Embeddings in spaces of Lipschitz type, entropy and approximation numbers, and applications.

Preprint series: 98-13, Analysis

The paper is published: J. Approx. Theory, 104 (2), 226-271, 2000.

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
47B06 Riesz operators; eigenvalue distributions; approximation numbers, $s$-numbers, Kolmogorov numbers, entropy numbers, etc. of operators

Abstract: We consider the embeddings of certain Besov and Triebel-Lizorkin spaces in spaces of Lipschitz type.
The prototype of such embeddings arises from the {\sc Br\'ezis-Wainger} result about the `almost' Lipschitz continuity of elements of the Sobolev spaces $H^{1+n/p}_p(R^n)$ when $1<p<\infty$.
Two-sided estimates are obtained for the entropy and approximation numbers of a variety of related embeddings.
The results are applied to give bounds for the eigenvalues of certain pseudo-differential operators and to provide information about the mapping properties of these operators.

Keywords: Lipschitz spaces, limiting embeddings, entropy numbers, approximation numbers, eigenvalues