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

## TL;DR

This paper investigates the relationship between Sobolev and Besov regularity of functions on Lipschitz domains, providing bounds and sharpness results for regularity in approximation theory and applying them to PDEs like Poisson and Stokes.

## Contribution

It establishes a method to derive upper bounds for Besov regularity from Sobolev regularity using classical embeddings and complex interpolation, with applications to PDEs.

## Key findings

- Bounds for Besov regularity in terms of Sobolev regularity.
- Sharpness of known Besov regularity results for the Poisson equation.
- Application of regularity bounds to PDEs like the p-Poisson and Stokes problems.

## Abstract

We study the interrelation between the limit $L_p(\Omega)$-Sobolev regularity $\overline{s}_p$ of (classes of) functions on bounded Lipschitz domains $\Omega\subseteq\mathbb{R}^d$, $d\geq 2$, 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, 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.