the corresponding operators. For each fixed $u \in V$, ${\phi := \mathop {{\varvec{\mathfrak {a}}}}(t; u, \cdot )}$ defines a continuous (anti-)linear functional on $V$, i.e., $\phi \in V'$, then it induces a linear operator ${\mathcal {A}}(t): V \rightarrow V'$ such that ${\mathop {{\varvec{\mathfrak {a}}}}(t; u, v) = \left\langle {\mathcal {A}}(t)u, v \right\rangle }$ for all $u, v \in V$. Observe that for $u \in V$,

$$\begin{aligned} {\Vert {\mathcal {A}}(t)u \Vert _{V'} = \sup _{v \in V, \Vert v \Vert _V=1} |\left\langle {\mathcal {A}}(t)u, v \right\rangle | = \sup _{v \in V, \Vert v \Vert _V=1} |\mathop {{\varvec{\mathfrak {a}}}}(t; u, v)| \le M \Vert u \Vert _V} \end{aligned}$$

so that ${\mathcal {A}}(t) \in \mathcal {B}(V, V')$. The operator ${\mathcal {A}}(t)$ can be seen as an unbounded operator on $V'$ with domain $V$ for all $t \in [0, \tau ]$. The operator $A(t)$ is then the part of ${\mathcal {A}}(t)$ on $H$, that is,

$$\begin{aligned} \mathcal {D}(A(t)) = \{ u \in V,\ {\mathcal {A}}(t) u \in H \}, \quad A(t) u = {\mathcal {A}}(t) u. \end{aligned}$$

It is a known fact that $-A(t)$ and $-{\mathcal {A}}(t)$ both generate holomorphic semigroups $(e^{-s \, A(t)})_{s \ge 0}$ and $(e^{-s\, {\mathcal {A}}(t)})_{s \ge 0}$ on $H$ and $V'$, respectively. For each $s \ge 0$, $e^{-s \, A(t)}$ is the restriction of $e^{-s \, {\mathcal {A}}(t)}$ to $H$. For all this, we refer to Ouhabaz [22, Chapter 1].

The notion of maximal $L_p$-regularity for the above Cauchy problem is defined as follows:

Definition 1

Fix $u_0$. We say that (P) has maximal $L_p$-regularity (in $H$) if for each $f \in L_p(0,\tau ; H)$ there exists a unique $u \in W^{1}_p(0, \tau ; H)$, such that $u(t) \in \mathcal {D}(A(t))$ for almost all $t$, which satisfies (P) in the $L_p$-sense. Here $W^{1}_p(0, \tau ; H)$ denotes the classical $L_p$-Sobolev space of order one of functions defined on $(0, \tau )$ with values in $H$.

Maximal regularity of an evolution equation on a Banach space $E$ depends on the operators involved in the equation, the space $E$ and the initial data $u_0$. The initial data has to be in an appropriate space. In the autonomous case, i.e., $A(t) = A$ for all $t \in [0, \tau ]$, maximal $L_p$-regularity is well understood and it is also known that $u_0$ has to be in the real-interpolation space , see [7]. We refer the reader further to the survey of Denk et al. [10] and the references given therein. Note also that maximal regularity turns to be an important tool to study quasi-linear equations, see e.g., the monograph of Amann [13].

For the non-autonomous case we consider here, we first recall that if the evolution equation is considered in $V'$, then Lions proved maximal $L_2$-regularity for all initial data $u_0 \in H$, see e.g., [17], [25, p.112]. This powerful result means that for every $u_0 \in H$ and $f \in L_2(0, \tau ; V')$, the equation

has a unique solution $u \in W^{1}_2(0,\tau ; V') \cap L_2(0, \tau ;V)$. Note that this implies the continuity of $u(\cdot )$ as an $H$-valued function, see [8, XVIIIChapter 3, p. 513]. It is a remarkable fact that Lions’s theorem does not require any regularity assumption (with respect to $t$) on the sesquilinear forms apart from measurability. The apparently additional information $u \in L_2(0, \tau ; V)$ follows from maximal regularity and the equation as follows: for almost all $t$, $u(t) \in V$. For these $t \in (0, \tau )$

$$\begin{aligned} \text {Re}\mathop {{\varvec{\mathfrak {a}}}}(t; u(t), u(t))&= - \text {Re}\left\langle u'(t), u(t) \right\rangle + \text {Re}\left\langle f(t), u(t) \right\rangle \\&\le \Vert u(t) \Vert _V \Vert u'(t) \Vert _{V'} + \Vert u(t) \Vert _V \Vert f(t) \Vert _{V'}. \end{aligned}$$

Suppose now that the forms are coercive (i.e., $\delta = 0$ in [H3]). Then for some constant $c> 0$ independent of $t$,

$$\begin{aligned} \Vert u(t) \Vert _V^2 \le c\left[ \Vert u'(t) \Vert _{V'}^2 + \Vert f(t) \Vert _{V'}^2\right] . \end{aligned}$$

(2)

Therefore, $u \in L_2(0, \tau ; V)$ whenever $u \in W^{1}_2(0,\tau ; V')$ and $f \in L_2(0,\tau ; V')$. If the forms are merely quasi-coercive, we note that if $u(t)$ is the solution of (P$^{\prime }$) then $u(t)e^{-\delta t}$ is the solution of the same problem with ${\mathcal {A}}(t) + \delta$ instead of ${\mathcal {A}}(t)$ and $f(t)e^{-\delta t}$ instead of $f(t)$. We apply now the previous estimate (2) to $u(t)e^{-\delta t}$ and $f(t) e^{-\delta t}$ and obtain

$$\begin{aligned} \Vert u(t) \Vert _V^2 \le c\left[ \Vert u'(t) \Vert _{V'}^2 + \Vert u(t) \Vert _{V'}^2 + \Vert f(t) \Vert _{V'}^2 \right] \end{aligned}$$

for some constant $c'$ independent of $t$.

Note however that maximal regularity in $V'$ is not satisfactory in applications to elliptic boundary value problems. For example, in order to identify the boundary condition one has to consider the evolution equation in $H$ rather than in $V'$. For symmetric forms (equivalently, self-adjoint operators $A(t)$) and $u_0 = 0$, Lions [17, p. 65], proved maximal $L_2$-regularity in $H$ under the additional assumption that ${t \mapsto \mathop {{\varvec{\mathfrak {a}}}}(t; u, v)}$ is $C^1$ on $[0,\tau ]$. For $u(0) = u_0 \in \mathcal {D}(A(0))$ Lions [17, p. 95] proved maximal $L_2$-regularity in $H$ for (P) provided ${t \mapsto \mathop {{\varvec{\mathfrak {a}}}}(t; u, v)}$ is $C^2$. If the forms are symmetric and $C^1$ with respect to $t$, Lions proved maximal $L_2$-regularity for $u(0) = u_0 \in V$ (one has to combine [17, Theorem 1.1, p. 129, Theorem 5.1, p. 138] to see this). He asked the following problems.

Problem A

Does the maximal $L_2$-regularity in $H$ hold for (P) with $u_0 = 0$ when ${t \mapsto \mathop {{\varvec{\mathfrak {a}}}}(t; u, v)}$ is continuous or even merely measurable?

Problem B

For $u(0) = u_0 \in \mathcal {D}(A(0))$, does the maximal $L_2$-regularity in $H$ hold under the weaker assumption that ${t \mapsto \mathop {{\varvec{\mathfrak {a}}}}(t; u, v)}$ is $C^1$ rather than $C^2$?

The Problem A is still open although some progress has been made. We mention here Ouhabaz and Spina [23] who prove maximal $L_p$-regularity for (P) when $u(0) = 0$ and ${t \mapsto \mathop {{\varvec{\mathfrak {a}}}}(t; u, v)}$ is $\alpha$-Hölder continuous for some $\alpha > \frac{1}{2}$. More recently, Arendt et al. [3] prove maximal $L_2$-regularity in $H$ for

$$\begin{aligned} \left\{ \begin{array}{l} B(t)u'(t) + A(t)\,u(t) = f(t), \quad t \in (0, \tau ] \\ u(0)=0 \end{array}\right. \end{aligned}$$

in the case where ${t \mapsto \mathop {{\varvec{\mathfrak {a}}}}(t; u, v)}$ is piecewise Lipschitz and $B(t)$ are bounded measurable operators satisfying $\gamma \Vert v \Vert _H^2 \le \text {Re}\left[ B(t)v\,|\,v\right] _H \le \gamma ' \Vert v \Vert _H^2$ for some positive constants $\gamma$ and $\gamma '$ and all $v \in H$. The multiplicative perturbation by $B(t)$ was motivated there by applications to some quasi-linear evolution equations.

Concerning the problem with $u_0 \ne 0$ and forms which are not necessarily symmetric, Bardos [5] gave a partial positive answer to Problem B in the sense that one can take the initial data $u_0$ in $V$ under the assumptions that the domains of both and coincide with $V$ as spaces and topologically with constants independent of $t$, and that is continuously differentiable with values in ${\mathcal L}(V,V')$. Note however that the property is not always true; this equality is equivalent to the Kato’s square root property: . The result of [5] was extended in Arendt et al. [3] by including the multiplication $B(t)$ above and also weakening the regularity of from continuously differentiable to piecewise Lipschitz. As in [5], it is also assumed in [3] that the domains of and coincide with $V$ as spaces and topologically with constants independent of $t$.

We emphasize that the above results from [3, 5, 17] on maximal $L_2$-regularity do not give any information on maximal $L_p$-regularity when $p \ne 2$ since the techniques used there are based on a representation lemma in (pre-) Hilbert spaces.

In the present paper we prove maximal $L_p$-regularity for (P) for all $p \in (1, \infty )$. We extend the results mentioned above and give a complete treatment of the problem with initial data $u_0 \not = 0$ even when the forms are not necessarily symmetric. In particular, we obtain a positive answer to Problem B under even more general assumptions.

Our main result is the following.

Theorem 2

Suppose that the forms ${(\mathop {{\varvec{\mathfrak {a}}}}(t; \cdot , \cdot ))_{0 \le t \le \tau }}$ satisfy the standing hypotheses [H1]–[H3] and the regularity condition

$$\begin{aligned} {\left| \mathop {{\varvec{\mathfrak {a}}}}(t; u, v) - \mathop {{\varvec{\mathfrak {a}}}}(s; u, v) \right| \le \omega (|t{-}s|)\, \Vert u \Vert _V \Vert v \Vert _V} \end{aligned}$$

(3)

where $\omega : [0, \tau ] \rightarrow [0, \infty )$ is a non-decreasing function such that

(4)

Then the Cauchy problem (P) with $u_0 = 0$ has maximal $L_p$-regularity in $H$ for all $p \in (1, \infty )$. If in addition $\omega$ satisfies the $p$-Dini condition

$$\begin{aligned} \int _0^\tau \left( \tfrac{\omega (t)}{t} \right) ^p \,{\mathrm {d}}t< \infty , \end{aligned}$$

(5)

then (P) has maximal $L_p$-regularity for all . Moreover there exists a positive constant $C$ such that

In this theorem, denotes the classical real-interpolation space, see [26, Chapter 1.13] or [18, Proposition 6.2].

We wish to point out that Yamamoto¹ states a result which resembles to our previous theorem in the setting of operators satisfying the so-called Acquistapace–Terreni conditions on the corresponding resolvents. Unfortunately it is difficult to follow his proofs.²

We have the following corollaries.

Corollary 3

Under the assumptions of the previous theorem, the Cauchy problem (P) with $u_0 = 0$ has maximal $L_2$-regularity in $H$. If in addition $\omega$ satisfies

$$\begin{aligned} \int _0^\tau \left( \tfrac{\omega (t)}{t} \right) ^2 \,{\mathrm {d}}t< \infty , \end{aligned}$$

(6)

then (P) has maximal $L_2$-regularity for all . In addition, there exists a positive constant $C$ such that

Obviously, if $A(0)$ is accretive then is well defined and . This corollary solves Problem B even under more general conditions than conjectured by J.L. Lions

Corollary 4

Assume additionally to the standing assumptions [H1]–[H3] that the form ${\mathop {{\varvec{\mathfrak {a}}}}(t; \cdot , \cdot )}$ is piecewise $\alpha$-Hölder continuous for some . That is, there exist $t_0 = 0 < t_1< \cdots < t_k = \tau$ such that on each interval $(t_i, t_{i+1})$ the form is the restriction of a $\alpha$-Hölder continuous form on $[t_i, t_{i+1}]$. Assume in addition that at the discontinuity points, we have . Then the Cauchy problem (P) has maximal $L_2$-regularity for all and there exists a positive constant $C$ such that

In this corollary, $A(t_j^-)$ is the operator associated with the extension of the form at the left of point $t_j$. Similarly for $A(t_j^+)$.

We mention that Fuje and Tanabe [12] constructed an evolution family associated with the non-autonomous problem considered here when the form ${\mathop {{\varvec{\mathfrak {a}}}}(t; \cdot , \cdot )}$ is $\alpha$-Hölder continuous for some . This is of independent interest but it is not clear if one obtains maximal regularity from any property of the corresponding evolution family.

Now we explain briefly the strategy of the proof. A starting point is a representation formula for the solution $u$ (recall that $u$ exists in $V'$ by Lions theorem), which already appeared in the work of Acquistapace and Terreni [1], namely

$$\begin{aligned} u(t)= & {} \; \int _0^t e^{-(t{-}s) {\mathcal {A}}(t)} ({\mathcal {A}}(t){-}{\mathcal {A}}(s)) u(s)\,{\mathrm {d}}s\nonumber \\&+ \int _0^t e^{-(t{-}s) {\mathcal {A}}(t) } f(s)\,{\mathrm {d}}s+ e^{-t \,{\mathcal {A}}(t)}u_0. \end{aligned}$$

(7)

This allows us to write ${\mathcal {A}}(t) u(t) = (Q {\mathcal {A}}(\cdot )u(\cdot ))(t) + (L f)(t) + (Ru_0)(t)$, where

$$\begin{aligned} (Q g)(t)&:= \int _0^t {\mathcal {A}}(t) e^{-(t{-}s) {\mathcal {A}}(t)} ({\mathcal {A}}(t) - {\mathcal {A}}(s)) {\mathcal {A}}(s)^{-1}g(s)\,{\mathrm {d}}s\\ (L g )(t)&:= {\mathcal {A}}(t) \int _0^t e^{-(t{-}s) {\mathcal {A}}(t)} g(s)\,{\mathrm {d}}s\quad \text {and} \quad (R u_0)(t) := {\mathcal {A}}(t) e^{-t {\mathcal {A}}(t)} u_0. \end{aligned}$$

Condition (4) allows us to prove invertibility of the operator $(I{-}Q)$ on $L_p(0, \tau ; H)$. The operator $L$ is seen as a pseudo-differential operator with an operator-valued symbol. We prove an $L_2$-boundedness result for such operators in Sect. 4, see Theorem 13. For operators with scalar-valued symbols, this result is due to Muramatu and Nagase [20]. We adapt their arguments to our setting of operator-valued symbols. This theorem is then used to prove $L_2(0, \tau ; H)$-boundedness of $L$. In order to extend $L$ to a bounded operator on $L_p(0, \tau ; H)$, for $p \in (1, \infty )$, we look at $L$ as a singular integral operator with an operator-valued kernel. We show that both $L$ and its adjoint $L^*$ satisfy the well known Hörmander integral condition. Finally, we treat the operator $R$ by taking the difference with ${\mathcal {A}}(0) e^{-t \,{\mathcal {A}}(0)} u_0$ and using the functional calculus for accretive operators on Hilbert spaces. In order to handle this difference we use (5), the remaining term, $t \mapsto A(0) e^{-t \, A(0)}u_0$ is then in $L_p(0, \tau ; H)$ if and only if .

Although most of the arguments outlined here use heavily the fact that $H$ is a Hilbert space, the strategy justifies some hope that the results extend to other situations such as $L_q$-spaces. One would then hope to prove $L_p(L_q)$ a priori estimates for parabolic equations with time dependent coefficients. In the last section of this paper we give some applications and prove $L_p(L_2)$—a priori estimates.

Maximal regularity may fail even for ordinary differential equations, letting $H = \mathbb {R}$. We illustrate this by an example which is essentially taken from Batty et al. [6].

Example 5

Consider . Then $\varphi$ in $L_{q, \text {loc}}(\mathbb {R})$ for $1\le q < p$ but $\varphi \not \in L_p([0,\varepsilon ])$ for $\varepsilon >0$. Chose a dense sequence $(t_n)$ of $[0,1]$ and a positive, summable sequence $(c_n)$. Define $a(t) := 1+ \sum c_n \varphi (t-t_n)$. Then $a \in L_q([0,1])$ for $1 \le q < p$ but $a \not \in L_p(I)$ for any interval $I \subset [0,1]$. Consider the non-autonomous equation

$$\begin{aligned} \left\{ \begin{array}{l} x'(t) + a(t) x(t) = 1\\ x(0)=0 \end{array}\right. \end{aligned}$$

(8)

Then by variation of constants formula, $x(t)=\int _0^t \exp \left( -\int _s^t a(r)\,{\mathrm {d}}r\right) \,{\mathrm {d}}s$. Since $a(r) \ge 0$,

$$\begin{aligned} |a(t) x(t)|&= a(t) \int _0^t \exp \left( - \int _s^t a(r)\,{\mathrm {d}}r\right) \,{\mathrm {d}}s\\&\ge a(t) \int _0^t \exp \left( - \int _0^1 a(r)\,{\mathrm {d}}r\right) \,{\mathrm {d}}s= C t\, \, a(t). \end{aligned}$$

Therefore, for $0<\alpha < \beta \le 1$ we have $|a(t)x(t)| \ge \alpha C\; a(t)$ on $[\alpha , \beta ]$ which implies that (8) cannot have maximal $L_p$-regularity.

On the other hand, if we replace the constant function 1 by $f$ we obtain

$$\begin{aligned} |a(t) x(t)|&= a(t) | \int _0^t f(s) \exp \left( - \int _s^t a(r)\,{\mathrm {d}}r\right) | \,{\mathrm {d}}s\\&\le a(t) \int _0^t | f(s) | \, {\mathrm {d}}s\le C a(t) \Vert f\Vert _{L_q} \end{aligned}$$

on $[0,1]$ and this shows that (8) has maximal $L_q$-regularity for $q<p$.

Notice however, that letting $p=2$ this example is not a counterexample to the questions we raise, since our standing hypothesis [H2] is not satisfied here.

Observe also that the operators in this example are all bounded and commute. Thus, these last two properties are not enough to obtain maximal $L_2$-regularity for non-autonomous evolution equations.

2 Preparatory lemmas

In this section we prove most of the main arguments which we will need for the proofs of our results. The only missing argument here concerns boundedness of pseudo-differential operators with operator-valued symbols which we write in a separate section for clarity of exposition. We formulate our arguments in a series of lemmata.

Throughout this section we will suppose that [H1]–[H3] are satisfied. Let $\mu \in \mathbb {R}$ and set $v(t) := e^{-\mu t} u(t)$. If $u$ satisfies (P), then $v$ satisfies the evolution equation

$$\begin{aligned} \left\{ \begin{array}{l} v'(t) + (\mu + A(t))\,v(t) = f(t)e^{-\mu t} \\ u(0)=u_0. \end{array}\right. \end{aligned}$$

In addition, $v \in W^{1}_p(0, \tau ; H)$ if and only if $u \in W^{1}_p(0, \tau ; H)$. This shows that we may replace ${\mathcal {A}}(t)$ (resp., $A(t)$) by ${\mathcal {A}}(t) + \mu$ (resp. $A(t) + \mu$). Therefore, we may suppose without loss of generality that $\delta =0$ in [H3]. In particular, we may suppose that $A(t)$ and ${\mathcal {A}}(t)$ are boundedly invertible by choosing $\mu > 0$ large enough. We will do so in the sequel without further mentioning it.

It is known that ${-}A(t)$ generates a bounded holomorphic semigroup on $H$. The same is true for ${-}{\mathcal {A}}(t)$ on $V'$. We write this explicitly in the next proposition in order to point out that the constants involved in the estimates are uniform with respect to $t$. The arguments are standard and can be found e.g., in [22, Sect. 1.4]. Denote by $S_\theta$ the open sector $S_{\theta } = \{ z \in \mathbb {C}^* :\;|\text {arg}(z) < \theta \}$ with vertex $0$.

Proposition 6

For any $t \in [0, \tau ]$, the operators $-A(t)$ and $-{\mathcal {A}}(t)$ generate strongly continuous analytic semigroups of angle on $H$ and $V'$, respectively. In addition, there exist constants $C$ and $C_\theta$, independent of $t$, such that

• (a) ,

• (b) $\displaystyle \Vert A(t) e^{-s \, A(t) } \Vert _{\mathcal {B}(H)} \le \tfrac{C}{s} \quad \mathrm{and}\quad \Vert {\mathcal {A}}(t) e^{-s \, {\mathcal {A}}(t) } \Vert _{\mathcal {B}(V')} \le \tfrac{C}{s}$,

• (c) $\Vert e^{-s \, A(t)} x \Vert _V \le \tfrac{C}{\sqrt{s}} \Vert x \Vert _H \quad \mathrm{and}\quad \Vert e^{-s \, {\mathcal {A}}(t)} \phi \Vert _H \le \tfrac{C}{\sqrt{s}} \Vert \phi \Vert _{V'}$,

• (d)

• (e) All the previous estimates hold for $A(t) + \mu$ with constants independent of $\mu$ for $\mu \ge 0$.

Proof

Fix $t \in [0, \tau ]$. By uniform boundedness and coercivity,

(9)

This means that $A(t)$ has numerical range contained in the closed sector $\overline{S_{\omega _0}}$ with . This implies the first part of assertion (a), see e.g., [22, Theorem 1.54]. Let $u \in V$ and set $\varphi = (\lambda + {\mathcal {A}}(t)) u \in V'$. Then

$$\begin{aligned} {\left\langle \varphi , u \right\rangle = \lambda \Vert u \Vert _H^2 + \mathop {{\varvec{\mathfrak {a}}}}(t; u, u)} \end{aligned}$$

and so coercivity yields

$$\begin{aligned} \Vert u \Vert _V^2\le & {} \tfrac{1}{\alpha } \text {Re}\mathop {{\varvec{\mathfrak {a}}}}(t; u, u)\le \tfrac{1}{\alpha } \left( |\left\langle \varphi , u \right\rangle | + |\lambda | \Vert u \Vert _H^2\right) \nonumber \\\le & {} \tfrac{1}{\alpha } \left( \Vert \varphi \Vert _{V'} \Vert u \Vert _V + |\lambda | \Vert u \Vert _H^2\right) . \end{aligned}$$

(10)

We aim to estimate $|\lambda | \Vert u \Vert _H^2$ against $\Vert u \Vert _V \Vert \varphi \Vert _{V'}$. Since the numerical range of ${\mathop {{\varvec{\mathfrak {a}}}}(t; \cdot , \cdot )}$ in contained in $\overline{S_{\omega _0}}$ we have

$$\begin{aligned} \text {dist}(\lambda , -S_{\omega _0}) \, \Vert u \Vert _H^2&\le \left| \lambda +\mathop {{\varvec{\mathfrak {a}}}}(t; \tfrac{u}{\Vert u \Vert }, \tfrac{u}{\Vert u \Vert }) \right| \,\Vert u \Vert _H^2\\&\le \left| \left\langle (\lambda I + {\mathcal {A}}(t))u, u \right\rangle \right| \le \Vert u \Vert _V \Vert \varphi \Vert _{V'}. \end{aligned}$$

Now let $\theta >\omega _0$ and $\lambda \not \in S_{\theta }$. Then $\text {dist}(\lambda , -S_{\omega _0}) \ge |\lambda | \sin (\theta {-}\omega _0)$ and therefore

$$\begin{aligned} |\lambda | \Vert u \Vert _H^2 \le \tfrac{1}{\sin (\theta {-}\omega _0)} \Vert u \Vert _V \Vert \varphi \Vert _{V'} \end{aligned}$$

as desired. From this and (10) it follows that

$$\begin{aligned} \Vert u \Vert _V \le \tfrac{1}{\alpha } \left( 1+ \tfrac{1}{\sin (\theta {-}\omega _0)}\right) \, \Vert (\lambda + {\mathcal {A}}(t)) u \Vert _{V'} \end{aligned}$$

(11)

uniformly for all $\lambda \not \in S_{\theta }$, $\theta >\omega _0$. This implies that $(\lambda +{\mathcal {A}}(t))$ is invertible with a uniform norm bound on $S_{\theta }^\complement$, $\theta >\omega _0$. This is equivalent to $-{\mathcal {A}}(t)$ being the generator of a bounded analytic semigroup on $V'$. The bound is independent of $t$. This proves assertion (a).

Assertion (b) follows from the analyticity of the semigroups $(e^{-s \, A(t)})_{s\ge 0}$ on $H$ and $(e^{-s \, {\mathcal {A}}(t)})_{s\ge 0}$ on $V'$ and the Cauchy formula as usual.

For assertion (c), observe that for $x \in H$

$$\begin{aligned} \alpha \Vert e^{-s \, A(t)} x \Vert _V^2&\le \text {Re}\mathop {{\varvec{\mathfrak {a}}}}(t; e^{-s \,{\mathcal {A}}(t)} x, e^{-s \, {\mathcal {A}}(t)} x)\\&= \text {Re}\left[ {\mathcal {A}}(t) e^{-s \, {\mathcal {A}}(t)} x\,|\,e^{-s \,{\mathcal {A}}(t)}x\right] _H \\&\le \tfrac{C}{s}\, \Vert x \Vert _H^2. \end{aligned}$$

The second inequality in (c) follows by duality.

The estimate (d) follows in a a natural way from (a) and (c) by writing the resolvent as the Laplace transform of the semigroup. Finally, in order to prove assertion (e), we notice that for a constant $\mu \ge 0$ we have

$$\begin{aligned} \Vert e^{-z \,(A(t) + \mu )} x \Vert _H \le \Vert e^{-z \, A(t)} x \Vert _H, \end{aligned}$$

for all . The same estimate also holds when replacing the norm of $H$ by the norm of $V$ or the norm of $V'$. Now we use the Cauchy formula to obtain (b) for $A(t) + \mu$. Assertion (d) for $A(t) + \mu$ follows again by the Laplace transform. The other estimates are obvious. $\square$

Finally we mention the following easy corollary of the proposition.

Corollary 7

Let $\omega : \mathbb {R}\rightarrow \mathbb {R}_+$ be some function and assume that

$$\begin{aligned} {| \mathop {{\varvec{\mathfrak {a}}}}(t; u, v)-\mathop {{\varvec{\mathfrak {a}}}}(s; u, v)| \le \omega (|t{-}s|) \Vert u \Vert _V \Vert v \Vert _V} \end{aligned}$$

for all $u, v \in V$. Then

$$\begin{aligned} \Vert R(z,A(t)) - R(z,A(s)) \Vert _{\mathcal {B}(H)} \le \tfrac{ c_\theta }{|z|} \omega (| t{-}s|) \end{aligned}$$

for all $z \notin S_{\theta }$ with any fixed .

Proof

Observe that for $u, v \in V$,

$$\begin{aligned}&\left| \left[ R(z,A(t)) u - R(z,A(s)) u\,|\,v\right] _H\right| \\&\quad = \left| \left[ R(z,A(t)) ( A(s)-A(t) ) R(z,A(s)) u\,|\,v\right] _H\right| \\&\quad = \left| \left[ A(s) R(z,A(s)) u\,|\, R(z,A(t))^* v\right] _H - \left[ A(t)R(z,A(s))u\,|\, R(z,A(t))^* v\right] _H\right| \\&\quad = \left| \mathop {{\varvec{\mathfrak {a}}}}(s; R(z,A(s)) u, R(z,A(t))^* v) - \mathop {{\varvec{\mathfrak {a}}}}(t; R(z,A(s)) u, R(z,A(t))^* v)\right| \\&\quad \le \tfrac{c_\theta }{|z|} \, \omega (| s{-}t |) \, \Vert u \Vert _H\,\Vert v \Vert _H, \end{aligned}$$

where we used Proposition 6(d). $\square$

Next we come to a formula for the solution $u$ of (P$^{\prime }$) in $V'$. Recall that $u$ exists by Lions’ theorem mentioned in the introduction. This formula already appears in Acquistapace–Terreni [1]. Fix $f \in C_c^\infty (0, \tau ; H)$ and $u_0 \in H$.

Lemma 8

For almost every $t \in (0, \tau )$, we have, in $V'$,

$$\begin{aligned} u(t)= & {} \int _0^t e^{-(t{-}s) {\mathcal {A}}(t)} ({\mathcal {A}}(t){-}{\mathcal {A}}(s)) u(s)\,{\mathrm {d}}s\nonumber \\&+ \int _0^t e^{-(t{-}s) {\mathcal {A}}(t) } f(s)\,{\mathrm {d}}s+ e^{-t \, {\mathcal {A}}(t)}u_0. \end{aligned}$$

(12)

Proof

Recall that $u \in W^1_2(0, \tau ; V')$ by Lions’ theorem and hence $u$ has a continuous representative. Fix $t \in (0, \tau )$ such that $\mathcal {D}({\mathcal {A}}(t)) = V$ (recall that this is true for almost all $t$). Set $v(s) := e^{-(t{-}s) \, {\mathcal {A}}(t) } u(s)$ for $0<s\le t<\tau$. Recall that $-{\mathcal {A}}(t)$ generates a bounded semigroup $e^{-s {\mathcal {A}}(t) }$ on $V'$, see Proposition 6(b). Since $u \in W^1_2(0,\tau ; V')$, $v$ has a distributional derivative in $V'$ which satisfies

$$\begin{aligned} v'(s)&= {\mathcal {A}}(t) e^{-(t{-}s) {\mathcal {A}}(t) } u(s) + e^{-(t{-}s) {\mathcal {A}}(t) } (f(s) - {\mathcal {A}}(s) u(s) )\\&= e^{-(t{-}s) {\mathcal {A}}(t) } ({\mathcal {A}}(t){-}{\mathcal {A}}(s)) u(s) + e^{-(t{-}s) {\mathcal {A}}(t) } f(s). \end{aligned}$$

Using the fact that $u \in L_2(0, \tau ; V)$, it follows that $v \in W^1_2(0,\tau ; V')$. Hence

$$\begin{aligned} v(t) - v(0)&= \int _0^t v'(s) \, {\mathrm {d}}s= \int _0^t e^{-(t{-}s) {\mathcal {A}}(t) } ({\mathcal {A}}(t){-}{\mathcal {A}}(s)) u(s)\,{\mathrm {d}}s\\&\quad + \int _0^t e^{-(t{-}s) {\mathcal {A}}(t) } f(s)\,{\mathrm {d}}s. \end{aligned}$$

This gives (12) by observing that $v(t) = u(t)$ and $v(0)=e^{-t \,{\mathcal {A}}(t)}u_0$. $\square$

Lemma 9

Suppose (4). Then for almost all $t \in [0, \tau ]$

$$\begin{aligned} {\mathcal {A}}(t) u(t) = (Q {\mathcal {A}}(\cdot )u(\cdot ))(t) + (L f)(t) + (Ru_0)(t) \end{aligned}$$

in $V'$, where

$$\begin{aligned} (Q g)(t) := \int _0^t {\mathcal {A}}(t) e^{-(t{-}s) {\mathcal {A}}(t)} ({\mathcal {A}}(t) - {\mathcal {A}}(s)) {\mathcal {A}}(s)^{-1}g(s)\,{\mathrm {d}}s\end{aligned}$$

(13)

and

$$\begin{aligned} (L f )(t) := {\mathcal {A}}(t) \int _0^t e^{-(t{-}s) \,{\mathcal {A}}(t)} f(s)\,{\mathrm {d}}s\quad \mathrm{and} \quad (R u_0)(t) := {\mathcal {A}}(t) e^{-t \, {\mathcal {A}}(t)} u_0. \end{aligned}$$

(14)

Proof

As in the proof of the previous lemma, we fix $t \in (0, \tau )$ such that $V = \mathcal {D}({\mathcal {A}}(t))$. It is enough to prove that each term in the sum (12) is in $V$. Observe that by analyticity, $e^{-t \,{\mathcal {A}}(t)}u_0 \in \mathcal {D}({\mathcal {A}}(t)) = V$. In passing we also note that since $u_0 \in H$, ${\mathcal {A}}(t) e^{-t \, {\mathcal {A}}(t)}u_0 = A(t) e^{-t \,A(t)}u_0$.

Concerning the first term, we recall that $u(s) \in V$ for almost all $s$ (note that $u \in L_2(0, \tau ; V)$). Therefore, $e^{-(t{-}s) {\mathcal {A}}(t)} ({\mathcal {A}}(t){-}{\mathcal {A}}(s)) u(s) \in V$ for almost every $s < t$ by the analyticity of the semigroup generated by $-{\mathcal {A}}(t)$. In addition,

$$\begin{aligned} \Vert {\mathcal {A}}(t) e^{(t{-}s) {\mathcal {A}}(t)} ({\mathcal {A}}(t){-}{\mathcal {A}}(s)) u(s) \Vert _{V'}&\le \tfrac{C}{t-s} \Vert ({\mathcal {A}}(t){-}{\mathcal {A}}(s)) u(s) \Vert _{V'} \\&= \tfrac{C}{t-s} \sup _{\Vert v \Vert _V = 1} |\mathop {{\varvec{\mathfrak {a}}}}(t; u(s), v) -\mathop {{\varvec{\mathfrak {a}}}}(s; u(s), v)|\\&\le \tfrac{C}{t-s} \, \omega (t{-}s) \Vert u(s) \Vert _V. \end{aligned}$$

Note that the operator

$$\begin{aligned} {\mathcal H}: \ h \mapsto \int _0^t \tfrac{\omega (t{-}s)}{t-s} \,h(s) \, {\mathrm {d}}s\end{aligned}$$

(15)

is bounded on $L_p(0, \tau ; \mathbb {R})$ for all $p \in [1, \infty ]$. The reason is that the associated kernel $(t,s) \mapsto 1\!\!1_{[0, t]}(s)\tfrac{\omega (t{-}s)}{t-s}$ is integrable with respect to each variable with a uniform bound with respect to the other variable as can be seen easily from (4). Recall again that $\Vert u(\cdot ) \Vert _V \in L_2(0, \tau ; \mathbb {R})$. Hence

$$\begin{aligned} s \mapsto 1\!\!1_{[0, t]}(s) \cdot {\mathcal {A}}(t) e^{-(t{-}s) \,{\mathcal {A}}(t)} ({\mathcal {A}}(t){-}{\mathcal {A}}(s)) u(s) \end{aligned}$$

is in $L_1(0, \tau ;V')$. Therefore, for every $\varepsilon > 0$

$$\begin{aligned} \int _0^{t-\varepsilon } e^{-(t{-}s) {\mathcal {A}}(t)} ({\mathcal {A}}(t){-}{\mathcal {A}}(s)) u(s)\,{\mathrm {d}}s\in \mathcal {D}({\mathcal {A}}(t)) \end{aligned}$$

and the fact that ${\mathcal {A}}(t)$ is a closed operator gives

$$\begin{aligned}&{\mathcal {A}}(t) \int _0^{t-\varepsilon } e^{-(t{-}s) {\mathcal {A}}(t)} ({\mathcal {A}}(t){-}{\mathcal {A}}(s)) u(s)\,{\mathrm {d}}s\\&\quad = \int _0^{t-\varepsilon } {\mathcal {A}}(t) e^{-(t{-}s) {\mathcal {A}}(t)} ({\mathcal {A}}(t){-}{\mathcal {A}}(s)) u(s)\,{\mathrm {d}}s. \end{aligned}$$

Since

$$\begin{aligned} s \mapsto 1\!\!1_{[0, t]}(s) \cdot {\mathcal {A}}(t) e^{-(t{-}s) {\mathcal {A}}(t)} ({\mathcal {A}}(t){-}{\mathcal {A}}(s)) u(s) \in L_1(0, \tau ;V'), \end{aligned}$$

we may let $\varepsilon \rightarrow 0$ and obtain $\int _0^{t} e^{-(t{-}s) {\mathcal {A}}(t)} ({\mathcal {A}}(t){-}{\mathcal {A}}(s)) u(s)\,{\mathrm {d}}s\in \mathcal {D}({\mathcal {A}}(t))$ and

$$\begin{aligned} {\mathcal {A}}(t) \int _0^t e^{-(t{-}s) {\mathcal {A}}(t)} ({\mathcal {A}}(t){-}{\mathcal {A}}(s)) u(s)\,{\mathrm {d}}s= \int _0^{t} {\mathcal {A}}(t) e^{-(t{-}s) {\mathcal {A}}(t)} ({\mathcal {A}}(t){-}{\mathcal {A}}(s)) u(s)\,{\mathrm {d}}s. \end{aligned}$$

Finally, the equality (12) and the fact that $u(t) \in V$ for almost all $t$ yields

$$\begin{aligned} \int _0^t e^{-(t{-}s) {\mathcal {A}}(t)} f(s) \, {\mathrm {d}}s\in \mathcal {D}({\mathcal {A}}(t)). \end{aligned}$$

This proves the lemma. $\square$

Recall the definition of the operator $L$,

$$\begin{aligned} L f(t) = {\mathcal {A}}(t) \int _0^t e^{-(t{-}s) A(t)} f(s)\,{\mathrm {d}}s. \end{aligned}$$

Let $f \in C_c^\infty (0, \tau ; H)$ and denote by $f_0$ its extension to the whole of $\mathbb {R}$ by $0$ outside $(0, \tau )$. Observe that $f_0$ is then in the Schwarz class $\mathcal {S}(\mathbb {R}; H)$. We denote for Fourier transform of $f_0$ by $\mathcal {F}f_0$ or $\widehat{f}_0$. Clearly,

$$\begin{aligned} \int _{-\infty }^t e^{-(t{-}s) A(t)} f_0(s)\,{\mathrm {d}}s= \tfrac{1}{2\pi } \; \int _{-\infty }^t e^{-(t{-}s) A(t)} \int _\mathbb {R}e^{\mathrm{i}s \xi } \widehat{f_0}(\xi )\,{\mathrm {d}}\xi \,{\mathrm {d}}s\end{aligned}$$

Now, exponential stability of $(e^{-s \, A(t)})_{s\ge 0}$ and the fact that $f_0 \in \mathcal {S}(\mathbb {R}; H)$ allows us to use Fubini’s theorem, giving

$$\begin{aligned} \int _{-\infty }^t e^{-(t{-}s) A(t)} \int _\mathbb {R}e^{\mathrm{i}s \xi } \widehat{f_0}(\xi )\,{\mathrm {d}}\xi \,{\mathrm {d}}s&= \int _\mathbb {R}\left( \int _{-\infty }^t e^{-(t{-}s) (\mathrm{i}\xi + A(t))} \,{\mathrm {d}}s\right) \widehat{f_0}(\xi ) e^{\mathrm{i}t \xi }\,{\mathrm {d}}\xi \\&= \int _\mathbb {R}(\mathrm{i}\xi +A(t))^{-1}\widehat{f_0}(\xi ) e^{\mathrm{i}t \xi }\,{\mathrm {d}}\xi . \end{aligned}$$

It follows that

$$\begin{aligned} \int _{-\infty }^t e^{-(t{-}s) A(t)} f_0(s)\,{\mathrm {d}}s= \tfrac{1}{2\pi } \int _\mathbb {R}(\mathrm{i}\xi +A(t))^{-1}\widehat{f_0}(\xi ) e^{\mathrm{i}t \xi }\,{\mathrm {d}}\xi . \end{aligned}$$

(16)

Observe that the right hand side of (16) converges in norm (as a Bochner integral) and that the same holds for

$$\begin{aligned} \int _\mathbb {R}A(t) (\mathrm{i}\xi +A(t))^{-1}\widehat{f_0}(\xi ) e^{\mathrm{i}t \xi }\,{\mathrm {d}}\xi \end{aligned}$$

since $\widehat{f_0} \in \mathcal {S}(\mathbb {R}; H)$. Thus, both terms in (16) take values in $\mathcal {D}(A(t))$. This shows that for $f\in C_c^\infty (0, \tau ; H)$, $(Lf)(t)$ is a well-defined function taking pointwise values in $H$. Hille’s theorem (see e.g., [11, II.2, Theorem 6]) then allows us to take the closed operator $A(t)$ inside the integral which finally gives the representation formula

$$\begin{aligned} L f(t) = \mathcal {F}^{-1}\left( \xi \mapsto \sigma (t, \xi ) \widehat{f_0}(\xi ) \right) (t), \end{aligned}$$

(17)

that allows us to see $L$ as a pseudo-differential operator with operator-valued symbol

$$\begin{aligned} \sigma (t, \xi ) = \left\{ \begin{array}{ll} A(0)(\mathrm{i}\xi +A(0))^{-1} &{} \text {if} \, t<0 \\ A(t)(\mathrm{i}\xi +A(t))^{-1} &{} \text {if} \, 0\le t \le \tau \\ A(\tau )(\mathrm{i}\xi +A(\tau ))^{-1} &{} \text {if} \, t > \tau \\ \end{array}\right. \end{aligned}$$

(18)

Lemma 10

Suppose that in addition to our standing assumptions [H1]–[H3] that (3) holds with $\omega : [0, \tau ] \rightarrow [0, \infty )$ a non-decreasing function such that

$$\begin{aligned} \int _0^\tau \tfrac{\omega (t)^2}{t}\,{\mathrm {d}}t< \infty . \end{aligned}$$

(19)

Then $L$ is a bounded operator on $L_2(0, \tau ; H)$.

Proof

We prove the Lemma by verifying the conditions of Theorem 13 below. Let $\sigma (t, \xi )$ be given by (18). We need to prove that

(20)

(21)

for $k = 0, 1, 2$. For $k=0$, (20) is just the sectoriality of $A(t)$, see Proposition 6 whereas (21) is precisely Corollary 7. Observe that a holomorphic function that satisfies

$$\begin{aligned} \Vert f(z) \Vert \le C \; \tfrac{1}{|z|} \end{aligned}$$

on the complement of a sector of angle $\theta$ will automatically satisfy

$$\begin{aligned} \Vert f^{(k)} (z) \Vert \le C_{\theta , k} \, C \; \tfrac{1}{|z|^{k+1}} \end{aligned}$$

on the complement of strictly larger sectors, simply by Cauchy’s integral formula for derivatives. Conditions (20) and (21) follow therefore for all $k \ge 1$. $\square$

Next we prove that the operator $L$ extends to a bounded operator on $L_p(0,\tau ;H)$.

Lemma 11

Under the assumptions of the previous lemma the operator $L$ is bounded on $L_p(0, \tau ; H)$ for all $p \in (1, \infty )$.

Proof

The operator $L$ is a singular integral operator with operator-valued kernel

$$\begin{aligned} K(t,s) = 1\!\!1_{\{0\le s \le t \le \tau \}} A(t) e^{-(t-s)A(t)}, \end{aligned}$$

where $1\!\!1$ denotes the indicator function. We prove that both $L$ and $L^*$ are of weak type $(1,1)$ operators and we conclude by the Marcinkiewicz interpolation theorem together with the previous lemma that $L$ is bounded on $L_p(0, \tau ; H)$ for all $p \in (1, \infty )$. It is known (see, e.g. [9, Theorems III.1.2, III.1.3]) that $L$ (respectively $L^*$) is of weak type $(1,1)$ if the corresponding kernel $K(t, s)$ satisfies the Hörmander integral condition. This means that we have to verify

$$\begin{aligned} \int _{|t-s|\ge 2|s'-s|} \Vert K(t,s) - K(t,s') \Vert _{\mathcal {B}(H)} \,{\mathrm {d}}t\le C \end{aligned}$$

(22)

and

$$\begin{aligned} \int _{|t-s|\ge 2|s'-s|} \Vert K(s,t) - K(s',t) \Vert _{\mathcal {B}(H)} \,{\mathrm {d}}t\le C \end{aligned}$$

(23)

for some constant $C$ independent of $s, s' \in (0, \tau )$. Note that the above mentioned theorems in [9] are formulated for integral operators on $L_p(\mathbb {R}; H)$ instead of $L_p(0, \tau ; H)$; however it is known that Hörmander’s integral condition works on any space satisfying the volume doubling condition, see [9, p. 15].

First consider the integral in (22). When $s \le s'$ and $2|s'{-}s| > \tau$ the integral vanishes. When $0 \le s \le s' \le t\le \tau$ and $2 s' {-}s \le \tau$, using that the semigroup $(e^{-s \,A(t)})_{s\ge 0}$ generated by $-A(t)$ is bounded holomorphic, with a norm bound independent of $t$, we have for some constant $C$

$$\begin{aligned}&\int _{|t{-}s|\ge 2|s'{-}s|} \Vert K(t,s) - K(t,s') \Vert _{\mathcal {B}(H)} \,{\mathrm {d}}t\\&\quad = \int _{2s'{-}s}^\tau \Vert A(t) e^{-(t{-}s)A(t)} - A(t) e^{-(t{-}s')A(t)} \Vert _{\mathcal {B}(H)} \, {\mathrm {d}}t\\&\quad = \int _{2s'{-}s}^\tau \int _s^{s'} \Vert A(t)^2 e^{-(t{-}r)A(t)} \,{\mathrm {d}}r \Vert _{\mathcal {B}(H)} \, {\mathrm {d}}t\\&\quad \le C \int _{2s'{-}s}^\tau \int _s^{s'} \tfrac{1}{(t{-}r)^2} \,{\mathrm {d}}r\,{\mathrm {d}}t= C \int _{2s'{-}s}^\tau \left[ \tfrac{1}{t{-}s'} - \tfrac{1}{t{-}s} \right] \, {\mathrm {d}}t\\&\quad = C \left[ \log \tfrac{t{-}s'}{t{-}s\,} \right] _{t = 2s'{-}s}^{t= \tau } \le C \log 2. \end{aligned}$$

When $s' < s$ and $3s-2s'>\tau$, the integral (22) vanishes. When $s'<s$ and $3s-2s' < \tau$, a similar calculation to the above shows that the integral is bounded by .

We now consider (23). When $s\le s'$, as above, we may assume that $3s{-}2s'>0$, since otherwise the integral in (23) vanishes. We have

$$\begin{aligned}&\int _{|t{-}s|\ge 2|s'{-}s|} \Vert K(s,t) - K(s',t) \Vert _{\mathcal {B}(H)} {\mathrm {d}}t\\&\quad = \int _{0}^{3s{-}2s'} \Vert A(s) e^{-(s{{-}}t) A(s)} - A(s') e^{-(s'{-}t) A(s')} \Vert _{\mathcal {B}(H)} \,{\mathrm {d}}t\\&\quad \le \int _{0}^{3s{-}2s'} \Vert A(s) e^{-(s{-}t) A(s)} - A(s) e^{-(s'{-}t) A(s)} \Vert _{\mathcal {B}(H)} \,{\mathrm {d}}t\\&\qquad + \int _{0}^{3s{-}2s'} \Vert A(s) e^{-(s'{-}t) A(s)} - A(s') e^{-(s'{-}t) A(s')} \Vert _{\mathcal {B}(H)} \,{\mathrm {d}}t=: I_1 + I_2. \end{aligned}$$

The first term $I_1$ is handled exactly as in the proof of (22). For the second term $I_2$, we write by the functional calculus

$$\begin{aligned} A(s) e^{-(s'{-}t) A(s)} - A(s') e^{-(s'{-}t) A(s')} = \tfrac{1}{2{\pi }i} \int _\Gamma z e^{-t z} \left[ R(z, A(s)) - R(z, A(s'))\right] \,{\mathrm {d}}z\end{aligned}$$

where $\Gamma$ is the boundary of an appropriate sector $S_{\theta }$. We apply Corollary 7 to deduce

$$\begin{aligned} \Vert A(s) e^{-(s'{-}t) A(s)} - A(s') e^{-(s'{-}t) A(s')} \Vert _{\mathcal {B}(H)}&\le C \int _0^\infty r e^{-(s'{-}t)r \cos \theta } \tfrac{\omega (s'{-}s)}{r} \,{\mathrm {d}}r\\&\le C \, \tfrac{\omega (s'{-}s)}{s'{-}t}. \end{aligned}$$

Therefore,

$$\begin{aligned}&\int _{0}^{3s{-}2s'} \Vert A(s) e^{-(s'{-}t) A(s)} - A(s') e^{-(s'{-}t) A(s')} \Vert _{\mathcal {B}(H)} \,{\mathrm {d}}t\\&\quad \le C \int _0^{3s{-}2s'} \tfrac{\omega (s'{-}s)}{s'{-}t} \,{\mathrm {d}}t\le C \int _0^\tau \omega (r) \,\tfrac{{\mathrm {d}}r}{r} = C', \end{aligned}$$

where we used the fact that $\omega$ is non-decreasing and $s'{-}s \le s'{-}t$ to write the second inequality. Finally, the integral (23) in the case $s'<s$ is treated similarly. Remark: A similar reasoning for the weak type $(1,1)$ estimate for $L$ and $L^*$ appears in [14, p.1051]. $\square$

Now we study the operator $R$.

Lemma 12

Assume (5). Then there exists $C > 0$ such that for every :

Proof

Recall that the operator $R$ is given by $(R g)(t) = A(t) e^{-t \,A(t)} g$ for $g \in H$. Let

$$\begin{aligned} (R_0 g)(t) := A(0) e^{-t \, A(0)}g. \end{aligned}$$

We aim to estimate the difference $(R-R_0)g$. Let $\Gamma = \partial S_{\theta }$ with and $\omega _0$ is as in the Proof of Proposition 6. Then, for $v \in V$, the functional calculus for the sectorial operators $A(t)$ and $A(0)$ gives

$$\begin{aligned}&\left\langle A(t) e^{-t \, A(t)} g - A(0) e^{-t \, A(0)}g , v \right\rangle \\&\quad = \tfrac{1}{2{\pi }i} \int _\Gamma \left\langle z e^{-t z} \left[ R(z, A(t)) - R(z, A(0))\right] g , v \right\rangle \, {\mathrm {d}}z\\&\quad = \tfrac{1}{2{\pi }i} \int _\Gamma \left\langle z e^{-t z} R(z, {\mathcal {A}}(t)) \left[ {\mathcal {A}}(0) - {\mathcal {A}}(t)\right] R(z, A(0))g , v \right\rangle \, {\mathrm {d}}z\\&\quad = \tfrac{1}{2{\pi }i} \int _\Gamma \left\langle z e^{-t z} \left[ {\mathcal {A}}(0) - {\mathcal {A}}(t)\right] R(z, A(0))g , R(z, A(t))^*v \right\rangle \, {\mathrm {d}}z\\&\quad = \tfrac{1}{2{\pi }i} \int _\Gamma z e^{-t z}\left[ \mathop {{\varvec{\mathfrak {a}}}}(0; R(z, A(0))g , R(z, A(t))^* v) \right. \\&\qquad \left. - \mathop {{\varvec{\mathfrak {a}}}}(t; R(z, A(0))g, R(z, A(t))^*v)\right] \,{\mathrm {d}}z. \end{aligned}$$

Now, taking the absolute value it follows from Proposition 6(d) that

$$\begin{aligned} \left| \left\langle (Rg - R_0 g)(t), v \right\rangle \right|&\le \tfrac{C_\alpha }{2{\pi }} \int _\Gamma \omega (t) |z| e^{-t \,\text {Re}(z)} \Vert R(z, A(0))g \Vert _V \Vert R(z, A(t))^*v \Vert _V \,|{\mathrm {d}}z|\\&\le \tfrac{C_{\alpha , \theta }}{2{\pi }} \omega (t) \Vert g \Vert _H \Vert v \Vert _H \int _\Gamma e^{-t \,\text {Re}z} \,|{\mathrm {d}}z| \\&\le C' \tfrac{\omega (t)}{t} \Vert g \Vert _H \Vert v \Vert _H. \end{aligned}$$

Since this true for all $v \in H$ we conclude that

$$\begin{aligned} \Vert (R u_0)(t) - (R_0 u_0)(t) \Vert _H \le C' \tfrac{\omega (t)}{t} \Vert u_0 \Vert _H. \end{aligned}$$

(24)

From the hypothesis (5) it follows that $Ru_0- R_0 u_0 \in L_p(0, \tau ; H)$. On the other hand, since $A(0)$ is invertible, it is well-known that $A(0) e^{-t \,A(0)}u_0 \in L_p(0, \tau ; H)$ if and only if (see Triebel [26, Theorem 1.14]). Therefore, $Ru_0 \in L_p(0, \tau ;H)$ and the lemma is proved. $\square$

3 Proofs of the main results

Proof of Theorem 2

Assume first that $u_0 = 0$ and let $f \in C_c^\infty (0, \tau ; H)$. From Lemma 9 it is clear that

$$\begin{aligned} (I-Q) A(\cdot )u(\cdot ) = L f(\cdot ). \end{aligned}$$

(25)

Recall that $L$ is bounded on $L_p(0, \tau ; H)$ by Lemma 11. We shall now prove that $Q$ is bounded on $L_p(0, \tau ; H)$. Let $g \in C_c^\infty (0, \tau ; H)$. By Proposition 6 we have

Since ${\Vert {\mathcal {A}}(t) x \Vert _{V'} = \sup _{\Vert v \Vert _V=1} \left| \mathop {{\varvec{\mathfrak {a}}}}(t; x, v)\right| }$, we use the regularity assumption (3) to bound $Qg$ further by

(26)

Now we estimate $\Vert {\mathcal {A}}(s)^{-1} g(s) \Vert _{V}$. By coercivity

$$\begin{aligned} \alpha \Vert {\mathcal {A}}(s)^{-1} g(s) \Vert _{V}^2&\le \text {Re}\mathop {{\varvec{\mathfrak {a}}}}(s; {\mathcal {A}}(s)^{-1}g(s), {\mathcal {A}}(s)^{-1}g(s))\\&= \text {Re}\langle {\mathcal {A}}(s) {\mathcal {A}}(s)^{-1}g(s), {\mathcal {A}}(s)^{-1}g(s) \rangle \\&= \text {Re}\left[ g(s)\,|\,{\mathcal {A}}(s)^{-1}g(s)\right] _H\\&\le \Vert g(s) \Vert _H^2 \Vert {\mathcal {A}}(s)^{-1} \Vert _{\mathcal {B}(H)}. \end{aligned}$$

We obtain from (26) that

(27)

Now, once we replace $A(t)$ by $A(t){+}\mu$, (26) is valid with a constant independent of $\mu \ge 0$ by Proposition 6(e). Using the estimate

$$\begin{aligned} \Vert ({\mathcal {A}}(s) + \mu )^{-1} \Vert _{\mathcal {B}(H)} \le \tfrac{1}{\mu }, \end{aligned}$$

in (27) for $A(s){+}\mu$ we see that

It remains to see that the operator $S$ defined by

is bounded on $L_p(0,\tau ; \mathbb {R})$. Observe that $S$ is an integral operator with kernel function . Hence by assumption (4) it is integrable with respect to each of the two variables with uniform bound with respect to the other variable. This implies that $S$ is bounded on $L_1(0, \tau ; H)$ and on $L_\infty (0, \tau ; H)$ and hence bounded on $L_p(0, \tau ;H)$.

It follows that $Q$ is bounded on $L_p(0, \tau ; H)$ with norm of at most $\tfrac{C''}{\sqrt{\mu }}$ for some constant $C''$. Taking then $\mu$ large enough makes $Q$ strictly contractive such that $(I-Q)^{-1}$ is bounded by the Neumann series. Then, for $f \in C_c^\infty (0, \tau ; H)$, (25) can be rewritten as

$$\begin{aligned} A(\cdot )u(\cdot ) = (I-Q)^{-1} L f(\cdot ). \end{aligned}$$

For general we suppose in addition to (4) that (5) holds. Lemma 12 shows that $R u_0 \in L_p(0, \tau ; H)$. As previously we conclude that

$$\begin{aligned} A(\cdot )u(\cdot ) = (I-Q)^{-1} (Lf + R u_0), \end{aligned}$$

whenever $f \in C_c^\infty (0, \tau ; H)$. Thus taking the $L_p$ norms we have

$$\begin{aligned} \Vert A(\cdot )u(\cdot ) \Vert _{L_p(0, \tau ;H)} \le C \Vert (Lf + R u_0) \Vert _{L_p(0, \tau ;H)}. \end{aligned}$$

We use again the previous estimates on $L$ and $R$ to obtain

Using the equation (P) we obtain a similar estimate for $u'$ and so

We write $u(t) = A(t)^{-1} A(t) u(t)$ and use one again the fact that the norms of $A(t)^{-1}$ on $H$ are uniformly bounded we obtain

We conclude therefore that the following a priori estimate holds

(28)

where the constant $C$ does not depend on $f \in C_c^\infty (0, \tau ; H)$.

Now let $f \in L_p(0, \tau ; H)$ and $(f_n) \subset C_c^\infty (0, \tau ; H)$ be an approximating sequence that converges in $L_p$ and pointwise almost everywhere. For each $n$, denote by $u_n$ the solution of (P) with right hand side $f_n$. We apply (28) to $u_n - u_m$ and we see that there exists $u \in W^1_p(0, \tau ; H)$ and $v \in L_p(0, \tau ; H)$ such that

$$\begin{aligned} u_n \mathop {\longrightarrow }\limits ^{L_{p}} u\qquad u_n' \mathop {\longrightarrow }\limits ^{L_{p}}u' \quad \text {and}\quad A(\cdot )u_n(\cdot )\mathop {\longrightarrow }\limits ^{L_{p}} v \end{aligned}$$

(29)

By extracting a subsequence, we may assume that these limits hold in the pointwise a.e. sense as well. For a fixed $t$, the operator $A(t)$ is closed and so $v (\cdot ) = A(\cdot ) u(\cdot )$. Passing to the limit in the equation

$$\begin{aligned} u_n'(t) + A(t)\, u_n(t) = f_n(t) \end{aligned}$$

shows that

$$\begin{aligned} u'(t) + A(t) u(t) = f(t) \end{aligned}$$

for a.e. $t \in (0, \tau )$. On the other hand, by Sobolev embedding, $(u_n)$ is bounded in $C([0,\tau ];H)$ and hence $u(0) = u_0$ since $u_n(0) = u_0$ by the definition of $u_n$. We conclude that $u$ satisfies

$$\begin{aligned} u'(t) + A(t)\, u(t) = f(t)\quad u(0)=u_0 \end{aligned}$$

in the $L_p$ sense. This means that $u$ is a solution to (P). Moreover, (28) transfers from $u_n$ to $u$. The uniqueness of the solution $u$ follows from the a priori estimate (28) as well. $\square$

Proof of Corollary 3

The result follows from Theorem 2 and the observation that

see e.g., [18, Corollaries 4.37, 4.30]. $\square$

Proof of Corollary 4

By the definition of maximal regularity, one may modify the operators $A(t), \ 0 \le t \le \tau$, on a set of Lebesgue measure zero. Therefore, we may assume without loss of generality that the mapping ${t \mapsto \mathop {{\varvec{\mathfrak {a}}}}(t; u, v)}$ is right continuous. We may assume again that the operators $A(\cdot )$ are all invertible. We apply Corollary 3 to the evolution equation

$$\begin{aligned} \left\{ \begin{array}{l} u_j'(t) + A(t) u_j(t) = f(t)\quad t \in (t_{j}, t_{j+1})\\ u_j(t_{j}) = u_{j-1}(t_{j}), \end{array} \right. \end{aligned}$$

since it is obvious that the assumed $\alpha$-Hölder continuity for some implies both (4) and (6). The solution $u_{j}$ is in $W^1_2(t_{j}, t_{j+1} ; H)$ provided the initial data satisfies

Note that the endpoint $u_{j-1}(t_{j})$ is well defined since $u_j \in C([t_{j}, t_{j+1}];H)$ by [8, XVIII Chapter3, p. 513]. In order to obtain a solution $u \in W^1_2(0, \tau ; H)$, we glue the solutions $u_j$. That is, we set $u(t) = u_j(t)$ for $t \in [t_j, t_{j+1}]$. What remains then to prove is that , where $u \in W^1_2(0, \tau ; V')$ is the solution in $V'$ given by Lions’ theorem.

Fix one of the discontinuity points $t_j$ and consider the autonomous equation

$$\begin{aligned} v'(s) + A(t_j^-) v(s) = f(s), \qquad v(0)=0. \end{aligned}$$

By maximal regularity of $A(t_j^-)$, its solution $v(s) = \int _0^s e^{-(s{-}r) A(t_j^-)} \, f(r)\,{\mathrm {d}}r$ satisfies $v(s) \in \mathcal {D}(A(t_j^-))$ for almost all $s$. Choose a sequence $(s_n)$ converging to $t_j$ from the left such that $v(s_n) \in \mathcal {D}(A(t_j^-))$. Since $A(t_j^-)$ is an accretive and sectorial operator it has a bounded $H^\infty$-calculus of some angle . Hence $A(t_j^-)$ and its adjoint admit square-function estimates of the form:

see e.g. [2, Sect. 8]. It follows that

(30)

Thus, (30) implies that the sequence $(v(s_n))_{n\ge 0}$ is bounded in the Hilbert space . It has a weakly convergent subsequence. By extracting a subsequence, we may assume that $(v(s_n))_{n\ge 0}$ converges weakly to some $v$ in . But the continuity of the solution $v(\cdot )$ implies that $v(s_n)$ tends also to $v(t_j)$ in $H$. Therefore,

In particular,

(31)

On the other hand, as in the proof of Lemma 8, we have for all $t> t_{j{-}1}$

$$\begin{aligned}&u(t) - e^{-(t{-}t_{j-1}) {\mathcal {A}}(t) } u(t_{j-1}) \nonumber \\&\quad = \int _{t_{j{-}1}}^t e^{-(t{-}s) {\mathcal {A}}(t)} ({\mathcal {A}}(t) - {\mathcal {A}}(s)) u(s)\,{\mathrm {d}}s+ \int _{t_{j{-}1}}^t e^{-(t{-}s) {\mathcal {A}}(t)} f(s)\,{\mathrm {d}}s. \end{aligned}$$

(32)

By analyticity of the semigroup $e^{-s \, {\mathcal {A}}(t_j^-) }$ it follows that . Now we prove that . It is enough to prove that

(33)

To this end, let $h \in H$ be such that $\Vert h \Vert _H \le 1$. By Proposition 6, (c), we have

(34)

Thus, since the form is $C^\alpha$ on $(t_{j-1}, t_j)$, we have for every small $\epsilon > 0$

Taking the supremum over all $h \in H$ of norm we obtain

Since this is true for all $\epsilon > 0$ we obtain (33). We conclude from this, (31) and (32) that . Finally, the latter space coincides with by the assumptions of the corollary. $\square$

4 Operator-valued pseudo-differential operators

Given a Hilbert space $H$, our aim in this section is to prove results on boundedness on $L_2(\mathbb {R}^n ; H)$ for pseudo-differential operators with minimal smoothness assumption on the symbol. The main results we will show here were proved in [20] in the scalar case (i.e., $H = \mathbb {C}$), see also [4]. The operator-valued version follows the lines in [20] and we give the details here for the sake of completeness. Let us mention the paper [15] where results on $L_p$-boundedness of pseudo-differential operators with operator-valued symbols are proved even when $H$ is not a Hilbert space. We do not appeal to the results from [15] in order to avoid assuming continuity and concavity assumptions on the function $\omega$ in Theorem 2.

Let $H$ be a Hilbert space on $\mathbb {C}$, with scalar product $\left[ \cdot \,|\,\cdot \right] _H$ and associated norm $\Vert \cdot \Vert _H$.

$$\begin{aligned} \sigma : \mathbb {R}^n \times \mathbb {R}^n \rightarrow \mathcal {B}(H) \end{aligned}$$

be bounded measurable. We define for $f$ in the Schwartz space $\mathcal {S}(\mathbb {R}^n; H)$

$$\begin{aligned} T_\sigma f(x) := \tfrac{1}{(2\pi )^n} \int _{\mathbb {R}^n} \sigma (x, \xi ) \widehat{f}(\xi ) e^{i x \xi }\ {\mathrm {d}}\xi . \end{aligned}$$

where we write $\widehat{f}$ for the Fourier transform of $f$. We shall also use the notation $|\xi |$ for the Euclidean distance in $\mathbb {R}^n$ and write henceforth $\langle \xi \rangle := \sqrt{1 + |\xi |^2}$. For the rest of this section, we will ignore the normalisation constant in the definition of the Fourier transform.

Theorem 13

Suppose that there exists a non-decreasing function $\omega : [0, \infty ) \rightarrow [0, \infty )$ such that

$$\begin{aligned} \Vert \partial _\xi ^\alpha \sigma (x, \xi ) \Vert _{\mathcal {B}(H)} \le C_\alpha \langle \xi \rangle ^{-|\alpha |} \end{aligned}$$

and

$$\begin{aligned} \Vert \partial _\xi ^\alpha \sigma (x, \xi ) - \partial _\xi ^\alpha \sigma (x', \xi ) \Vert _{\mathcal {B}(H)} \le C_\alpha \langle \xi \rangle ^{-| \alpha | } \omega (| x-x'|) \end{aligned}$$

for all and some positive constant $C_\alpha$. Suppose in addition that

$$\begin{aligned} \int _0^1 \omega (t)^2 \, \tfrac{{\mathrm {d}}t}{t} < \infty , \end{aligned}$$

then $T_\sigma$ is a bounded operator on $L_2(\mathbb {R}^n ; H)$.³

Proof

Let $\varphi \in C^\infty (\mathbb {R}^n)$ be a non-negative function with support in the unit ball such that $\int _{\mathbb {R}^n} \varphi (x) \,{\mathrm {d}}x= 1$. Fix a constant $\delta \in (0,1)$ and define the symbols

$$\begin{aligned} \sigma _1(x, \xi ) := \int _{\mathbb {R}^n} \varphi (y) \, \sigma (x - \frac{y}{\langle \xi \rangle ^{\delta }}, \xi ) \, {\mathrm {d}}y\end{aligned}$$

and

$$\begin{aligned} \sigma _2(x, \xi ) := \sigma (x, \xi ) - \sigma _1(x, \xi ). \end{aligned}$$

It is clear that

$$\begin{aligned} \sigma _1(x, \xi ) = \int _{\mathbb {R}^n} \varphi (\langle \xi \rangle ^\delta (x-y)) \, \sigma (y,\xi )\langle \xi \rangle ^{n \delta } \,{\mathrm {d}}y\end{aligned}$$

and one checks that

$$\begin{aligned} \Vert \partial _x^\beta \partial _{\xi }^\alpha \sigma _1(x, \xi ) \Vert _{\mathcal {B}(H)} \le c_{\alpha , \beta } \langle \xi \rangle ^{- |\alpha | + \delta | \beta |} \le c_{\alpha , \beta } \langle \xi \rangle ^{-\delta (|\alpha | - | \beta |)} \end{aligned}$$

(35)

and

$$\begin{aligned} \Vert \partial _\xi ^\alpha \sigma _2(x,\xi ) \Vert _{\mathcal {B}(H)} \le c_\alpha \omega (\langle \xi \rangle ^{-\delta }) \langle \xi \rangle ^{-|\alpha |} \end{aligned}$$

(36)

for $|\alpha | \le [\frac{n}{2}] + 2$ and all $\beta$. Using (36) we conclude by the next theorem that $T_{\sigma _2}$ is bounded on $L_2(\mathbb {R}^n; H)$. The boundedness of $T_{\sigma _1}$ on $L_2(\mathbb {R}^n; H)$ follows from (35) and Theorem 1 in [19]. Note that it is assumed there that the symbol is $C^\infty$ but the estimate needed in Theorem 1 is exactly (35) with $|\alpha | \le [\frac{n}{2}] + 2$.

Theorem 14

Let $\delta \in (0, 1)$ and $w: [0, 1]\rightarrow \mathbb {R}_+$ be a non-decreasing measurable function satisfying

$$\begin{aligned} \int _0^1 \omega (t)^2\,\tfrac{{\mathrm {d}}t}{t} < \infty . \end{aligned}$$

If a bounded strongly measurable symbol $\sigma : \mathbb {R}^n \times \mathbb {R}^n \rightarrow \mathcal {B}(H)$ satisfies

$$\begin{aligned} \Vert \partial _\xi ^\alpha \sigma (x, \xi ) \Vert _{\mathcal {B}(H)} \le C_\alpha \langle {\xi }\rangle ^{-\alpha } \, \omega ( \langle {\xi }\rangle ^{-\delta }) \end{aligned}$$

(37)

for , then $T_\sigma$ is bounded on $L_2(\mathbb {R}^n; H )$.

Proof

Let $\varphi$ be a non-negative $C_c^\infty$ function satisfying $\varphi (\xi ) = 1$ for $|\xi | \le 2$ and $\varphi (\xi ) = 0$ for $|\xi |>3$. Then we may rewrite

$$\begin{aligned} \sigma (x, \xi ) = \varphi (\xi ) \sigma (x, \xi ) + (1{-}\varphi (\xi )) \sigma (x, \xi ) = \sigma _1(x, \xi ) + \sigma _2(x, \xi ) \end{aligned}$$

and treat both parts separately. For the first part, let $f \in \mathcal {S}(\mathbb {R}^n ; H)$ and $h \in H$. Then

$$\begin{aligned} \left[ (T_{\sigma _1} f)(x) \,|\,h\right] _H&= \int _{\mathbb {R}^n} e^{ix\cdot \xi } \left[ \sigma _1(x, \xi ) \widehat{f}(\xi )\,|\,h\right] _H\,{\mathrm {d}}\xi \\&= \int _{\mathbb {R}^n} \left[ f(y) \,|\,\int _{\mathbb {R}^n} e^{i(x-y)\cdot \xi } \sigma _1(x, \xi )^*h \,{\mathrm {d}}\xi \right] _H\,{\mathrm {d}}y\\&=: \int _{\mathbb {R}^n} \left[ f(y) \,|\, K(x, x{-}y) \right] _H\,{\mathrm {d}}y. \end{aligned}$$

By Plancherel’s theorem,

$$\begin{aligned} \int _{\mathbb {R}^n} \Vert \langle {z}\rangle ^{2\kappa } K(x, z) \Vert ^2\,{\mathrm {d}}z&\le \sum _{|\alpha |\le 2\kappa } c_\alpha \int _{\mathbb {R}^n} \Vert z^\alpha K(x, z) \Vert ^2\,{\mathrm {d}}z\\&= \sum _{|\alpha |\le 2\kappa } c_\alpha \int _{\mathbb {R}^n} \Vert \partial _\xi ^\alpha \sigma _1(x, \xi )^* h \Vert ^2\,{\mathrm {d}}z=: C_1 \Vert h \Vert ^2, \end{aligned}$$

where $C_1$ is finite due to the support of $\sigma _1$. By the Cauchy-Schwarz inequality,

$$\begin{aligned}&\Vert T_{\sigma _1} f \Vert _{L_2(\mathbb {R}^n; H) }^2 \\&\quad = \int _{\mathbb {R}^n} \sup _{h \in H, \Vert h \Vert \le 1} \left| \int _{\mathbb {R}^n} \left[ f(y)\,|\,K(x, x{-}y)\right] _H\,{\mathrm {d}}y\right| ^2 \,{\mathrm {d}}x\\&\quad \le \int _{\mathbb {R}^n} \sup _{h \in H, \Vert h \Vert \le 1} \left( \int _{\mathbb {R}^n} \langle {x{-}y}\rangle ^{-2\kappa } \Vert f(y) \Vert _H^2 \, {\mathrm {d}}y\right) \\&\qquad \times \, \left( \int _{\mathbb {R}^n} \Vert \langle {x{-}y}\rangle ^{2\kappa } K(x, x{-}y) \Vert _H^2 \,{\mathrm {d}}y\right) \,{\mathrm {d}}x\\&\quad \le C_1 \int _{\mathbb {R}^n} \int _{\mathbb {R}^n} \langle {x{-}y}\rangle ^{-2\kappa } \,{\mathrm {d}}x\;\Vert f(y) \Vert _H^2 \,{\mathrm {d}}y= C_1 C_2 \Vert f \Vert _{L_2(\mathbb {R}^n; H)}^2. \end{aligned}$$

Thus, $T_{\sigma _1}$ is bounded on $L_2(\mathbb {R}^n; H)$.

Next we show boundedness of $T_{\sigma _2}$. Recall that $\text {supp}(\sigma _2) \subset \{ (x, \xi ) :\;|\xi | \ge 2 \}$. Let $\phi \in C_c^\infty$ such that $\text {supp}(\phi ) \subseteq [1,2]$ and $\int _0^\infty |\phi (t)|^2\,\tfrac{{\mathrm {d}}t}{t} = 1$. Let $f \in \mathcal {S}(\mathbb {R}^n; H)$ and $h \in H$.

Recall that $\phi \in \mathcal {S}(\mathbb {R})$, so that by Plancherel’s theorem

$$\begin{aligned} \left[ (T_{\sigma _2} f)(x)\,|\,h\right] _H = \int _0^{1} \int _{\mathbb {R}^n} \left[ K_1(t, x, z)\,|\, K_2(t, x, z)\right] _H \,{\mathrm {d}}z\, \tfrac{{\mathrm {d}}t}{t}, \end{aligned}$$

where $K_1$ and $K_2$ are the respective Fourier transforms

and

Now, by the Cauchy–Schwarz inequality,

$$\begin{aligned}&\left| \left[ (T_{\sigma _2} f)(x)\,|\,h\right] _H \right| ^2 \\&\quad = \left| \int _0^{1} \int _{\mathbb {R}^n} \left[ \langle {z}\rangle ^{-\kappa }K_1(t, x,z)\,|\,\langle {z}\rangle ^{\kappa } K_2(t, x, z) \right] _H \,{\mathrm {d}}z\,\tfrac{{\mathrm {d}}t}{t} \right| ^2 \\&\quad \le \left( \int _0^{1} \int _{\mathbb {R}^n} \left\| \langle {z}\rangle ^{-\kappa } K_1(t, x,z) \right\| _H^2 \,{\mathrm {d}}z\,\tfrac{{\mathrm {d}}t}{t}\right) \left( \int _0^{1} \int _{\mathbb {R}^n} \left\| \langle {z}\rangle ^{\kappa } K_2(t, x, z) \right\| _H^2 \,{\mathrm {d}}z\,\tfrac{{\mathrm {d}}t}{t}\right) . \end{aligned}$$

Observe that . Recall that $\phi \in C_c^\infty (\mathbb {R})$ has its support in $[1, 2]$, so that, for $|\xi | \ge 1$, and using the growth assumption (37) on derivatives of $\sigma$,

Note that we used here the monotonicity of $\omega$. The Dini type condition on $\omega$ then gives

$$\begin{aligned} \int _0^{1} \int _{\mathbb {R}^n} \left\| \langle {z}\rangle ^{\kappa } K_2(t, x, z) \right\| _H^2 {\mathrm {d}}z\tfrac{{\mathrm {d}}t}{t} {\le } C'' \Vert h \Vert _H^2 \int _0^{1} \int _{1\le |\xi |^2 \le 2} \omega (t^{+\delta })^2\,{\mathrm {d}}\xi \,\tfrac{{\mathrm {d}}t}{t} =: C_2 \Vert h \Vert _H^2. \end{aligned}$$

We conclude by observing that, again by Plancherel,

$$\begin{aligned}&\int _{\mathbb {R}^n} \sup _{h\in H, \Vert h \Vert \le 1} \left| \left[ (T_{\sigma _2} f)(x)\,|\,h\right] _H \right| ^2\,{\mathrm {d}}x\\&\quad \le C_2 \int _{\mathbb {R}^n} \int _0^{1} \int _{\mathbb {R}^n} \langle {z}\rangle ^{-2\kappa } \left\| K_1(t, x,z) \right\| _H^2 \,{\mathrm {d}}z\,\tfrac{{\mathrm {d}}t}{t} \,{\mathrm {d}}x\\&\quad = C_2 \int _0^{1} \int _{\mathbb {R}^n} \langle {z}\rangle ^{-2\kappa } \int _{\mathbb {R}^n} \left\| K_1(t, x,z) \right\| _H^2 \,{\mathrm {d}}x\,{\mathrm {d}}z\,\tfrac{{\mathrm {d}}t}{t}\\&\quad = C_2' \int _0^{1} \int _{\mathbb {R}^n} \langle {z}\rangle ^{-2\kappa } \int _{\mathbb {R}^n} |\phi (|t\xi |)|^2 \Vert \widehat{f}(\xi ) \Vert _H^2 \,{\mathrm {d}}\xi \,{\mathrm {d}}z\,\tfrac{{\mathrm {d}}t}{t}\\&\quad \le c_n C_2' \int _{\mathbb {R}^n} \Vert \widehat{f}(\xi ) \Vert _H^2 \int _0^{\infty } |\phi (|t\xi |)|^2 \,\tfrac{{\mathrm {d}}t}{t} \,{\mathrm {d}}\xi \\&\quad = c_n C_2' \int _{\mathbb {R}^n} \Vert \widehat{f}(\xi ) \Vert _H^2\, {\mathrm {d}}\xi .\\ \end{aligned}$$

Therefore, $T_{\sigma _2}$ and hence $T_\sigma$ are bounded on $L_2(\mathbb {R}^n; H)$. $\square$

5 Examples

In this section we discuss some examples and applications of our results. We shall focus on two simple but relevant linear problems which involve elliptic operators. Note that following [3], we may also consider quasi-linear evolution equations. Our maximal regularity results can be used to improve some results in [3] on existence of solutions to quasi-linear problems in the sense that we assume less regularity with respect to $t$ of the coefficients of the equations. We shall not pursue this direction here.

5.1 Elliptic operators

Define on $H = L_2(\mathbb {R}^d, {\mathrm {d}}x)$ the sesquilinear forms

$$\begin{aligned} \mathop {{\varvec{\mathfrak {a}}}}(t; u, v) = \sum _{k,j=1}^d \int _{\mathbb {R}^d}a_{kj}(t,x) \partial _k u\overline{\partial _j v} \ {\mathrm {d}}x\quad \text {for}\quad u, v \in W^{1}_2(\Omega ). \end{aligned}$$

We assume that $a_{kj}: [0, \tau ] \times \mathbb {R}^d \rightarrow \mathbb {C}$ such that:

$$\begin{aligned} a_{kj} \in L_\infty ([0,\tau ]\times \mathbb {R}^d) \quad \text {for}\;1 \le k, j \le d, \end{aligned}$$

and

$$\begin{aligned} \text {Re}\sum _{k,j=1}^d a_{kj}(t,x) \xi _k \overline{\xi _j} \ge \nu |\xi |^2 \quad \text {for all }\quad \xi \in \mathbb {C}^d \quad \text {and a.e. }\quad (t,x) \in [0, \tau ]\times \mathbb {R}^d. \end{aligned}$$

Here $\nu > 0$ is a constant independent of $t$.

It is easy to check that ${\mathop {{\varvec{\mathfrak {a}}}}(t; \cdot , \cdot )}$ is $W^{1}_2(\mathbb {R}^d)$-bounded and quasi-coercive. The associated operator with ${\mathop {{\varvec{\mathfrak {a}}}}(t; \cdot , \cdot )}$ is the elliptic operator given by the formal expression

$$\begin{aligned} A(t) u = - \sum _{k,j=1}^d \partial _j \left( a_{kj}(t,.) \partial _k u \right) . \end{aligned}$$

In addition to the above assumptions we assume that for some constant $M$ and

$$\begin{aligned} |a_{kj}(t,x) \!-\! a_{kj}(s, x) |\le M |t\!-\!s|^\alpha \quad \text {for a.e.} \quad x \in \mathbb {R}^d \quad \text {and all}\quad t, s \in [0, \tau ].\qquad \end{aligned}$$

(38)

By the Kato square root property, it is known that , see [24]. Therefore, applying Corollary 4 we conclude that for every $f \in L_2(0,\tau ; H)$ the problem

$$\begin{aligned} \left\{ \begin{array}{l} u'(t) - \sum _{k,j=1}^d \partial _j \left( a_{kj}(t,.) \partial _k u(t) \right) = f(t), \quad t \in (0, \tau ] \\ u(0)= u_0 \in W^{1}_2(\mathbb {R}^d) \end{array} \right. \end{aligned}$$

has a unique solution $u \in W^{1}_2(0,\tau ; H) \cap L_2(0,\tau ; W^{1}_2(\mathbb {R}^d))$.

5.2 Time-dependent Robin boundary conditions

We consider here the Laplacian on a domain $\Omega$ with a time dependent Robin boundary condition

$$\begin{aligned} \partial _\nu u(t) + \beta (t,.) u = 0 \quad \text {on}\quad \Gamma = \partial \Omega , \end{aligned}$$

(39)

for some function $\beta : [0,\tau ] \times \Gamma \rightarrow \mathbb {R}$. This example is taken from [3]. The difference here is that we assume less regularity on $\beta$ and also that we can treat maximal $L_p$-regularity for $p \in (1, \infty )$ whereas the results in [3] are restricted to the case $p=2$.

Let $\Omega$ be a bounded domain of $\mathbb {R}^d$ with Lipschitz boundary $\Gamma = \partial \Omega$ and denote by $\sigma$ the $(d{-}1)$-dimensional Hausdorff measure on $\Gamma$. Let $\beta : [0,\tau ] \times \Gamma \rightarrow \mathbb {R}$ be a bounded measurable function which is Hölder continuous w.r.t. the first variable, i.e.,

$$\begin{aligned} |\beta (t,x) - \beta (s, x) |\le M |t-s|^\alpha \end{aligned}$$

(40)

for some constants $M$, and all $t, s \in [0, \tau ], \ x \in \Gamma$. We consider the symmetric form

$$\begin{aligned} {\mathop {{\varvec{\mathfrak {a}}}}(t; u, v) = \int _\Omega \nabla u \nabla v\ d x + \int _\Gamma \beta (t, .) u v\ d\sigma , \, u, v \in W^{1}_2(\Omega ).} \end{aligned}$$

(41)

The form ${\mathop {{\varvec{\mathfrak {a}}}}(t; \cdot , \cdot )}$ is $W^{1}_2(\Omega )$-bounded and quasi-coercive. The first statement follows from the continuity of the trace operator and the boundedness of $\beta$. The second one is a consequence of the inequality

$$\begin{aligned} \int _\Gamma |u |^2 \ d\sigma \le \varepsilon \Vert u \Vert _{ W^{1}_2}^2 + c_\varepsilon \Vert u \Vert _{L_2(\Omega )}^2, \end{aligned}$$

(42)

which is valid for all $\varepsilon > 0$ ($c_\varepsilon$ is a constant depending on $\varepsilon$). Note that (42) is a consequence of compactness of the trace as an operator from $W^{1}_2(\Omega )$ into $L_2(\Gamma , d \sigma )$, see [21, Chapter 2 §6, Theorem 6.2].

The operator $A(t)$ associated with ${\mathop {{\varvec{\mathfrak {a}}}}(t; \cdot , \cdot )}$ on $H:= L_2(\Omega )$ is (minus) the Laplacian with time dependent Robin boundary conditions (39). As in [3], we use the following weak definition of the normal derivative. Let $v \in W^{1}_2(\Omega )$ such that $\Delta v \in L_2(\Omega )$. Let $h \in L_2(\Gamma , d \sigma )$. Then $\partial _\nu v = h$ by definition if $\int _\Omega \nabla v \nabla w + \int _\Omega \Delta v w = \int _\Gamma h w \, d \sigma$ for all $w \in W^{1}_2(\Omega )$. Based on this definition, the domain of $A(t)$ is the set

$$\begin{aligned} \mathcal {D}(A(t)) = \{ v \in W^{1}_2(\Omega ): \Delta v \in L_2(\Omega ), \partial _\nu v + \beta (t) v\vert _\Gamma = 0 \}, \end{aligned}$$

and for $v\in \mathcal {D}(A(t))$ the operator is given by $A(t)v = - \Delta v$.

Observe that the form ${\mathop {{\varvec{\mathfrak {a}}}}(t; \cdot , \cdot )}$ is symmetric, so that . From Corollary 4 it follows that the heat equation

$$\begin{aligned} \left\{ \begin{array}{ll} u'(t) - \Delta u(t) = f(t)&{}\\ u(0) = u_0 &{}\quad u_0 \in W^{1}_2(\Omega )\\ \partial _\nu u(t) + \beta (t,.) u = 0 &{}\quad \text {on}\;\Gamma \end{array} \right. \end{aligned}$$

has a unique solution $u \in W^{1}_2(0,\tau ; L_2(\Omega ))$ whenever $f \in L_2(0,\tau ; L_2(\Omega ))$. This example is also valid for more general elliptic operators than the Laplacian.

Note that in both examples we have assumed $\alpha$-Hölder continuity in (38) and (40). We could replace this assumption by piecewise $\alpha$-Hölder continuity as authorised by Corollary 4.