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

We study the interrelation between the limit $L_p(Ω)$-Sobolev regularity $\overline{s}_p$ of (classes of) functions on bounded Lipschitz domains $Ω\subseteq\mathbb{R}^d$, $d\geq 2$, and the limit regularity $\overlineα_p$ within the corresponding adaptivity scale of Besov spaces $B^α_{τ,τ}(Ω)$, where $1/τ=α/d+1/p$ and $α>0$ ($p>1$ 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 $\overlineα_p$ 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.