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

摘要

We study the interrelation between the limit Lp(Omega)-Sobolev regularity sp of (classes of) functions on bounded Lipschitz domains Omega subset of Rd, d >= 2, and the limit regularity 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 (alpha) over barp in terms of (S) over barp 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.