On the limit regularity in Sobolev and Besov scales related to approximation theory

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 $B_{\tau ,\tau }^{\alpha }(\Omega )$, 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, Mayboroda, and Mitrea (Contemp. Math. 445). 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. Keywords: Non-linear approximation, adaptive methods, Besov space, Triebel-Lizorkin space, regularity of solutions, stationary Stokes equation, Poisson equation, $p$-Poisson equation, Lipschitz domain.

Dedicated to Prof. Dr. Stephan Dahlke on the occasion of his 60th birthday; 26 pages, 3 figures