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On the Limit Regularity in Sobolev and Besov Scales Related to Approximation Theory

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Abstract

We study the interrelation between the limit \(L_p(\Omega )\)-Sobolev regularity \(\overline{s}_p\) of (classes of) functions on bounded Lipschitz domains \(\Omega \subseteq \mathbb {R}^d\), \(d\ge 2\), and the limit regularity \(\overline{\alpha }_p\) within the corresponding adaptivity scale of Besov spaces \(B^\alpha _{\tau ,\tau }(\Omega )\), where \(1/\tau =\alpha /d+1/p\) and \(\alpha >0\) (\(p>1\) fixed). The former determines the convergence rate of uniform numerical methods, whereas the latter corresponds to the convergence rate of best N-term approximation. We show how additional information on the Besov or Triebel–Lizorkin regularity may be used to deduce upper bounds for \(\overline{\alpha }_p\) in terms of \(\overline{s}_p\) simply by means of classical embeddings and the extension of complex interpolation to suitable classes of quasi-Banach spaces due to Kalton et al. (in: De Carli and Milman (ed) Interpolation theory and applications, American Mathematical Society, Providence, 2007). The results are applied to the Poisson equation, to the p-Poisson problem, and to the inhomogeneous stationary Stokes problem. In particular, we show that already established results on the Besov regularity for the Poisson equation are sharp.

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Acknowledgements

The authors are grateful to the anonymous reviewers for their valuable comments and their constructive suggestions which helped to improve the manuscript.

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Correspondence to Markus Weimar.

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Dedicated to Prof. Dr. Stephan Dahlke on the occasion of his 60th birthday.

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Communicated by Dahlke.

Appendix: Basics from Function Space Theory

Appendix: Basics from Function Space Theory

In this supplementary section we collect the main definitions and assertions concerning function spaces on domains which are needed throughout the paper. Here ‘domain’ always means ‘non-empty, connected, open set’. Special attention is paid to bounded Lipschitz domains \(\Omega \subseteq \mathbb {R}^d\), \(d\in \mathbb {N}\), as defined, e.g., in Triebel [32, Sect. 1.11.4].

Besov and Triebel–Lizorkin Spaces

In accordance with Triebel [31] we use the Fourier analytic approach towards Besov and Triebel–Lizorkin spaces on \(\mathbb {R}^d\) and define the corresponding spaces on domains by restriction.

Let \(d\in \mathbb {N}\). By \(\mathcal {S}(\mathbb {R}^d)\) we denote the Schwartz space of all complex-valued rapidly decreasing \(\mathcal {C}^\infty \) functions on \(\mathbb {R}^d\) and \(\mathcal {S}'(\mathbb {R}^d)\) denotes its dual space of tempered distributions. Moreover, for domains \(\Omega \subseteq \mathbb {R}^d\) we let \(\mathcal {D}(\Omega ):=\mathcal {C}_0^\infty (\Omega )\) denote the collection of all complex-valued \(\mathcal {C}^\infty \) functions in \(\mathbb {R}^d\) with compact support in \(\Omega \) and denote by \(\mathcal {D}'(\Omega )\) its dual space of distributions on \(\Omega \). As usual, we say two functionals f and g equal each other in \(\mathcal {S}'(\mathbb {R}^d)\) or \(\mathcal {D}'(\Omega )\) if

$$\begin{aligned} f(\varphi )=g(\varphi ) \qquad \text {for all }\, \varphi \,\text { from }\, \mathcal {S}(\mathbb {R}^d) \,\text { or }\, \mathcal {D}(\Omega ), \,\text { respectively}. \end{aligned}$$

For \(g\in \mathcal {S}'(\mathbb {R}^d)\) we denote by \(g_{|_{\Omega }}\) the restriction of g to \(\Omega \) which means that

$$\begin{aligned} g_{|_{\Omega }} \in \mathcal {D}'(\Omega ) \qquad \text {and} \qquad (g_{|_{\Omega }})(\varphi ):=g(\varphi ) \qquad \text {for all} \qquad \varphi \in \mathcal {D}(\Omega ). \end{aligned}$$

Note that this is meaningful since \(\mathcal {D}(\Omega )\subseteq \mathcal {D}(\mathbb {R}^d)\subseteq \mathcal {S}(\mathbb {R}^d)\).

In addition, let \(\mathcal {F}\) and \(\mathcal {F}^{-1}\) denote the (extension of the) Fourier transform, respectively its inverse, on \(\mathcal {S}'(\mathbb {R}^d)\). Fix an arbitrary \(\phi _0 \in \mathcal {S}(\mathbb {R}^d)\) such that

$$\begin{aligned} \phi _0(x)=1 \quad \text {if} \quad \left| x \right| _2\le 1 \qquad \text {and} \qquad \phi _0(x)=0 \quad \text {if} \quad \left| x \right| _2\ge \frac{3}{2}. \end{aligned}$$

Then the collection \(\Phi :=(\phi _k)_{k\in \mathbb {N}_0}\), with

$$\begin{aligned} \phi _k(x):=\phi _0(2^{-k}x)-\phi _0(2^{-k+1}x), \qquad x\in \mathbb {R}^d,\qquad k\in \mathbb {N}, \end{aligned}$$

defines a smooth dyadic resolution of unity and we have

$$\begin{aligned} f = \sum _{k=0}^\infty \mathcal {F}^{-1}[\phi _k\, \mathcal {F}f] \qquad \text {(convergence in }\mathcal {S}'(\mathbb {R}^d)) \end{aligned}$$

for all \(f\in \mathcal {S}'(\mathbb {R}^d)\). Due to the celebrated Paley-Wiener-Schwartz-Theorem, the building blocks \(\mathcal {F}^{-1}[\phi _k\, \mathcal {F}f]\), \(k\in \mathbb {N}_0\), are actually entire analytic functions; see, for instance, Triebel [31, Sect. 1.2.1]. As usual, for \(0<q<\infty \), \(\ell _q(\mathbb {N}_0)\) is the space of q-summable scalar-valued sequences over \(\mathbb {N}_0\) (bounded sequences, if \(q=\infty \)).

Definition A.1

For \(d\in \mathbb {N}\) choose \(\Phi \) as above and let \(\Omega \subsetneq \mathbb {R}^d\) denote an arbitrary domain. Moreover, let \(s\in \mathbb {R}\) and \(0<p,q\le \infty \).

  1. (i)

    The set \(B^s_{p,q}(\mathbb {R}^d):=\left\{ f\in \mathcal {S}'(\mathbb {R}^d) \; \big | \;\left\| f \; \big | \;B^s_{p,q}(\mathbb {R}^d) \right\| <\infty \right\} \), quasi-normed by

    $$\begin{aligned} \left\| f \; \big | \;B^s_{p,q}(\mathbb {R}^d) \right\| := \left\| \left( 2^{ks} \left\| \mathcal {F}^{-1}[\phi _k\, \mathcal {F}f](\cdot ) \; \big | \;L_p(\mathbb {R}^d) \right\| \right) _{k\in \mathbb {N}_0} \; \big | \;\ell _q(\mathbb {N}_0) \right\| , \end{aligned}$$

    is called Besov space.

  2. (ii)

    If \(p<\infty \), then the set \(F^s_{p,q}(\mathbb {R}^d):=\left\{ f\in \mathcal {S}'(\mathbb {R}^d) \; \big | \;\left\| f \; \big | \;F^s_{p,q}(\mathbb {R}^d) \right\| <\infty \right\} \), quasi-normed by

    $$\begin{aligned} \left\| f \; \big | \;F^s_{p,q}(\mathbb {R}^d) \right\| := \left\| \left\| \left( 2^{ks} \left| \mathcal {F}^{-1}[\phi _k\, \mathcal {F}f](\cdot ) \right| \right) _{k\in \mathbb {N}_0} \; \big | \;\ell _q(\mathbb {N}_0) \right\| \; \big | \;L_p(\mathbb {R}^d) \right\| , \end{aligned}$$

    is called Triebel–Lizorkin space.

  3. (iii)

    If \(A\in \{B,F\}\) with \(p<\infty \) for \(A=F\), then the set

    $$\begin{aligned} A^s_{p,q}(\Omega ):=\left\{ f \in \mathcal {D}'(\Omega ) \; \big | \;\text {there exists } g\in A^{s}_{p,q}(\mathbb {R}^d) \text { with } g_{|_{\Omega }} = f \text { in } \mathcal {D}'(\Omega ) \right\} , \end{aligned}$$

    quasi-normed by

    $$\begin{aligned} \left\| f \; \big | \;A^s_{p,q}(\Omega ) \right\| := \inf _{\begin{array}{c} g\in A^{s}_{p,q}(\mathbb {R}^d)\\ g_{|_{\Omega }} = f \text { in } \mathcal {D}'(\Omega ) \end{array}} \left\| g \; \big | \;A^{s}_{p,q}(\mathbb {R}^d) \right\| , \end{aligned}$$

    is called Besov resp. Triebel–Lizorkin space on \(\Omega \).

Standard proofs show that the spaces introduced above are quasi-Banach spaces (Banach iff \(\min \{p,q\}\ge 1\) and Hilbert iff \(p=q=2\)) and that different \(\Phi \) provide equivalent quasi-norms, see, e.g., Triebel [31, Sect. 2.3.2]. Furthermore, these scales of spaces cover a variety of classical function spaces—such as, e.g., Lebesgue, Sobolev(-Slobodeckij), Bessel potential, Lipschitz, Hölder(-Zygmund), or Hardy spaces—as special cases. Besides our Fourier analytic definition, there is a big variety of other descriptions of these spaces which are equivalent at least for large ranges of parameters. To give an example, we note that at least for

$$\begin{aligned} s > \sigma _{p}:= d \, \max \left\{ \frac{1}{p}-1,0\right\} \end{aligned}$$

the spaces \(A^s_{p,q}(\mathbb {R}^d)\) (and also \(A^{s}_{p,q}(\Omega )\) for bounded Lipschitz domains \(\Omega \subseteq \mathbb {R}^d\)) exclusively contain regular distributions, i.e., functions, which makes it possible to characterize them as subspaces of some Lebesgue space by means of iterated differences. For details we refer to Triebel [32, Sect. 1.11.9].

Sobolev Spaces

We follow the usual approach and define the following Sobolev-type spaces based on Besov and Triebel–Lizorkin spaces.

Definition A.2

For \(d\in \mathbb {N}\) let \(\Omega \subseteq \mathbb {R}^d\) denote a bounded Lipschitz domain. Then we set

$$\begin{aligned}&W^m_p(\Omega ):=F^m_{p,2}(\Omega ),&m\in \mathbb {N}_0, 1<p<\infty ,&\quad \text {(Sobolev)}\\&W^s_p(\Omega ):=F^s_{p,p}(\Omega )=B^s_{p,p}(\Omega ),&0<s\notin \mathbb {N}, 1\le p< \infty ,&\quad \text {(Sobolev}\!-\!\text {Slobodeckij)}\\&W^s_p(\Omega ):=\left[ W^{-s}_{p',0}(\Omega ) \right] ',&s<0, 1<p<\infty ,&\\&H^s_p(\Omega ):=F^s_{p,2}(\Omega ),&s\in \mathbb {R}, 1<p<\infty ,&\quad \text {(Bessel potential)}\\&H^s(\Omega ):=H^s_2(\Omega )=F^s_{2,2}(\Omega )=B^s_{2,2}(\Omega ),&s\in \mathbb {R},&\quad \text {(Sobolev}\!-\!\text {Hilbert)} \end{aligned}$$

where for \(1<p<\infty \) the index \(p'\) is given by \(1/p+1/p'=1\) and \(W_{p,0}^s(\Omega )\) denotes the closure of \(\mathcal {C}_0^\infty (\Omega )\) w.r.t. the norm \(\left\| \cdot \; \big | \;W_p^s(\Omega ) \right\| \) if \(s>0\).

It is worth noting that these definitions are equivalent with the common definitions of Sobolev(-Slobodeckij) and Bessel potential spaces: For \(s=m\in \mathbb {N}_0\) we have

$$\begin{aligned} W^m_p(\Omega )=\bigg \{f\in L_p(\Omega ) \,\bigg |\, \left\| f \; \big | \;W^m_p(\Omega ) \right\| :=\bigg [ \sum _{\left| \alpha \right| _1\le m} \left\| D^\alpha f \; \big | \;L_p(\Omega ) \right\| ^p \bigg ]^{1/p}<\infty \bigg \}, \end{aligned}$$

see Triebel [32, Theorem 1.122], while \(W^s_p(\Omega )=B^s_{p,p}(\Omega )\) for \(0<s\notin \mathbb {N}\) coincides with the definition of Sobolev-Slobodeckij spaces as real interpolation space of \(L_p(\Omega )\) with \(W^m_p(\Omega )\) for some \(m\in \mathbb {N}\) with \(m>s\) and suitable parameters; see, e.g., DeVore [13, Sect. 4.6].

Embeddings

The scales of Besov and Triebel–Lizorkin spaces \(A^s_{p,q}(\Omega )\) on bounded Lipschitz domains satisfy various embeddings. Let us mention a few of them:

Proposition A.3

For \(d\in \mathbb {N}\) let \(\Omega \subseteq \mathbb {R}^d\) denote a bounded Lipschitz domain. Further assume \(s,s_0,s_1\in \mathbb {R}\) and let \(0<p,p_0,p_1,q,q_0,q_1\le \infty \).

  1. (i)

    Assume additionally that \(p<\infty \). Then

    $$\begin{aligned} B^{s}_{p,q_0}(\Omega ) \hookrightarrow F^{s}_{p,q}(\Omega ) \hookrightarrow B^{s}_{p,q_1}(\Omega ). \end{aligned}$$

    holds if, and only if, we have \(q_0 \le \min \{p,q\} \le \max \{p,q\}\le q_1\).

  2. (ii)

    If additionally \(p_0<p_1<\infty \) and \(s_0-d/p_0=s_1-d/p_1\), then

    $$\begin{aligned} F^{s_0}_{p_0,q_0}(\Omega ) \hookrightarrow F^{s_1}_{p_1,q_1}(\Omega ). \end{aligned}$$
  3. (iii)

    If additionally \(A\in \{B,F\}\) (and \(p<\infty \) if \(A=F\)), as well as \(q_0\le q_1\), then

    $$\begin{aligned} A^s_{p,q_0}(\Omega ) \hookrightarrow A^{s}_{p,q_1}(\Omega ). \end{aligned}$$
  4. (iv)

    If additionally \(X,Y\in \{B,F\}\) and

    $$\begin{aligned} s_0-s_1 > d \, \max \left\{ \frac{1}{p_0} -\frac{1}{p_1}, 0 \right\} , \end{aligned}$$

    then

    $$\begin{aligned} X^{s_0}_{p_0,q_0}(\Omega ) \hookrightarrow Y^{s_1}_{p_1,q_1}(\Omega ) \end{aligned}$$

    (with finite integrability parameter for F-spaces).

  5. (v)

    Assume additionally that \(p_0<p<p_1\) and

    $$\begin{aligned} s_0-\frac{d}{p_0} = s-\frac{d}{p} =s_1-\frac{d}{p_1}. \end{aligned}$$

    Then

    $$\begin{aligned} B^{s_0}_{p_0,q_0}(\Omega ) \hookrightarrow F^s_{p,q}(\Omega ) \hookrightarrow B^{s_1}_{p_1,q_1}(\Omega ) \end{aligned}$$

    holds if, and only if, we have \(q_0\le p\le q_1\).

Proof

For (i), (ii), and (v) see, e.g., Triebel [32, p. 60] and the references therein. For (iii) and (iv) additionally consult Triebel [31, Proposition 2 in Sect. 2.3.2], as well as [33, Theorem 4.33 and Remark 4.34]. \(\square \)

Note that Proposition A.3(iv) particularly implies that for \(A\in \{B,F\}\) we have

$$\begin{aligned}&A^{s_0}_{p_0,q}(\Omega ) \hookrightarrow W^{s_1}_{p_1}(\Omega ) \qquad \text {if} \qquad s_0>s_1\ge 0, \text { as well as } 1<p_1 \le p_0 \le \infty ,\\&\quad \text { and } 0<q\le \infty \end{aligned}$$

with \(p_0<\infty \) if \(A=F\), since \(W^{s_1}_p(\Omega )\) can be identified with \(F^{s_1}_{p,2}(\Omega )\) (if \(s_1\in \mathbb {N}\)) or \(F^{s_1}_{p,p}(\Omega )\) (if \(0<s_1 \notin \mathbb {N}\)).

Complex Interpolation

For some open set \(\Omega \) let \(X(\Omega )\) and \(Y(\Omega )\) denote quasi-normed spaces of complex-valued functions or distributions on \(\Omega \). Then, under certain conditions, the (extended) complex interpolation method is applicable and yields further quasi-normed spaces of functions on \(\Omega \). Besides other useful properties these spaces, usually denoted by \([X(\Omega ),Y(\Omega )]_\theta \), \(\theta \in (0,1)\), satisfy

$$\begin{aligned} X(\Omega )\cap Y(\Omega ) \hookrightarrow [X(\Omega ),Y(\Omega )]_\theta \hookrightarrow X(\Omega )+ Y(\Omega ). \end{aligned}$$

Thus, in particular, any set \(S(\Omega )\subset X(\Omega )\cap Y(\Omega )\) is also contained in \([X(\Omega ),Y(\Omega )]_\theta \) for all \(\theta \in (0,1)\). For details we refer to Bergh and Löfström [2] and Kalton et al. [24].

It turns out that the scales of Besov and Triebel–Lizorkin spaces \(A^s_{p,q}(\Omega )\) on bounded Lipschitz domains behave well w.r.t. this method:

Proposition A.4

(Kalton et al. [24, Theorem 9.4]) For \(d\in \mathbb {N}\) let \(\Omega \subseteq \mathbb {R}^d\) denote a bounded Lipschitz domain and assume \(\theta \in (0,1)\). Moreover, let \(A\in \{B,F\}\), as well as \(s,s_0,s_1\in \mathbb {R}\), and \(0<p,p_0,p_1,q,q_0,q_1\le \infty \) (with \(p_0,p_1<\infty \) for \(A=F\)), and \(\min \{q_0,q_1\}<\infty \). Then

$$\begin{aligned} s=(1-\theta )\,s_0 + \theta \, s_1, \qquad \frac{1}{p}=\frac{1-\theta }{p_0}+\frac{\theta }{p_1}, \qquad \text {and}\qquad \frac{1}{q}=\frac{1-\theta }{q_0}+\frac{\theta }{q_1} \end{aligned}$$

implies

$$\begin{aligned} \left[ A^{s_0}_{p_0,q_0}(\Omega ), A^{s_1}_{p_1,q_1}(\Omega ) \right] _{\theta } = A^{s}_{p,q}(\Omega ) \end{aligned}$$

in the sense of equivalent quasi-norms.

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Cioica-Licht, P.A., Weimar, M. On the Limit Regularity in Sobolev and Besov Scales Related to Approximation Theory. J Fourier Anal Appl 26, 10 (2020) doi:10.1007/s00041-019-09707-8

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Keywords

  • Non-linear approximation
  • Adaptive methods
  • Besov space
  • Triebel–Lizorkin space
  • Regularity of solutions
  • Poisson equation

Mathematics Subject Classification

  • 35B35
  • 35J92
  • 41A25
  • 46E35
  • 65M99