**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*

**Upload:** 1999-01-21

**Update:** 2000-04-25

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