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Spaces of Lipschitz type, embeddings and entropy numbers. (English) Zbl 0932.46026

The aim of the paper is twofold. First limiting embeddings of Besov spaces \(B^{1+{n\over p}}_{pq}(\mathbb{R}^n)\) and \(F^{1+{n\over p}}_{pq}(\mathbb{R}^n)\), with the fractional Sobolev spaces \(H^{1+{n\over p}}_p(\mathbb{R}^n)\) as a subclass, in \(\text{Lip}^{1,-\alpha}_p(\mathbb{R}^n)\) normed by \[ \|f|L_\infty(\mathbb{R}^n)\|+ \sup{|f(x)- f(y)|\over|x-y||\log|x-y||^\alpha},\quad \alpha>0, \] are considered. Sharp exponents \(\alpha\) in dependence on \(p\) and \(q\) are obtained (Theorem 2.1). Secondly, restricted to bounded domains the related embeddings (with exception of limiting cases) are compact. It is the second aim of the paper to estimate this compactness in terms of entropy numbers.
Reviewer: H.Triebel (Jena)

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
47B06 Riesz operators; eigenvalue distributions; approximation numbers, \(s\)-numbers, Kolmogorov numbers, entropy numbers, etc. of operators
41A46 Approximation by arbitrary nonlinear expressions; widths and entropy
46E15 Banach spaces of continuous, differentiable or analytic functions
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