Spaces of Lipschitz type, embeddings and entropy numbers.

Preprint series: 98-12, Analysis

The paper is published: Dissertationes Math., Vol. 380, 1-43, 1999

MSC:
26A16 Lipschitz (Holder) classes
46E35 Sobolev spaces and other spaces of smooth'' functions, embedding theorems, trace theorems
41A46 Approximation by arbitrary nonlinear expressions; widths and entropy
46E15 Banach spaces of continuous, differentiable or analytic functions

Abstract: We establish the sharpness of the embedding of certain Besov and Triebel-Lizorkin spaces in spaces of Lipschitz type.
In particular, this proves the sharpness of the {\sc Br\'ezis-Wainger} result concerning the `almost' Lipschitz continuity of elements of the Sobolev space $H^{1+n/p}_p(R^n)$, where $1<p<\infty$.
Upper and lower estimates are obtained for the entropy numbers of related embeddings of Besov spaces on bounded domains.

Keywords: Lipschitz spaces, limiting embeddings, entropy numbers