On the lack of interior regularity of the $p$-Poisson problem with $p>2$

TL;DR

This paper demonstrates that solutions to the $p$-Poisson problem with $p>2$ can have limited interior regularity, showing the optimality of existing regularity theorems through explicit constructions.

Contribution

It constructs specific right-hand sides to show the sharpness of known regularity results for the $p$-Poisson problem, confirming their optimality.

Findings

Solutions have limited Besov regularity depending on the right-hand side.

The results confirm the optimality of Savaré's shift theorem.

The findings apply to all integrability parameters and are of local nature.

Abstract

In this note we are concerned with interior regularity properties of the $p$-Poisson problem $\Delta_p(u)=f$ with $p>2$. For all $0<\lambda\leq 1$ we constuct right-hand sides $f$ of differentiability $-1+\lambda$ such that the (Besov-) smoothness of corresponding solutions $u$ is essentially limited to $1+\lambda / (p-1)$. The statements are of local nature and cover all integrability parameters. They particularly imply the optimality of a shift theorem due to Savar\'e [J. Funct. Anal. 152:176-201, 1998], as well as of some recent Besov regularity results of Dahlke et al. [Nonlinear Anal. 130:298-329, 2016]. Keywords: Nonlinear and adaptive approximation, Besov space, regularity of solutions, $p$-Poisson problem.

Equations

$$-{\Delta }_p(u)=f$$

$${\Delta }_p(u):=\mathrm{div}(A(\nabla u)),\text{where}A(\nabla u):={\left|\nabla u\right|}^{p-2}\nabla u$$

$${\int }_{\Omega }{⟨A(\nabla u)(x),\nabla \psi (x)⟩}_{{\mathbb{R}}^d}\mathrm{d}x=f(\psi ),\psi \in \mathcal{D}(\Omega ),$$

$$\left|g\right|_{B^{s}_{\varrho,q}(\Omega)}:=\begin{cases}\displaystyle\bigg{(}\int_{0}^{\overline{t}}\bigg{[}t^{-s}\sup_{\begin{subarray}{c}h\in\mathbb{R}^{d},\\ \left|h\right|_{2}\leq t\end{subarray}}\left\|\Delta_{h}^{k}g\;\vrule\;L_{\varrho}(\Omega_{h,k})\right\|\bigg{]}^{q}\frac{\,\mathrm{d}t}{t}\bigg{)}^{1/q},&0<q<\infty,\\ \displaystyle\sup_{0<t\leq\overline{t}}t^{-s}\sup_{\begin{subarray}{c}h\in\mathbb{R}^{d},\\ \left|h\right|_{2}\leq t\end{subarray}}\left\|\Delta_{h}^{k}g\;\vrule\;L_{\varrho}(\Omega_{h,k})\right\|,&q=\infty,\end{cases}$$

$$\Vert g\text{vrule}{B}_{\varrho ,q}^s(\Omega )\Vert :=\Vert g\text{vrule}{L}_{\varrho }(\Omega )\Vert +{\left|g\right|}_{B_{\varrho ,q}^s(\Omega )}$$

$${\Omega }_{h,k}:=\{x\in {\mathbb{R}}^d\text{vrule}x+\ell h\in \Omega \text{ for all }\ell =0,\dots ,k\}.$$

$$W_{\varrho }^s(\Omega )=B_{\varrho ,\varrho }^s(\Omega )\text{for all}1\le \varrho <\infty ,$$

$$f\in W_{p^{\prime }}^{-1+\lambda }(\Omega )⟹u\in W_p^{1+\frac{\lambda }{p-1}}(\Omega )$$

$$f\in B_{p^{\prime },1}^{-1+\frac{1}{p^{\prime }}}(\Omega )⟹u\in B_{p,\infty }^{1+\frac{1/p^{\prime }}{p-1}}(\Omega ).$$

$$f=f_{\lambda ,\mu }\in B_{\mu ,\infty }^{-1+\lambda }(\Omega )\cap W_{p^{\prime }}^{-1}(\Omega )$$

$$\displaystyle u\in\left\{\begin{array}[]{rclcrlcr}\!\!B^{1+\frac{\lambda}{p-1}-\varepsilon}_{\varrho,q}(\Omega)\!\!\!&\setminus&\!\!\!B^{1+\frac{\lambda}{p-1}}_{\varrho,q}(\Omega)&\;\text{if}&\mu(p-1)<\varrho\!\!\!&\leq\infty&\text{and}&0<q\leq\infty,\\ \!\!B^{1+\frac{\lambda}{p-1}}_{\varrho,\infty}(\Omega)\!\!\!&\setminus&\!\!\!B^{1+\frac{\lambda}{p-1}}_{\varrho,q}(\Omega)&\;\text{if}&\varrho\!\!\!&=\mu(p-1)&\text{and}&0<q<\infty,\\ \!\!B^{1+\frac{\lambda}{p-1}}_{\varrho,q}(\Omega)\!\!\!&\setminus&\!\!\!B^{1+\frac{\lambda}{p-1}+\varepsilon}_{\varrho,q}(\Omega)&\;\text{if}&1\leq\varrho\!\!\!&<\mu(p-1)&\text{and}&0<q\leq\infty.\end{array}\right.$$

$$\displaystyle A(\nabla u)\in\left\{\begin{array}[]{rclcrlcr}\big{(}B^{\lambda-\varepsilon}_{\varrho,q}(\Omega)\!\!\!&\setminus&\!\!\!B^{\lambda}_{\varrho,q}(\Omega)\big{)}^{d}&\quad\text{if}&\mu<\varrho\!\!\!&\leq\infty&\text{and}&0<q\leq\infty,\\ \big{(}B^{\lambda}_{\varrho,\infty}(\Omega)\!\!\!&\setminus&\!\!\!B^{\lambda}_{\varrho,q}(\Omega)\big{)}^{d}&\quad\text{if}&\varrho\!\!\!&=\mu&\text{and}&0<q<\infty,\\ \big{(}B^{\lambda}_{\varrho,q}(\Omega)\!\!\!&\setminus&\!\!\!B^{\lambda+\varepsilon}_{\varrho,q}(\Omega)\big{)}^{d}&\quad\text{if}&1\leq\varrho\!\!\!&<\mu&\text{and}&0<q\leq\infty.\end{array}\right.$$

$$u\in B_{\varrho ,q}^{1+\frac{\lambda }{p-1}}(\Omega )\setminus B_{\varrho ,q}^{1+\frac{\lambda }{p-1}+c_{\varrho }\epsilon }(\Omega )\text{for all}0<\varrho <1\text{and}0<q\le \infty$$

$$f=f_{\mu }\in L_{\nu }(\Omega )\cap W_{p^{\prime }}^{-1}(\Omega )\text{for all}\{\begin{array}{ll}0<\nu <\mu & \text{if}\mu <\infty ,\\ 0<\nu \le \infty & \text{if}\mu =\infty ,\end{array}$$

$$A(\nabla u)\in\big{(}W^{1}_{\varrho}(\Omega)\big{)}^{d}\qquad\text{if and only if}\qquad\varrho<\mu.$$

$$u\in B_{{\tau }_{\sigma },{\tau }_{\sigma }}^{\sigma }(\Omega )\text{for all}0<\sigma <\overline{\sigma }:=1+\frac{1}{p-1}\text{and}\frac{1}{{\tau }_{\sigma }}=\frac{\sigma }{2}+\frac{1}{p}.$$

$$a_{n,\theta }:=4\sum _{j=1}^{n-1}{j}^{-\theta },n\in \mathbb{N}\setminus \{1\}.$$

$$4=a_{2,\theta }<\dots <a_{n,\theta }<a_{n+1,\theta }=a_{n,\theta }+4n^{-\theta }<\dots <a_{\infty ,\theta }:=\underset{n\to \infty }{\lim }{a}_{n,\theta }=4\zeta (\theta )<\infty .$$

$$w_{\sigma,\theta}(\xi):=\left\{\begin{array}[]{lcrl}(\xi-a_{n,\theta})^{\sigma}&\quad\text{if}&a_{n,\theta}\leq\xi\!\!\!&<a_{n,\theta}+n^{-\theta},\\ n^{-\theta\sigma}&\quad\text{if}&a_{n,\theta}+n^{-\theta}\leq\xi\!\!\!&<a_{n,\theta}+2\,n^{-\theta},\\ (a_{n,\theta}+3\,n^{-\theta}-\xi)^{\sigma}&\quad\text{if}&a_{n,\theta}+2\,n^{-\theta}\leq\xi\!\!\!&<a_{n,\theta}+3\,n^{-\theta},\\ 0&\quad\text{if}&a_{n,\theta}+3\,n^{-\theta}\leq\xi\!\!\!&<a_{n+1,\theta},\end{array}\right.$$

$$w_{\sigma ,\theta }(\xi ):=0\text{on}\mathbb{R}\setminus [a_{2,\theta },a_{\infty ,\theta }).$$

$$\mathscr{P}_{\theta}:=\big{\{}\xi\in[a_{2,\theta},a_{\infty,\theta})\;\vrule\;\xi=a_{n,\theta}+k\,n^{-\theta}\text{ for some }n\geq 2\text{ and }k\in\{0,\ldots,3\}\big{\}}\cup\{a_{\infty,\theta}\}\subset\mathbb{R}.$$

$$\Vert g\text{vrule}{B}_{\varrho ,q}^{1+s}(\Omega )\Vert \sim \sum _{\begin{array}{c}\alpha \in {\mathbb{N}}_0^d:\\ {|\alpha |}_1\le 1\end{array}}\Vert D^{\alpha }g\text{vrule}{B}_{\varrho ,q}^s(\Omega )\Vert .$$

$$\sigma \ge 1,\text{or}0<\sigma <1\text{and}\frac{1-\sigma }{1-1/\theta }<\frac{1}{\varrho }.$$

$$0\le \frac{1}{\varrho }<\min \left\{\theta (1+\sigma ),\frac{1-\sigma }{1-1/\theta }\right\}.$$

$$s=\sigma +\frac{1-1/\theta }{\varrho }\text{and}q=\infty ,\text{or}s<\sigma +\frac{1-1/\theta }{\varrho }\text{and}0<q\le \infty .$$

$${\Vert w_{\sigma ,\theta }^{\prime }\text{vrule}{L}_{\varrho }(\mathbb{R})\Vert }^{\varrho }$$

$$=2{\sigma }^{\varrho }\sum _{n=2}^{\infty }{\int }_0^{n^{-\theta }}{x}^{(\sigma -1)\varrho }\mathrm{d}x.$$

$${\Vert w_{\sigma ,\theta }^{\prime }\text{vrule}{L}_{\varrho }(\mathbb{R})\Vert }^{\varrho }$$

$$\max \left\{1-\sigma ,\frac{1-\sigma }{1-1/\theta }\right\}<\frac{1}{\varrho }.$$

$$s=\sigma +\frac{1-1/\theta }{\varrho }\text{and all}q\in {\mathbb{R}}_{+}.$$

$${\left|w_{\sigma ,\theta }\right|}_{B_{\varrho ,q}^s(\mathbb{R})}<\infty \text{if and only if}q=\infty .$$

Full text

Keywords: Nonlinear and adaptive approximation, Besov space, regularity of solutions, $p$-Poisson problem.

2010 Mathematics Subject Classification: 35B35, 35J92, 41A46, 46E35, 65M99.

1 Introduction and main results

In what follows we deal with interior regularity properties of solutions $u\in W^{1}_{p}(\Omega)$ to the $p$-Poisson problem

[TABLE]

on bounded Lipschitz domains $\Omega\subset\mathbb{R}^{d}$ for $d\in\mathbb{N}$ and $1<p<\infty$. Here the $p$-Laplace operator $\Delta_{p}$ is given by

[TABLE]

is called the naturally associated vector field. For distributions $f\in W^{-1}_{p^{\prime}}(\Omega):=(W^{1}_{p,0}(\Omega))^{\prime}$, with $1/p+1/p^{\prime}=1$, the corresponding variational formulation is given by

[TABLE]

where $\mathcal{D}(\Omega):=C_{0}^{\infty}(\Omega)$ is the set of test functions on $\Omega$, the space $W^{1}_{p,0}(\Omega)$ is the closure of $\mathcal{D}(\Omega)$ w.r.t. the first order $L_{p}$-Sobolev norm, and $\left\langle\cdot,\cdot\right\rangle_{\mathbb{R}^{d}}$ denotes the inner product on $\mathbb{R}^{d}$.

Equations of type (1) arise in various applications such as non-Newtonian fluid theory, rheology, radiation of heat and many others. In fact the quasi-linear operator $\Delta_{p}$ has a similar model character for nonlinear problems as the ordinary Laplacian (i.e., the case $p=2$) for linear problems. Meanwhile, many results concerning existence and uniqueness of solutions are known. For details we refer to [8] and the references therein. However, most of these results deal with classical function spaces of Hölder or Sobolev type. On the other hand, in view of strong relations to nonlinear approximation classes and adaptive numerical algorithms, regularity results in more general smoothness spaces of Besov type became more and more important in recent times; see, e.g., [2, 5]. For the $p$-Poisson equation (1) and related problems only few results are known in this direction, see [1, 3, 6], as well as [4, 10, 12].

Let us recall that for $0<\varrho,q\leq\infty$ and $s\in\mathbb{R}$ Besov spaces $B^{s}_{\varrho,q}(\mathbb{R}^{d})$ are quasi-Banach spaces which can be defined as subsets of tempered distributions $f\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$ by means of harmonic analysis. The corresponding spaces on domains (i.e. connected open subsets of $\mathbb{R}^{d}$) are then defined by restriction such that we obtain subsets of $\mathcal{D}^{\prime}(\Omega)$.

Remark \therem (Function spaces).

We assume that the reader is familiar with the basics of function space theory as it can be found, e.g., in the monographic series of Triebel [14, 15, 16, 17]. Anyhow, let us mention that by now various equivalent characterizations, embeddings, interpolation and duality assertions for the scale of Besov spaces are known. Without going into further details, let us recall the following results, valid for bounded Lipschitz domains $\Omega\subset\mathbb{R}^{d}$ or $\Omega=\mathbb{R}^{d}$ itself:

• (i)

For $0<\varrho,q\leq\infty$, and $s>d\,\max\{0,1/\varrho-1\}$ the Besov space $B^{s}_{\varrho,q}(\Omega)$ can be characterized as collection of all $g\in L_{\varrho}(\Omega)$ for which

[TABLE]

is finite [16, Sect. 1.11.9].

In fact, the expression

[TABLE]

provides a quasi-norm on $B^{s}_{\varrho,q}(\Omega)$. Here we assume $\overline{t}>0$ and $k>s$ to be fixed. Further, $\Delta_{h}^{k}g$ denotes the $k$-th order finite difference of $g$ with step size $h\in\mathbb{R}^{d}$ and

[TABLE]

• (ii)

For $0<s\notin\mathbb{N}$ we have $B_{\infty,\infty}^{s}(\Omega)=C^{s}(\Omega)$ (Hölder spaces) and

[TABLE]

(Sobolev-Slobodeckij spaces) in the sense of equivalent norms.

So, roughly speaking, in $B^{s}_{\varrho,q}(\Omega)$ we collect all $g$ such that their weak partial derivatives $D^{\alpha}g$ up to order $s$ belong to the Lebesgue space $L_{\varrho}(\Omega)$. The third parameter $0<q\leq\infty$ acts as a minor important fine index. $\square$

In his seminal paper [10] Savaré developed a variational argument which allows to show the following shift theorem for the $p$-Poisson problem:

Proposition \theprop (see [10, Thm. 2]).

For $d\in\mathbb{N}$ let $\Omega\subset\mathbb{R}^{n}$ be a bounded Lipschitz domain. For $2<p<\infty$ and $f\in W^{-1}_{p^{\prime}}(\Omega)$ let $u$ be the unique weak solution to (1) in $W_{p,0}^{1}(\Omega)$. Then for all $\lambda\in(0,1/p^{\prime})$ the following implications hold:

[TABLE]

and

[TABLE]

In addition, Savaré claims that (2) is “optimal” [10, Rem. 4.3] and refers to Simon [12]. But Simon’s optimality results refer to a (slightly) different equation on the whole of $\mathbb{R}^{d}$ with right hand sides in $L_{r}$ or $C^{\infty}$ and hence they do not cover Savaré’s claim at all.

However, it is possible to use similar ideas in order to show the following Theorem 1.1 which constitutes the main result of this note. It states that for $p>2$ and all $0<\lambda<1$ there are right-hand sides of smoothness $-1+\lambda$ such that the smoothness of corresponding solutions to the $p$-Poisson problem (1) is essentially limited by $1+\lambda/(p-1)$. Moreover, this actually holds independently of the integrability parameter. That is, in sharp contrast to point singularities, we do not gain smoothness when derivatives are measured in weaker $L_{\varrho}$-norms. For a (constructive) proof we refer to Section 2.2 below.

Theorem 1.1**.**

Let $d\in\mathbb{N}$ and $2\leq p<\infty$ be fixed. Further let $\Omega\subseteq\mathbb{R}^{d}$ be either $\mathbb{R}^{d}$ itself, a bounded Lipschitz domain, or an interval (if $d=1$). Moreover, assume $0<\varepsilon<1/p$. Then for all $\varepsilon\,(p-1)<\lambda<1-\varepsilon$ and $1<\mu\leq\infty$ there exists a right-hand side

[TABLE]

*with compact support in $\Omega$ such that the corresponding weak solution $u\in W^{1}_{p,0}(\Omega)$ to (1) is compactly supported as well and satisfies *

[TABLE]

In addition, the naturally associated vector field satisfies

[TABLE]

Before we proceed some general comments are in order:

Remark \therem.

First of all, let us stress the point that, due to the compact support of $f$ and $u$, Theorem 1.1 is of local nature.

Secondly, we note that the restriction to $\varrho\geq 1$ is for notational convinience only. Using standard embeddings (see Section 2.1(iii) below) and complex interpolation (see, e.g., Kalton et al. [7, Theorem 5.2]) we can easily extend (6) by

[TABLE]

with some $c_{\varrho}\sim 1/\varrho$. Likewise, the same arguments can be used to extend also (10). $\square$

Observe that Theorem 1.1 applied for $\mu:=p^{\prime}$ indeed shows optimality of Savaré’s result in some sense:

Corollary \thecor.

In Section 1 the smoothness of $u$ w.r.t. $L_{p}$ cannot be improved without strengthening the assumptions on $f$.

Proof.

Choosing $\lambda:=\widetilde{\lambda}+(p-1)\delta$ with some $\widetilde{\lambda}\in(0,1/p^{\prime})$ and $\delta>0$ arbitrarily small, Theorem 1.1 allows to find a right-hand side $f\in B_{\mu,\infty}^{-1+\lambda}(\Omega)\hookrightarrow B_{p^{\prime},p^{\prime}}^{-1+\widetilde{\lambda}}(\Omega)=W_{p^{\prime}}^{-1+\widetilde{\lambda}}(\Omega)$ such that the corresponding solution of the Dirichlet problem for the $p$-Poisson equation satisfies $u\notin B_{p,p}^{1+\frac{\widetilde{\lambda}}{p-1}+\delta}(\Omega)=W_{p}^{1+\frac{\widetilde{\lambda}}{p-1}+\delta}(\Omega)$. Similarly for $\lambda:=1/p^{\prime}+\delta$ with $\delta>0$, there exists $f\in B_{\mu,\infty}^{-1+\lambda}(\Omega)\hookrightarrow B_{p^{\prime},1}^{-1+\frac{1}{p^{\prime}}}(\Omega)$ such that $u\notin B_{p,q}^{1+\frac{\lambda}{p-1}}(\Omega)\hookleftarrow B_{p,\infty}^{1+\frac{1/p^{\prime}}{p-1}+\delta}(\Omega)$. In view of Section 1, these examples remain valid also on smooth domains. ∎

Furthermore, Theorem 1.1 shows that regarding regularity questions it seems better to look at the the mapping $f\mapsto A(\nabla u)$, rather than $f\mapsto u$. In fact, in view of the case $\varrho=\mu$ in (10), one might conjecture the existence of a $p$-independent mechanism which (for some range of parameters) locally transfers exactly one order of regularity from the right-hand side $f$ to the naturally associated vector filed $A(\nabla u)$. For the case $d=2$ this already has been verified in [1]. In other words, Theorem 1.1 shows that also their results cannot be improved.

Theorem 1.1 is complemented by

Theorem 1.2**.**

Let $d\in\mathbb{N}$ and $2<p<\infty$. Further let $\Omega\subseteq\mathbb{R}^{d}$ be either $\mathbb{R}^{d}$ itself, a bounded Lipschitz domain, or an interval (if $d=1$). Moreover, let $0<\varepsilon<\min\{1/(p-1),1-1/(p-1)\}$. Then for all $1<\mu\leq\infty$ there exists a compactly supported right-hand side

[TABLE]

such that the corresponding weak solution $u\in W^{1}_{p,0}(\Omega)$ to (1) is compactly supported as well and satisfies (6) with $\lambda=1$. Moreover, then for $1<\varrho<\infty$ there holds

[TABLE]

Here Section 1 applies likewise. Moreover, also this result implies certain optimality statements:

Remark \therem.

At first, setting $\mu:=\varrho:=\infty$ in Theorem 1.2, we recover the well-known assertion that for bounded right-hand sides the local Hölder regularity of the gradient $\nabla u$ of solutions to the $p$-Poisson equation (1) with $p>2$ is bounded by $1/(p-1)$.

Secondly, in [3] it has been shown that for $p>2$, bounded Lipschitz domains $\Omega\subset\mathbb{R}^{2}$, and right-hand sides $f\in L_{\infty}(\Omega)$ the unique solution $u\in W^{1}_{p,0}(\Omega)$ to (1) satisfies

[TABLE]

In view of Theorem 1.2 and Section 1, $\overline{\sigma}$ cannot be replaced by any larger number. $\square$

The rest of this note is devoted to the proofs of Theorem 1.1 and Theorem 1.2, respectively. Section 2.1 collects some quite technical preparations. Afterwards, the statements are proven easily in Section 2.2 and Section 2.3.

Notations: In the sequel $\mathbb{N}$ denotes the natural numbers without zero and we use $\mathbb{R}_{+}$ for the set of strictly positive reals. For families $\{a_{j}\;\vrule\;j\in\mathcal{J}\}$ and $\{b_{j}\;\vrule\;j\in\mathcal{J}\}$ of non-negative reals over a common index set $\mathcal{J}$ we write $a_{j}\lesssim b_{j}$ if there exists a constant $c>0$ (independent of the context-dependent parameters $j$) such that $a_{j}\leq c\cdot b_{j}$ holds uniformly in $j\in\mathcal{J}$. Consequently, $a_{j}\sim b_{j}$ means $a_{j}\lesssim b_{j}$ and $b_{j}\lesssim a_{j}$. In addition, the symbol $\hookrightarrow$ is used to denote continuous embeddings.

2 Proofs

Our main proofs given in Section 2.2 and Section 2.3 below require some preparations. The basic idea will be based on a construction given by Simon [13, Sect. 4].

2.1 Preparations

For $1<\theta<\infty$ define the sequence

[TABLE]

Then for all $n\geq 3$

[TABLE]

Further, with $\sigma\in\mathbb{R}_{+}$ let $w_{\sigma,\theta}\colon\mathbb{R}\to[0,\infty)$ be defined piecewise by

[TABLE]

on $[a_{n,\theta},a_{n+1,\theta})$, $n\geq 2$, and

[TABLE]

Moreover, let us define $\mathscr{S}_{\theta}:=[4,4\,\zeta(\theta)]\subset\mathbb{R}$, as well as the set of transition points

[TABLE]

Lemma \thelem (Properties of $w_{\sigma,\theta}$).

Let $\sigma\in\mathbb{R}_{+}$ and $1<\theta<\infty$, as well as $0<\varrho\leq\infty$. Then

1. (i)

$w_{\sigma,\theta}$* is continuous with compact support $\operatorname{supp}(w_{\sigma,\theta})\subseteq\mathscr{S}_{\theta}$.* 2. (ii)

$\mathscr{P}_{\theta}$* is countable and $w_{\sigma,\theta}$ is continuously differentiable on $\mathbb{R}\setminus\mathscr{P}_{\theta}$, i.e., $w_{\sigma,\theta}^{\prime}$ exists a.e.* 3. (iii)

$w_{\gamma\sigma,\theta}=(w_{\sigma,\theta})^{\gamma-1}w_{\sigma,\theta}$* for all $\gamma\in\mathbb{R}_{+}$.* 4. (iv)

$0\leq w_{\sigma,\theta}(\xi)\leq 2^{-\sigma\theta}$* for all $\xi\in\mathbb{R}$.*

Proof.

All statements are obvious consequences of the definition of $w_{\sigma,\theta}$. In order to see (iv), note that $0\leq w_{\sigma,\theta}(\xi)\leq n^{-\sigma\theta}$ on $[a_{n,\theta},a_{n+1,\theta}]$ with $n\geq 2$. ∎

In the sequel, we will need sharp regularity assertions for $w_{\sigma,\theta}$. Before we state and prove them, let us recall some well-known embedding results for Besov spaces which are proven here for the sake of completeness.

Proposition \theprop.

For $d\in\mathbb{N}$ let $\Omega\subseteq\mathbb{R}^{d}$ be either $\mathbb{R}^{d}$ itself, a bounded Lipschitz domain, or an interval (if $d=1$). Further let $0<\varrho,\varrho_{0},\varrho_{1},q,q_{0},q_{1}\leq\infty$ and $s,s_{0},s_{1}\in\mathbb{R}$. Then

1. (i)

B^{1+s}_{\varrho,q}(\Omega)=\big{\{}g\in B^{s}_{\varrho,q}(\Omega)\;\vrule\;\nabla g\in(B^{s}_{\varrho,q}(\Omega))^{d}\big{\}}* with*

[TABLE] 2. (ii)

If $s_{1}<s_{0}$, then $g\in B^{s_{0}}_{\varrho,q_{0}}(\Omega)$ implies $g\in B^{s_{1}}_{\varrho,q_{1}}(\Omega)$. 3. (iii)

If $\varrho_{1}\leq\varrho_{0}$ and $g\in B^{s}_{\varrho_{0},q}(\Omega)$ has compact support in $\mathbb{R}^{d}$, then $g\in B^{s}_{\varrho_{1},q}(\Omega)$.

Proof.

Assertion (i) is a special instance of [17, Prop. 4.21]. So, let us prove (ii) and (iii). By means of Rychkov’s extension operator [9] we can w.l.o.g. assume that $\Omega=\mathbb{R}^{d}$. Further, let $c(g)$ denote the sequence of wavelet coefficients of $g$ w.r.t. a sufficiently smooth Daubechies wavelet system on $\mathbb{R}^{d}$. Then the wavelet isomorphism from [16, Thm. 3.5] implies that $\left\|g\;\vrule\;B^{s}_{\varrho,q}(\mathbb{R}^{d})\right\|\sim\left\|c(g)\;\vrule\;b^{s}_{\varrho,q}(\nabla)\right\|$ for all $0<\varrho,q\leq\infty$ and $s\in\mathbb{R}$ with $b^{s}_{\varrho,q}(\nabla)$ being suitable sequence spaces. Now (ii) follows from the standard embedding $b^{s_{0}}_{\varrho_{0},q_{0}}(\nabla)\hookrightarrow b^{s_{1}}_{\varrho_{0},q_{1}}(\nabla)$ if $s_{1}<s_{0}$ which can be found, e.g., in [18, Prop. 2.5].

In order to prove (iii), we note that the compact support of $g$ implies that $\left\|c(g)\;\vrule\;b^{s}_{p,q}(\nabla)\right\|$ equals \left\|c(g)\big{|}_{\widetilde{\nabla}}\;\vrule\;b^{s}_{p,q}(\widetilde{\nabla})\right\|, where $b^{s}_{p,q}(\widetilde{\nabla})$ refers to the corresponding sequence space for some bounded domain. For these spaces there holds the embedding $b^{s_{0}}_{\varrho_{0},q_{0}}(\widetilde{\nabla})\hookrightarrow b^{s_{0}}_{\varrho_{1},q_{0}}(\widetilde{\nabla})$ if $\varrho_{1}\leq\varrho_{0}$, see [18, Prop. 2.5] again. ∎

Lemma \thelem (Regularity of $w_{\sigma,\theta}$).

Let $\sigma\in\mathbb{R}_{+}$ and $1<\theta<\infty$, as well as $0<\varrho\leq\infty$. Then

1. (i)

$w_{\sigma,\theta}\in L_{\varrho}(\mathbb{R})$, 2. (ii)

$w_{\sigma,\theta}^{\prime}\in L_{\varrho}(\mathbb{R})$* holds if and only if*

[TABLE] 3. (iii)

Additionally assume $0<\sigma<1/\theta<1$ and

[TABLE]

Then $w_{\sigma,\theta}\in B^{s}_{\varrho,q}(\mathbb{R})$ holds if and only if

[TABLE]

Proof.

In view of Section 2.1(i) assertion (i) is obvious.

Let us show (ii). Clearly $w_{\sigma,\theta}^{\prime}\in L_{\infty}(\mathbb{R})$ is equivalent to $\sigma\geq 1$. On the other hand, for $0<\varrho<\infty$ we have

[TABLE]

The latter integral is finite if and only if $1+(\sigma-1)\varrho>0$. In this case there holds

[TABLE]

which is finite if only if the argument of the Riemann zeta function $\zeta$ is strictly larger than one. Thus, for $0<\varrho<\infty$ we have $w_{\sigma,\theta}^{\prime}\in L_{\varrho}(\mathbb{R})$ if and only if $1+(\sigma-1)\varrho>0$ and $\theta(1+(\sigma-1)\varrho)>1$ which is equivalent to

[TABLE]

For $\sigma\geq 1$ this condition holds for all $\varrho$. On the other hand, if $0<\sigma<1$, then $0<1-\sigma<1$ and $1/(1-1/\theta)=\theta/(\theta-1)>1$ implies that the maximum in (13) is attained by its second entry. Hence, $w_{\sigma,\theta}^{\prime}\in L_{\varrho}(\mathbb{R})$ is equivalent to (11).

It remains to show assertion (iii). We split its proof into several steps.

Step 1 (Preparations). Note that for (iii) it suffices to show that $w_{\sigma,\theta}\in B^{s}_{\varrho,\infty}(\mathbb{R})\setminus B^{s}_{\varrho,q}(\mathbb{R})$ for

[TABLE]

Indeed, according to Section 2.1(ii), $w_{\sigma,\theta}\in B^{s}_{\varrho,\infty}(\mathbb{R})$ implies $w_{\sigma,\theta}\in B^{s^{\prime}}_{\varrho,q}(\mathbb{R})$ for all $s^{\prime}<s$ and $0<q\leq\infty$. Similarly, $w_{\sigma,\theta}\in B^{s^{\prime\prime}}_{\varrho,q}(\mathbb{R})$ for some $s^{\prime\prime}>s$ and some $0<q\leq\infty$ would yield $w_{\sigma,\theta}\in B^{s}_{\varrho,1}(\mathbb{R})$.

From (i) we know that $\left\|w_{\sigma,\theta}\;\vrule\;L_{\varrho}(\mathbb{R})\right\|<\infty$, so that it remains to prove that

[TABLE]

To this end, note that $0<\sigma$ and $1/\theta<1$ implies $s>0$, while $1/\varrho<\theta\,(1+\sigma)$ holds if and only if $s>1/\varrho-1$. Moreover, the assumption $1/\varrho<(1-\sigma)/(1-1/\theta)$ is equivalent to $s<1$. Hence,

[TABLE]

so that we can use first order differences. Therefore it is enough to show that

[TABLE]

because then

[TABLE]

Of course, we may assume w.l.o.g. that $h>0$.

Step 2 (Case $\varrho=\infty$). We prove “$\lesssim$” for $\varrho=\infty$ in (14). For this purpose, it suffices to show that

[TABLE]

So let $x,y\in\mathbb{R}$ with $h:=y-x>0$ be fixed. Note that it is enough to consider $a_{2,\theta}\leq x<a_{\infty,\theta}$, because $x<a_{2,\theta}$ implies

[TABLE]

while $x\geq a_{\infty,\theta}$ would lead to $w_{\sigma,\theta}(x)=w_{\sigma,\theta}(y)=0$. For $x\in[a_{2,\theta},a_{\infty,\theta})$ the quantity

[TABLE]

is well-defined. In case $h=y-x\geq M^{-\theta}$, we have

[TABLE]

as claimed. So let us turn to the case $0<h<M^{-\theta}$. If $y>a_{M+1,\theta}$, then again $w_{\sigma,\theta}(x)=0$. Moreover, in this case $a_{M+1,\theta}<y=x+h<a_{M+1,\theta}+h$, i.e.,

[TABLE]

Similarly, if $y\in[a_{M+1,\theta}-M^{-\theta},a_{M+1,\theta}]$, then $w_{\sigma,\theta}(y)=0$ and

[TABLE]

Hence, we are left with the case $a_{M,\theta}\leq x<y<a_{M,\theta}+3\,M^{-\theta}$ and $0<h=y-x<M^{-\theta}$, but for this situation (15) is obvious.

For the corresponding lower bound let $0<h<\overline{t}$. Then

[TABLE]

Step 3 (Case $\varrho<\infty$). In order to prove (14) for $\varrho<\infty$, consider the disjoint union

[TABLE]

where we set $\mathscr{L}_{\theta}:=(-\infty,4)$, $\mathscr{R}_{\theta}:=[4\,\zeta(\theta),\infty)$, as well as

[TABLE]

and $\mathscr{R}_{n,\theta}:=[a_{n,\theta}+3\,n^{-\theta},a_{n,\theta}+4\,n^{-\theta})$ for all $n\in\mathbb{N}$ with $n\geq 2$. Now let $0<h\leq\overline{t}=(1/6)^{\theta}$ be arbitrarily fixed. Then $N(h,\theta):=\left\lceil h^{-1/\theta}/3\right\rceil\in\mathbb{N}$ satisfies $N(h,\theta)\geq 2$ and

[TABLE]

due to the assumption $\theta>1$. Further, for all $n\in\mathbb{N}$ with $2\leq n\leq N(h,\theta)$ it holds $n\leq h^{-1/\theta}/3+1$, i.e.,

[TABLE]

In this case

[TABLE]

which yields the desired lower bound

[TABLE]

Let us show the corresponding upper bound. Using Step 2 and log-convexity of $L_{\varrho}$-norms, we see that the bound for $L_{\varrho}$ implies the respective bound for all $L_{p}$ with $0<1/p<1/\varrho$:

[TABLE]

Therefore, since $(1-\sigma)/(1-1/\theta)>1$ if and only if $\sigma<1/\theta$, we may assume w.l.o.g.

[TABLE]

So, let $0<\varrho\leq 1$ and consider $2\leq n\leq N(h,\theta)$. Then Hölder’s inequality (with $1/r:=1-\varrho$ and $1/r^{\prime}=\varrho$) and the monotonicity of $w_{\sigma,\theta}$ imply

[TABLE]

i.e., $\left\|\Delta_{h}w_{\sigma,\theta}\;\vrule\;L_{\varrho}(\mathscr{L}_{n,\theta})\right\|^{\varrho}\leq h^{\varrho}\,n^{-\theta(\sigma\varrho+1-\varrho)}$, as well as

[TABLE]

so that $\left\|\Delta_{h}w_{\sigma,\theta}\;\vrule\;L_{\varrho}(\mathscr{T}_{n,\theta})\right\|^{\varrho}\lesssim h^{\varrho}\,n^{-\theta(\sigma\varrho+1-\varrho)}$. Further we have $\sigma\varrho+1-\varrho>0$ because $1/\varrho\geq 1\geq 1-\sigma$. So, (16) and $h\leq n^{-\theta}$ yield that also

[TABLE]

Combining the latter estimates shows that

[TABLE]

Now additionally assume $1/\varrho<(1-\sigma)/(1-1/\theta)$. Then there holds $0<\theta(\sigma\varrho+1-\varrho)<1$ and hence

[TABLE]

where

[TABLE]

Therefore, we arrive at

[TABLE]

Moreover, (15) and $\theta>1$ yield that also

[TABLE]

is bounded by

[TABLE]

Finally, we clearly have $w_{\sigma,\theta}(x)=0$ on $\mathscr{L}_{\theta}\cup\mathscr{R}_{\theta}$ and hence $\left\|\Delta_{h}w_{\sigma,\theta}\;\vrule\;L_{\varrho}(\mathscr{R}_{\theta})\right\|=0$, as well as

[TABLE]

Altogether, this shows (14) and thus the proof is complete. ∎

Remark \therem.

We stress that some parameter restrictions in Section 2.1(iii) are stronger than required. If $\varrho=\infty$, our proof actually works for all $0<\sigma<1<\theta<\infty$. Moreover, the upper bound on $1/\varrho$ in (12) seems to be an artifact of our proof technique. At least for the “only if” part it can be dropped, as can be seen easily using complex interpolation. $\square$

In order to proceed, again let $\sigma\in\mathbb{R}_{+}$ and $1<\theta<\infty$. Then, based on $w_{\sigma,\theta}$ as defined above, let us set

[TABLE]

Lemma \thelem (Properties of $v_{\sigma,\theta}$ and $u_{\sigma,\theta}$).

Let $\sigma\in\mathbb{R}_{+}$ and $1<\theta<\infty$. Then

1. (i)

the supports of $u_{\sigma,\theta}$ and $v_{\sigma,\theta}$ are contained in $\widetilde{\mathscr{S}}_{\theta}:=[1/4,3/4]$. 2. (ii)

$v_{\gamma\sigma,\theta}=\left|v_{\sigma,\theta}\right|^{\gamma-1}v_{\sigma,\theta}$* for all $\gamma\in\mathbb{R}_{+}$.* 3. (iii)

$u_{\sigma,\theta}\in C^{1}(\mathbb{R}_{+})$* with $u^{\prime}_{\sigma,\theta}=v_{\sigma,\theta}$.* 4. (iv)

for all $0<\varrho,q\leq\infty$ and $s<1$ we have

[TABLE] 5. (v)

for all $1<\varrho<\infty$ we have

[TABLE]

Proof.

We use $\operatorname{supp}(w_{\sigma,\theta})\subseteq\mathscr{S}_{\theta}=[a_{2,\theta},a_{\infty,\theta}]=[4,4\,\zeta(\theta)]$, as shown in Section 2.1(i), to deduce the representation

[TABLE]

This proves (i) for $v_{\sigma,\theta}$. Moreover, for $0<r<1/4$ we have $u_{\sigma,\theta}(r)=\int_{0}^{r}0\,\mathrm{d}\xi=0,$ while for $r>3/4$ we may write

[TABLE]

which shows (i) for $u_{\sigma,\theta}$. Further, (ii) directly follows from (19) and Section 2.1(iii).

We are left with proving the regularity assertions (iii)–(v). The fact that $u_{\sigma,\theta}\in C^{1}(\mathbb{R}_{+})$ with $u_{\sigma,\theta}^{\prime}=v_{\sigma,\theta}$ is a consequence of the fundamental theorem of calculus and the continuity of $v_{\sigma,\theta}$, cf. (18) and Section 2.1(i) again. This shows (iii).

If we assume that $w_{\sigma,\theta}\in B_{\varrho,q}^{s}(\mathbb{R})$, then also

[TABLE]

because $B_{\varrho,q}^{s}(\mathbb{R})$ is invariant under diffeomorphic coordinate transformations; see, e.g., Triebel [14, Sect. 2.10.2]. Since v_{\sigma,\theta}=\widetilde{v_{\sigma,\theta}}\big{|}_{\mathbb{R}_{+}} this yields $v_{\sigma,\theta}\in B_{\varrho,q}^{s}(\mathbb{R}_{+})$. On the other hand, $v_{\sigma,\theta}\in B_{\varrho,q}^{s}(\mathbb{R}_{+})$ implies that there exists $g\in B_{\varrho,q}^{s}(\mathbb{R})$ such that v_{\sigma,\theta}=g\big{|}_{\mathbb{R}_{+}}. Now let $\chi\in C^{\infty}(\mathbb{R})$ with

[TABLE]

Then, according to a multiplication theorem by Triebel [15, Sect. 4.2.2], we conclude that $w_{\sigma,\theta}=\chi\,g\in B_{\varrho,q}^{s}(\mathbb{R})$. Due to (iii), this shows that for all $0<\varrho,q\leq\infty$ and $s\in\mathbb{R}$

[TABLE]

Moreover, we may extend $u_{\sigma,\theta}\in C^{1}(\mathbb{R}_{+})$ by zero in order to obtain $\widetilde{u_{\sigma,\theta}}\in C^{1}(\mathbb{R})$. Using the characterization of Besov spaces in terms of first order differences, we see that this gives $\widetilde{u_{\sigma,\theta}}\in B^{1-\varepsilon}_{\infty,q}(\mathbb{R})$ for all $0<\varepsilon<1$. Choosing $\varepsilon$ small enough such that $s<1-\varepsilon$ then shows $\widetilde{u_{\sigma,\theta}}\in B^{s}_{\infty,q}(\mathbb{R})\hookrightarrow B^{s}_{\varrho,q}(\mathbb{R})$, i.e., $u_{\sigma,\theta}\in B^{s}_{\varrho,q}(\mathbb{R}_{+})$, where we used Section 2.1 and the compact support of $\widetilde{u_{\sigma,\theta}}$. Therefore, (iv) follows from Section 2.1(i).

Since Sobolev spaces $W^{k}_{\varrho}$ can be identified as special Triebel-Lizorkin spaces $F^{k}_{\varrho,2}$, we can argue similarly for this case. Instead of (20) we now have that for every $k\in\mathbb{N}_{0}$ (particularly for $k=1$) and $1<\varrho<\infty$ there holds $u_{\sigma,\theta}^{\prime}\in W^{k}_{\varrho}(\mathbb{R}_{+})$ if and only if $w_{\sigma,\theta}\in W^{k}_{\varrho}(\mathbb{R})$. Further from (i) and (iii) we clearly have $u_{\sigma,\theta}\in W^{1}_{\varrho}(\mathbb{R}_{+})$. Together this shows (v) and hence the proof is complete. ∎

Next, for $d\in\mathbb{N}$ let $\Omega\subseteq\mathbb{R}^{d}$ be either $\mathbb{R}^{d}$ itself, a bounded Lipschitz domain, or an interval (if $d=1$), and assume that $\Omega$ contains the Euclidean unit ball $B_{1}(0):=\{x\in\mathbb{R}^{d}\;\vrule\;\left|x\right|_{2}<1\}$. Given $\sigma\in\mathbb{R}_{+}$, as well as $1<p,\theta<\infty$, we then let

[TABLE]

for all test functions $\psi\in\mathcal{D}(\Omega)$. It is easily seen that we actually deal with distributions $u_{\sigma,\theta,d},f_{\sigma,\theta,d}^{[p]}\in\mathcal{D}^{\prime}(\Omega)$ since Section 2.1(iii) implies $u_{\sigma,\theta},v_{(p-1)\sigma,\theta}\in L_{\infty}(\mathbb{R}_{+})$. These distributions are closely related:

Lemma \thelem.

Let $d\in\mathbb{N}$ and $1<p,\theta<\infty$, as well as $\sigma\in\mathbb{R}_{+}$. Further let $\Omega\subseteq\mathbb{R}^{d}$ be either $\mathbb{R}^{d}$ itself, a bounded Lipschitz domain, or an interval (if $d=1$), and assume $B_{1}(0)\subseteq\Omega$. Then

1. (i)

$\operatorname{supp}(u_{\sigma,\theta,d}),\operatorname{supp}(f^{[p]}_{\sigma,\theta,d})\subseteq B_{4/5}(0)$, 2. (ii)

for all $x\in\Omega$ there holds

[TABLE] 3. (iii)

$f_{\sigma,\theta,d}^{[p]}\in W_{p^{\prime}}^{-1}(\Omega)$* and $u_{\sigma,\theta,d}\in W^{1}_{p,0}(\Omega)$ constitutes a weak solution $u$ to*

[TABLE]

Proof.

In view of Section 2.1 assertion (i) is obvious. Moreover, it is clear that $u_{\sigma,\theta,d}\in\mathcal{D}^{\prime}(\Omega)$ is regular and can be identified with the function \big{(}u_{\sigma,\theta}\circ r_{d}\big{)}\big{|}_{\Omega}\in C^{1}(\Omega), where we set

[TABLE]

With this interpretation we have

[TABLE]

for all $x\in B_{1/4}(0)$, while on $\Omega\setminus B_{1/8}(0)$ the chain rule and Section 2.1(iii) give

[TABLE]

where

[TABLE]

Together this shows

[TABLE]

and hence

[TABLE]

since $u_{\sigma,\theta},v_{\sigma,\theta}\in L_{\infty}(\mathbb{R}_{+})$ with compact support and $\left|x_{j}/\left|x\right|_{2}\right|\leq 1$. So, we can conclude $u_{\sigma,\theta,d}\in W^{1}_{p,0}(\Omega)$. Further, as a direct consequence of (24), we obtain

[TABLE]

such that by Section 2.1(ii) with $\gamma:=p-1$ we have that

[TABLE]

holds for all $x\in\Omega$. Together with (24) this proves (22), as well as

[TABLE]

for each $\psi\in\mathcal{D}(\Omega)$. In other words, there holds $-\Delta_{p}(u_{\sigma,\theta,d})=f_{\sigma,\theta,d}^{[p]}$ in the weak sense. Finally, Hölder’s inequality on $B_{1}(0)\subseteq\Omega$ proves

[TABLE]

Since by definition $\mathcal{D}(\Omega)$ is dense $W^{1}_{p,0}(\Omega)$ we therefore have $f_{\sigma,\theta,d}^{[p]}\in(W^{1}_{p,0}(\Omega))^{\prime}=W^{-1}_{p^{\prime}}(\Omega)$ and the proof is complete. ∎

In order to provide further regularity assertions for $u_{\sigma,\theta,d}$ and $f_{\sigma,\theta,d}^{[p]}$, we will need the subsequent result which characterizes the smoothness and integrability of rotationally invariant functions. Therein $r_{d}$ has the same meaning as in (23).

Proposition \theprop.

Let $0<a<b<\infty$. Assume that $g\colon\mathbb{R}_{+}\to\mathbb{C}$ is measurable with support $\operatorname{supp}(g)\subseteq[a,b]$ and let $g_{d}:=g\circ r_{d}$. Then

1. (i)

$g_{d}\colon\mathbb{R}^{d}\to\mathbb{C}$* is well-defined almost everywhere.* 2. (ii)

for $0<\varrho,q\leq\infty$ and $s>d\,\max\{0,1/\varrho-1\}$ there holds

[TABLE]

as well as

[TABLE] 3. (iii)

for $1<\varrho<\infty$ and $k\in\mathbb{N}$ there holds

[TABLE]

Proof.

With $g$ and $r_{d}$ also $g_{d}$ is measurable such that it can be represented as an a.e. convergent pointwise limit of simple functions. This proves (i).

If $d=1$, then the equivalences in (ii) and (iii) are trivial, as $g$ vanishes in a neighborhood of the origin. So let us assume $d\geq 2$. Then for $\varrho<\infty$ the first assertion in (ii) follows from a simple transformation into (generalized) polar coordinates $x=r\,\vartheta(\phi)$, $(r,\phi)\in[0,\infty)\times\Phi$:

[TABLE]

where we used that $\operatorname{supp}(g)\subseteq[a,b]$ and that $T$ is some tensor product of trigonometric functions defined on $\Phi\subset[-\pi,\pi]^{d-1}$. For $\varrho=\infty$ the equivalence (26) is obvious.

It remains to prove the equivalences for multivariate Besov and Sobolev spaces. In case of $B^{s}_{\varrho,q}$ and $\varrho=\infty$, this follows from results due to Sickel et al. [11, Thm. 2], while the case $0<\varrho<\infty$ is covered by [11, Cor. 1 & 2]. However, [11, Cor. 1 & 2] also covers the assertion for Sobolev spaces since $W^{k}_{\varrho}=F^{k}_{\varrho,2}$ if $1<\varrho<\infty$. ∎

Lemma \thelem (Regularity of $u_{\sigma,\theta,d}$, $A(\nabla u_{\sigma,\theta,d})$, and $f^{[p]}_{\sigma,\theta,d}$).

Let $d\in\mathbb{N}$ and $1<p,\theta<\infty$, as well as $\sigma\in\mathbb{R}_{+}$. Further let $\Omega\subseteq\mathbb{R}^{d}$ be either $\mathbb{R}^{d}$ itself, a bounded Lipschitz domain, or an interval (if $d=1$), and assume $B_{1}(0)\subseteq\Omega$. Moreover, let $0<\varrho,q\leq\infty$ and $s\in\mathbb{R}$ with $d\,\max\{0,1/\varrho-1\}<s<1$. Then

1. (i)

there holds

[TABLE]

and $1<\varrho<\infty$ implies that

[TABLE] 2. (ii)

there holds

[TABLE]

and $1<\varrho<\infty$ implies that

[TABLE]

Additionally assume that $\min\{\varrho,q\}>1$. Then

1. (iii)

A(\nabla u_{\sigma,\theta,d})\in\big{(}B^{s}_{\varrho,q}(\Omega)\big{)}^{d}* implies $f_{\sigma,\theta,d}^{[p]}\in B^{-1+s}_{\varrho,q}(\Omega)$,* 2. (iv)

$w^{\prime}_{(p-1)\sigma,\theta}\in L_{\varrho}(\mathbb{R})$* implies $f_{\sigma,\theta,d}^{[p]}\in L_{\varrho}(\Omega)$.*

Proof.

Recall that $u_{\sigma,\theta,d}\in\mathcal{D}^{\prime}(\Omega)$ can be identified with the function $u_{\sigma,\theta}\circ r_{d}$ restricted to $\Omega$. Hence, $u_{\sigma,\theta}\circ r_{d}\in B^{1+s}_{\varrho,q}(\mathbb{R}^{d})$ implies $u_{\sigma,\theta,d}\in B^{1+s}_{\varrho,q}(\Omega)$. On the other hand, if $u_{\sigma,\theta,d}\in B^{1+s}_{\varrho,q}(\Omega)$, then there exists $\widetilde{u}\in B^{1+s}_{\varrho,q}(\mathbb{R}^{d})$ with \widetilde{u}\big{|}_{\Omega}=u_{\sigma,\theta,d}. Now let $\chi\in C^{\infty}(\mathbb{R}^{d})$ with

[TABLE]

Then $u_{\sigma,\theta}\circ r_{d}=\chi\,\widetilde{u}\in B^{1+s}_{\varrho,q}(\mathbb{R}^{d})$ due to [15, Sect. 4.2.2]. Therefore, $u_{\sigma,\theta,d}\in B^{1+s}_{\varrho,q}(\Omega)$ is equivalent to $u_{\sigma,\theta}\circ r_{d}\in B^{1+s}_{\varrho,q}(\mathbb{R}^{d})$. By Section 2.1(i) and Section 2.1(ii) this holds if and only if $u_{\sigma,\theta}\in B^{1+s}_{\varrho,q}(\mathbb{R}_{+})$. Since we assume $s<1$, we can use Section 2.1(iv) to see that this is equivalent to $w_{\sigma,\theta}\in B^{s}_{\varrho,q}(\mathbb{R})$. Thus, we have shown (i) in the case of Besov spaces. For Sobolev spaces we can argue similarly.

Next we apply (i) to deduce that $w_{(p-1)\sigma,\theta}\in B^{s}_{\varrho,q}(\mathbb{R})$ is equivalent to $u_{(p-1)\sigma,\theta,d}\in B^{1+s}_{\varrho,q}(\Omega)$. By Section 2.1(i) this holds if and only if $u_{(p-1)\sigma,\theta,d}\in B^{s}_{\varrho,q}(\Omega)$ and $\nabla u_{(p-1)\sigma,\theta,d}\in(B^{s}_{\varrho,q}(\Omega))^{d}$. Since we assume that $s<1$, the first condition is always fulfilled (cf. the proof of Section 2.1!), and by Section 2.1(ii) $\nabla u_{(p-1)\sigma,\theta,d}$ is nothing but $A(\nabla u_{\sigma,\theta,d})$. Also here the proof for Sobolev spaces is essentially the same.

Let us prove (iii). To this end, we note that A(\nabla u_{\sigma,\theta,d})\in\big{(}B^{s}_{\varrho,q}(\Omega)\big{)}^{d} implies that for every $j=1,\ldots,d$ we have

[TABLE]

according to Section 2.1, where we set

[TABLE]

cf. Triebel [17, Def. 2.1]. Further from [17, Thm. 3.30] and $\min\{\varrho,q\}>1$ it follows that $\widetilde{B}^{s}_{\varrho,q}(\Omega)=(B^{-s}_{\varrho^{\prime},q^{\prime}}(\Omega))^{\prime}$ such that for all $j=1,\ldots,d$ we can estimate

[TABLE]

Therefore the representation formula (25) together with Section 2.1(i) yield that for all $\psi\in\mathcal{D}(\Omega)$ there holds

[TABLE]

Now we again employ [17, Thm. 3.30] to see that $\mathcal{D}(\Omega)$ is dense in $\widetilde{B}^{1-s}_{\varrho^{\prime},q^{\prime}}(\Omega)$ and hence f_{\sigma,\theta,d}^{[p]}\in\big{(}\widetilde{B}^{1-s}_{\varrho^{\prime},q^{\prime}}(\Omega)\big{)}^{\prime}=B^{-1+s}_{\varrho,q}(\Omega), as claimed.

It remains to prove (iv). For this purpose, note that for $f_{\sigma,\theta,d}^{[p]}\in L_{\varrho}(\Omega)=(L_{\varrho^{\prime}}(\Omega))^{\prime}$ it is sufficient to find $f_{d}\in L_{\varrho}(\Omega)$ such that

[TABLE]

since $\mathcal{D}(\Omega)$ is dense in $L_{\varrho^{\prime}}(\Omega)$ if $1\leq\varrho^{\prime}<\infty$. We claim that this $f_{d}$ is given by the restriction of $f_{d}=f\circ r_{d}$ to $\Omega$, where

[TABLE]

Recall that due to Section 2.1 $v_{(p-1)\sigma,\theta}$ is continuous with support in $[1/4,3/4]$. Hence, the function $r\mapsto g(r):=v_{(p-1)\sigma,\theta}(r)/r$ belongs to $L_{\varrho}(\mathbb{R}_{+})$. Moreover, it is clear that our assumption $w_{(p-1)\sigma,\theta}^{\prime}\in L_{\varrho}(\mathbb{R})$ implies that also $v_{(p-1)\sigma,\theta}^{\prime}\in L_{\varrho}(\mathbb{R}_{+})$. Therefore, we have $f\in L_{\varrho}(\mathbb{R}_{+})$ and from Section 2.1 we conclude $f_{d}\in L_{\varrho}(\mathbb{R}^{d})$. This shows that indeed $f_{d}\in L_{\varrho}(\Omega)$. Thus, we are left with proving (27). To this end, let $\psi\in\mathcal{D}(\Omega)$ be arbitrarily fixed and assume for a moment that $d\geq 2$. Then $B_{1}(0)\subseteq\Omega$ and a transformation into polar coordinates $x=r\,\vartheta(\phi)$, $(r,\phi)\in[0,\infty)\times\Phi$, yields

[TABLE]

(cf. the proof of Section 2.1). Since $v^{\prime}_{(p-1)\sigma,\theta}$ belongs to $L_{1}(\mathbb{R}_{+})$ and $\psi$ is smooth, we may use integration by parts to see that

[TABLE]

is finite, because the boundary term vanishes and

[TABLE]

as well as $v_{(p-1)\sigma,\theta}$ are bounded on $[1/4,3/4]$. Hence, for the inner integral in (28) we find

[TABLE]

and thus (25) shows that we indeed have (27):

[TABLE]

Finally, a similar calculation shows that (29) remains valid also for $d=1$. So, the proof is complete. ∎

Now we are well-prepared to give profound proofs of our main results stated in Theorem 1.1 and Theorem 1.2.

2.2 Proof of Theorem 1.1

Proof.

Let $d\in\mathbb{N}$, as well as $2\leq p<\infty$, and $0<\varepsilon<1/p$ be given fixed. Further let $\Omega\subseteq\mathbb{R}^{d}$ be either $\mathbb{R}^{d}$ itself, a bounded Lipschitz domain, or an interval (if $d=1$). Since $\Omega$ is open it contains inner points. By a simple translation and dilation argument (see, e.g. [3, Sect. 4]) we may w.l.o.g. assume that the Euclidean ball of radius one, $B_{1}(0)$, is contained in $\Omega$. In what follows we will choose specific values $\sigma\in(0,1)$ as well as $\theta\in(1,\infty)$ and define $u:=u_{\sigma,\theta,d}$ and $f:=f_{\sigma,\theta,d}^{[p]}$ according to (21). From Section 2.1 it then follows that $u\in W^{1}_{p,0}(\Omega)$ is a weak solution to (1) with right-hand side $f\in W^{-1}_{p^{\prime}}(\Omega)$ and that the supports of $u$ and $f$ are contained in $B_{4/5}(0)$.

Given $1<\mu\leq\infty$ and $\varepsilon\,(p-1)<\lambda<1-\varepsilon$ we choose $\theta$ such that $0<1-\varepsilon<1/\theta<1$ and define

[TABLE]

Then it is easily seen that

[TABLE]

Indeed, the lower bound on $\lambda$ shows that $\lambda/(p-1)-\varepsilon$ is strictly positive. If $1<\mu<\infty$, we note that $p\geq 2$ implies $1-\varepsilon\,(p-1)\,\mu<1-\varepsilon<1/\theta$ and hence

[TABLE]

while the corresponding estimate for $\mu=\infty$ is trivial since $\varepsilon>0$. Moreover, $1/\theta<1$ yields

[TABLE]

which completes the proof of (30).

Next we note that (30) particularly implies that

[TABLE]

So, we can employ Section 2.1(iii) to see that for $1\leq\varrho\leq\infty$ there holds

[TABLE]

where

[TABLE]

Note that our assumptions imply that $0<1-1/\theta<\varepsilon$,

[TABLE]

as well as $0<\lambda/(p-1)\pm\varepsilon<1$. If $\varrho$ is large, we have $\lambda/(p-1)-\varepsilon<s_{\varrho}<\lambda/(p-1)$. Thus (31) and Section 2.1(i) prove

[TABLE]

for this case. The regularity statements for $u$ in the remaining cases are obtained likewise.

Similarly, (30) and Section 2.1(iii) show that for $1\leq\varrho\leq\infty$ there holds

[TABLE]

where now (depending on the relation of $\varrho$ and $\mu$ to each other)

[TABLE]

Therefore, we can use Section 2.1(ii) to deduce the regularity statements for $A(\nabla u)$. In particular we have A(\nabla u)\in\big{(}B_{\mu,\infty}^{\lambda}(\Omega)\big{)}^{d} such that by Section 2.1(iii) $f=f_{\sigma,\theta,d}^{[p]}\in B^{-1+\lambda}_{\mu,\infty}(\Omega)$, as claimed. ∎

2.3 Proof of Theorem 1.2

In order to show Theorem 1.2 we essentially follow the lines of the proof of Theorem 1.1. So let us focus on the necessary modifications only.

Proof.

Given $1<\mu\leq\infty$ and $0<\varepsilon<\min\{1/(p-1),1-1/(p-1)\}$ (note that this time $p>2$!) we choose $\theta$ such that $0<1-\varepsilon<1/\theta<1$ and define

[TABLE]

Then there holds

[TABLE]

Indeed, $\mu>1$ and $0<1/\theta<1$ show that $0\leq(1-1/\theta)/\mu<1-1/\theta<1$. This proves $0<\sigma$ as well as $1/\theta<(p-1)\,\sigma\leq 1$. Hence, we also have $\sigma\leq 1/(p-1)<1-\varepsilon<1/\theta$ due to our assumption on $\varepsilon$.

Now the claimed regularity of $u$ follows exactly as in the proof of Theorem 1.1, where this time our restrictions on $\varepsilon$ ensure that $0<1/(p-1)\pm\varepsilon<1$.

In order to prove the regularity statement for $A(\nabla u)$ we like to apply Section 2.1(ii). To this end, we have to show that $w_{(p-1)\sigma,\theta}\in W^{1}_{\varrho}(\mathbb{R})$ for $1<\varrho<\infty$ if and only if $\varrho<\mu$. By Section 2.1(i) this reduces to the claim $w^{\prime}_{(p-1)\sigma,\theta}\in L_{\varrho}(\mathbb{R})$. If $\mu=\infty$, we actually have $(p-1)\,\sigma=1$. Therefore, from Section 2.1(ii) it follows that $w^{\prime}_{(p-1)\sigma,\theta}\in L_{\varrho}(\mathbb{R})$ for all $0<\varrho\leq\infty$. On the other hand, if $\mu<\infty$, then $0<(p-1)\,\sigma<1$. Hence, in this case we have $w^{\prime}_{(p-1)\sigma,\theta}\in L_{\varrho}(\mathbb{R})$ if and only if

[TABLE]

In conclusion, A(\nabla u)\in\big{(}W^{1}_{\varrho}(\Omega)\big{)}^{d} for $1<\varrho<\infty$ is equivalent to $\varrho<\mu$, as claimed.

It remains to prove that $f=f_{\sigma,\theta,d}^{[p]}$ belongs to $L_{\nu}(\Omega)$. If $\nu>1$, this follows from Section 2.1(iv) and the calculations above. However, in view of the compact support of $f$, this lower bound on $\nu$ can be dropped. ∎