Elliptic and Parabolic Boundary Value Problems in Weighted Function Spaces

Abstract

In this paper we study elliptic and parabolic boundary value problems with inhomogeneous boundary conditions in weighted function spaces of Sobolev, Bessel potential, Besov and Triebel-Lizorkin type. As one of the main results, we solve the problem of weighted L_q-maximal regularity in weighted Besov and Triebel-Lizorkin spaces for the parabolic case, where the spatial weight is a power weight in the Muckenhoupt $A_{\infty }$-class. In the Besov space case we have the restriction that the microscopic parameter equals to q. Going beyond the A_p-range, where p is the integrability parameter of the Besov or Triebel-Lizorkin space under consideration, yields extra flexibility in the sharp regularity of the boundary inhomogeneities. This extra flexibility allows us to treat rougher boundary data and provides a quantitative smoothing effect on the interior of the domain. The main ingredient is an analysis of anisotropic Poisson operators.

Appendices

Appendix A: A Weighted Version of a Theorem due to Clément and Prüss

The following theorem is a weighted version of a result from [17] (see [46, Theorem 5.3.15]). For its statement we need some notation that we first introduce.

Let X be a Banach space. We write $\widehat {C^{\infty }_{c}}(\mathbb {R}^{n};X) := {\mathscr{F}}^{-1}C^{\infty }_{c}(\mathbb {R}^{n};X)$ and $\widehat {L^{1}}(\mathbb {R}^{n};X) := {\mathscr{F}}^{-1}L^{1}(\mathbb {R}^{n};X)$. Then

$$ L_{1,\text{loc}}(\mathbb{R}^{n};\mathcal{B}(X)) \times \widehat{C^{\infty}_{c}}(\mathbb{R}^{n};X) \longrightarrow \widehat{L^{1}}(\mathbb{R}^{n};X), (m,f) \mapsto \mathcal{F}^{-1}[m\hat{f}] =: T_{m}f. $$

For $p \in (1,\infty )$ and $w \in A_{p}(\mathbb {R}^{n})$ we define $\mathfrak {M}L_{p}(\mathbb {R}^{n},w;X)$ as the space of all $m \in L_{1,\text {loc}}(\mathbb {R}^{n};{\mathscr{B}}(X))$ for which T_m extends to a bounded linear operator on $L_{p}(\mathbb {R}^{n},w;X)$, equipped with the norm

$$ \|{m}\|_{\mathfrak{M}L_{p}(\mathbb{R}^{n},w;X)} := \|{T_{m}}\|_{\mathcal{B}(L_{p}(\mathbb{R}^{n},w;X))}. $$

Theorem A.1

Let X be a Banach space, $p \in (1,\infty )$ and $w \in A_{p}(\mathbb {R}^{n})$. For all $m \in \mathfrak {M}L_{p}(\mathbb {R}^{n},w;X)$ it holds that

$$ \{ m(\xi) : \xi\ \text{is a Lebesgue point of}\ m\} $$

is R-bounded with

$$ \begin{array}{@{}rcl@{}} {m}_{L_{\infty}(\mathbb{R}^{n};\mathcal{B}(X))} &\leq \mathcal{R}_{p}\left( \{ m(\xi) : \xi \ \text{is a Lebesgue point of}\ m \}\right) \lesssim_{p,w} \|{m}\|_{\mathfrak{M}L_{p}(\mathbb{R}^{n},w;X)}. \end{array} $$

Proof

This can be shown as in [46, Theorem 5.3.15]. Let us comment on some modifications that have to be made for the second estimate. Modifying the Hölder argument given there according to Eq. 2-2, the implicit constant C_p,w of interest can be estimated by

$$ C_{p,w} \leq \liminf_{\epsilon \to 0}\epsilon^{d}\|{\phi(\epsilon \cdot )}\|_{L_{p}(\mathbb{R}^{n},w)} \|{\psi(\epsilon \cdot )}\|_{L_{p^{\prime}}(\mathbb{R}^{n},w^{\prime}_{p})}, $$

where $\phi ,\psi \in \mathcal {S}(\mathbb {R}^{n})$ are such that $\hat {\phi }$, $\check {\psi }$ are compactly supported with the property that ${\int \limits } \hat {\phi } \check {\psi } d\xi = 1$. By a change of variable,

$$ \epsilon^{d}\|{\phi(\epsilon \cdot )}\|_{L_{p}(\mathbb{R}^{n},w)} \|{\psi(\epsilon \cdot )}\|_{L_{p^{\prime}}(\mathbb{R}^{n},w^{\prime}_{p})} = \|{\phi}\|_{L_{p}(\mathbb{R}^{n},w(\epsilon \cdot ))}\|{\psi}\|_{L_{p^{\prime}}(\mathbb{R}^{n},w^{\prime}_{p}(\epsilon \cdot ))}. $$

Since $\mathcal {S}(\mathbb {R}^{n}) \hookrightarrow L_{p}(\mathbb {R}^{n},w)$ with norm estimate only depending on n, p and $[w]_{A_{p}}$ (as a consequence of [69, Lemma 4.5]) and since the A_p-characteristic is invariant under scaling, the desired result follows. □

Appendix B: Pointwise Multiplication

Lemma B.1

Let ${\mathscr{O}}$ be either $\mathbb {R}^{d}_{+}$ or a $C^{\infty }$-domain in $\mathbb {R}^{d}$ with a compact boundary $\partial {\mathscr{O}}$, let X be a Banach space, $U \in \{\mathbb {R}^{d},{\mathscr{O}}\}$ and let either

• $p \in [1,\infty )$, $q \in [1,\infty ]$, $\gamma \in (-1,\infty )$ and ${\mathscr{A}} \in \{B,F\}$; or

• X be reflexive, $p,q \in (1,\infty )$, $\gamma \in (-\infty ,p-1)$ and ${\mathscr{A}} \in \{{\mathscr{B}},\mathcal {F}\}$.

Let $s_{0},s_{1} \in \mathbb {R}$ and $\sigma \in \mathbb {R}$ satisfy $\sigma > \sigma _{s_{0},s_{1},p,\gamma }$. Then for all $m \in B^{\sigma }_{\infty ,1}(U;{\mathscr{B}}(X))$ and $f \in {\mathscr{A}}^{s_{0} \vee s_{1}}_{p,q}(U,w^{\partial {\mathscr{O}}}_{\gamma };X)$ there is the estimate

$$ \begin{array}{@{}rcl@{}} \|{mf}\|_{\mathscr{A}^{s_{1}}_{p,q}(U,w^{\partial\mathscr{O}}_{\gamma};X)} &\lesssim& \|{m}\|_{L_{\infty}(U;\mathcal{B}(X))+B^{-(s_{0}-s_{1})_{+}}_{\infty,1}(U;\mathcal{B}(X))} \|{f}\|_{\mathscr{A}^{s_{0} \vee s_{1}}_{p,q}(U,w^{\partial\mathscr{O}}_{\gamma};X)} \\&&+ \|{m}\|_{B^{\sigma}_{\infty,1}(U;\mathcal{B}(X))} \|{f}\|_{\mathscr{A}^{s_{0}}_{p,q}(U,w^{\partial\mathscr{O}}_{\gamma};X)}. \end{array} $$

(B-1)

Proof

The proof of [62, Lemma 3.1] carries over verbatim to the X-valued setting. □

Remark B.2

In connection to the above lemma, note that

$$ L_{\infty}(U;\mathcal{B}(X)) + B^{-(s_{0}-s_{1})_{+}}_{\infty,1}(U;\mathcal{B}(X)) = \left\{\begin{array}{ll} L_{\infty}(U;\mathcal{B}(X)),& s_{1} \geq s_{0},\\ B^{s_{1}-s_{0}}_{\infty,1}(U;\mathcal{B}(X)), & s_{1} < s_{0}, \end{array}\right. $$

(B-2)

as a consequence of $B^{0}_{\infty ,1} \hookrightarrow L_{\infty } \hookrightarrow B^{0}_{\infty ,\infty }$ and $B^{s}_{\infty ,\infty } \hookrightarrow B^{s-\epsilon }_{\infty ,1}$, $s \in \mathbb {R}$, 𝜖 > 0. Furthermore,

$$ B^{\sigma}_{\infty,1}(U;\mathcal{B}(X)) \hookrightarrow L_{\infty}(U;\mathcal{B}(X)) + B^{-(s_{0}-s_{1})_{+}}_{\infty,1}(U;\mathcal{B}(X)) $$

(B-3)

as σ > s₁ − s₀ ≥−(s₀ − s₁)₊.

Remark B.3

Lemma B.1 has a version for more general weights: $A_{\infty }$-weights in case (i) and $[A_{\infty }]'_{p}$-weights in case (ii). The condition $\sigma >\sigma _{s_{0},s_{1},p,\gamma }$ then has to be replaced by

$$ \sigma > \max\left\{ \left( \frac{1}{\rho_{w,p}}-1\right)_{+}-s_{0},-\left( \frac{1}{\rho_{w^{\prime}_{p},p^{\prime}}}-1\right)_{+}+s_{1} ,s_{1}-s_{0}\right\}, $$

where $\rho _{w,p} := \sup \{ r \in (0,1) : w \in A_{p/r}\}$ with the convention that $\sup \emptyset = \infty$ and $\frac {1}{\infty }=0$.

Definition B.4

Let $(S,{\mathscr{A}},\mu )$ be a measure space and X a Banach space. Then we define the space $\mathcal {R}L_{\infty }(S;{\mathscr{B}}(X))$ as the space of all strongly measurable functions $f\colon S\to {\mathscr{B}}(X)$ such that

$$ \|f\|_{\mathcal{R}L_{\infty}(S;\mathcal{B}(X))}:=\inf_{g}\mathcal{R}\{g(\omega):\omega\in S\}<\infty $$

where the infimum is taken over all strongly measurable $g\colon S\to {\mathscr{B}}(X)$ such that f = g almost everywhere.

Lemma B.5

Let ${\mathscr{O}}$ be either $\mathbb {R}^{d}_{+}$ or a $C^{\infty }$-domain in $\mathbb {R}^{d}$ with a compact boundary $\partial {\mathscr{O}}$, let X be a UMD Banach space, $U \in \{\mathbb {R}^{d},{\mathscr{O}}\}$, $p\in (1,\infty )$ and $w\in A_{p}(\mathbb {R}^{n})$. Let further $s_{0},s_{1} \in \mathbb {R}$ and $\sigma \in \mathbb {R}$ satisfy $\sigma > \max \limits \{-s_{0},s_{1},s_{1}-s_{0}\}$. Then for all $m \in B^{\sigma }_{\infty ,1}(U;{\mathscr{B}}(X))$ and $f \in H^{s_{1}\vee s_{0}}_{p}(U,w;X)$ there is the estimate

$$ \|{mf}\|_{H^{s_{1}}_{p}(U,w;X)} \lesssim \|{m}\|_{\mathcal{R}L_{\infty}(U;\mathcal{B}(X))} \|{f}\|_{H^{s_{1}}_{p}(U,w;X)} + \|{m}\|_{B^{\sigma}_{\infty,1}(U;\mathcal{B}(X))} \|{f}\|_{H^{s_{0}}_{p}(U,w;X)}. $$

(B-4)

Proof

It suffices to consider the case $U=\mathbb {R}^{d}$. We use paraproducts as in [82, Section 4.4] and [72, Section 4.2].

By [72, Lemma 4.4], the paraproduct π₁ : (m,f)↦π₁(m,f) gives rise to bounded bilinear mapping

$$ {\Pi}_{1}: \mathcal{R}L_{\infty}(\mathbb{R}^{d};\mathcal{B}(X)) \times {H^{s}_{p}}(\mathbb{R}^{d},w;X) \longrightarrow {H^{s}_{p}}(\mathbb{R}^{d},w;X). $$

By a slight modification of [72, Lemma 4.6] (see [62, Lemma 3.1]), for i ∈{2, 3}, the paraproduct π_i : (m,f)↦π_i(m,f) gives rise to bounded bilinear mapping

$$ {\Pi}_{i}: B^{\sigma}_{\infty,1}(\mathbb{R}^{d};\mathcal{B}(X)) \times F^{s_{0}}_{p,\infty}(\mathbb{R}^{d},w;X) \longrightarrow F^{s_{1}}_{p,1}(\mathbb{R}^{d},w;X) $$

and thus a bounded bilinear mapping

$$ {\Pi}_{i}: B^{\sigma}_{\infty,1}(\mathbb{R}^{d};\mathcal{B}(X)) \times H^{s_{0}}_{p}(\mathbb{R}^{d},w;X) \longrightarrow H^{s_{1}}_{p}(\mathbb{R}^{d},w;X). $$

□

Proposition B.6

Under the conditions of Lemma B.1 with s₀ ≥ s₁, we have the continuous bilinear mapping

$$ B^{\sigma}_{\infty,1}(U;\mathcal{B}(X)) \times \mathscr{A}^{s_{0}}_{p,q}(U,w^{\partial\mathscr{O}}_{\gamma};X) \longrightarrow \mathscr{A}^{s_{1}}_{p,q}(U,w^{\partial\mathscr{O}}_{\gamma},X), (m,f) \mapsto mf. $$

(B-5)

Under the conditions of Lemma B.5 with s₀ ≥ s₁, we have the continuous bilinear mapping

$$ B^{\sigma}_{\infty,1}(U;\mathcal{B}(X)) \times H^{s_{0}}_{p}(U,w;X) \longrightarrow H^{s_{1}}_{p}(U,w,X), (m,f) \mapsto mf. $$

(B-6)

Proof

Equation B-5 is a direct consequence of Lemma B.1. The case s = 0 in Eq. B-6 follows from [72, Proposition 3.8]. The case s₀ > s₁ in Eq. B-6 follows from the A_p-version of Eq. B-5 (see Remark B.3) as $\sigma > \sigma _{s_{0}-\epsilon ,s_{1}+\epsilon ,p,w}$ for sufficiently small 𝜖 > 0. □

Appendix C: Comments on the Localization and Perturbation Procedure

The localization and perturbation arguments are quite technical but standard, let us just say the following. The localization in Theorem 5.3 can be carried out as in [67, Sections 2.3 & 2.4] and [60, Appendix B], where we need to use some of the pointwise estimates from Appendix B as well some of the localization and rectification results for weighted Besov and Triebel-Lizorkin spaces from [62, Section 4] (which extend to the vector-valued situation) in order to perform all the arguments. Furthermore, the localization in Theorem 6.2 can be carried out as in [62, Theorem 9.2]. The results in [62, Section 4] are a generalization of results on the invariance of Besov- and Triebel-Lizorkin spaces under diffeomorphic transformations such as [84, Theorem 4.16] to the weighted anisotropic mixed-norm setting. They lead to the conditions (SO) in Section 5 and (SO)_s in Section 6.

We would also like to mention [24] and [25], where the authors treat maximal L_q-L_p-regularity for parabolic boundary value problems on the half-space in which the elliptic operators have top order coefficients in the VMO class in both time and space variables. In their proofs, they do not use localization for the results on VMO coefficients, but they extend some techniques by Krylov as well as Dong and Kim.

While the geometric steps of the localization procedure in our setting are the same as in the standard L_p-setting, there are some differences in what kind of perturbation results we need. The main difference lies in the treatment of the top order perturbation of the differential operator on the domain. More precisely, the following lemma is a useful tool in the localization procedure for our setting.

Lemma C.1

Let E be a Banach space and $A\colon E\supset D(A)\to E$ a closed linear operator. Suppose that there is a constant C > 0 such that for all λ > 0 and all u ∈ D(A) it holds that

$$ \begin{array}{@{}rcl@{}} \| u\|_{D(A)} + \lambda \|u\|_{E}\leq C \|(\lambda+A) u\|_{E}. \end{array} $$

(C-1)

Let $\||{ \cdot }\||\colon E\to [0,\infty )$ a mapping and 𝜃 ∈ (0, 1) such that

$$ \begin{array}{@{}rcl@{}} \||{u}\|| \leq \|u\|_{E}^{1-\theta}\|u\|^{\theta}_{D(A)} \end{array} $$

(C-2)

holds for all u ∈ D(A). Let further P : D(A) → E and suppose that there are constants $\delta , C^{\prime }\in (0,\infty )$ such that

$$ \begin{array}{@{}rcl@{}} \|P(u)\|_{E}\leq \delta\|u\|_{D(A)}+C^{\prime}\||{u}\|| \end{array} $$

(C-3)

for all u ∈ D(A). Then there is $\lambda _{0}\in (0,\infty )$ only depending on $\delta ,C^{\prime }$ and 𝜃 such that for all λ ≥ λ₀ and all u ∈ D(A) we have the estimate

$$ \begin{array}{@{}rcl@{}} \|P(u)\|_{E}\leq 2\delta C\|(\lambda+A)u\|_{E}. \end{array} $$

(C-4)

Proof

For u ∈ D(A) we have that

$$ \begin{array}{@{}rcl@{}} \|P(u)\|_{E}&\leq& \delta\|u\|_{D(A)}+C^{\prime}\||{u}\|| \leq \delta\|u\|_{D(A)}+C^{\prime} \|u\|_{E}^{1-\theta}\|u\|^{\theta}_{D(A)} \\ &\leq& 2\delta\|u\|_{D(A)} +\delta C_{\delta} \|u\|_{E} \end{array} $$

with $C_{\delta }:=(\frac {\delta }{C^{\prime }\theta })^{\theta /(1-\theta )}(1-\theta )$. Here, we used Young’s inequality with the Peter-Paul trick. Using Eq. C-1 with λ ≥ C_δ/2, we can further estimate

$$ \begin{array}{@{}rcl@{}} \|P(u)\|_{E}&\leq 2\delta C \| (\lambda+A)u\|_{E} \end{array} $$

so that λ₀ = C_δ/2 is the asserted parameter. □

If one wants to apply Lemma C.1 for a localization procedure, then one can treat the top order perturbation as follows: Suppose that the differential operator has the form $1+{\sum }_{|\alpha |=2m} (a_{\alpha }+p_{\alpha }(x))D^{\alpha }$ with p_α being small in a certain norm. The mapping P in Lemma C.1 can be chosen to be $P(u)(x)={\sum }_{|\alpha |=2m}p_{\alpha }(x)D^{\alpha } u(x)$ and A can be chosen to be the realization of $1+{\sum }_{|\alpha |=2m}a_{\alpha }D^{\alpha }$ in $\mathbb {E}$ with vanishing boundary conditions. Now one can use Lemma B.1 in combination with Remark B.2 in the Besov-Triebel-Lizorkin case and Lemma B.5 in the Bessel potential case to obtain an estimate of the form Eq. C-3. In order to do this for example in the parabolic case, one chooses 𝜖 ∈ (0, 2m) such that σ > σ_s,p,γ + 𝜖 ≥ σ_{s−𝜖,𝜖,p,γ}. Then one chooses s₀ = s − 𝜖 and s₁ = s. These choices lead to the estimate

$$ \|p_{\alpha}D^{\alpha}u\|_{\mathbb{E}}\lesssim \|p_{\alpha}\|_{L_{\infty}} \|u\|_{\mathbb{E}^{2m}}+\|p_{\alpha}\|_{BUC^{\sigma}}\|u\|_{\mathbb{E}^{2m-\epsilon}}\quad(|\alpha|=2m) $$

in the Besov and Triebel-Lizorkin cases and

$$ \|p_{\alpha}D^{\alpha}u\|_{\mathbb{E}}\lesssim \|p_{\alpha}\|_{\mathcal{R}L_{\infty}} \|u\|_{\mathbb{E}^{2m}}+\|p_{\alpha}\|_{BUC^{\sigma}}\|u\|_{\mathbb{E}^{2m-\epsilon}}\quad(|\alpha|=2m) $$

in the Bessel potential case. It holds that $\|\cdot \|_{D(A)}\eqsim \|\cdot \|_{\mathbb {E}^{2m}}$ on $D(A)\subset \mathbb {E}^{2m}$. Hence, if $\theta =1-\frac {\epsilon }{2m}$ and $E=\mathbb {E}$, then these estimates would correspond to Eq. C-3 in Lemma C.1 where $\cdot =M\|\cdot \|_{\mathbb {E}^{2m-\epsilon }}$ for a suitable constant M > 0 such that Eq. C-2 holds. Note that Eq. C-1 follows from the sectoriality of A. Therefore, if p_α is small in $L_{\infty }$ or $\mathcal {R}L_{\infty }$-norm, respectively, then Lemma C.1 shows that P is just a small perturbation of a suitable shift of the operator A.