On more general Lipschitz spaces

Preprint series: 99-47, Analysis

The paper is published: Z. Anal. Anwendungen, 19(3), 781-799, 2000.

MSC:
26A16 Lipschitz (Holder) classes
46E35 Sobolev spaces and other spaces of smooth'' functions, embedding theorems, trace theorems
26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.), {For properties determined by Fourier coefficients, See 42A16; for those determined by approximation properties, See 41A25, 41A27}

Abstract: The present paper deals with (logarithmic) Lipschitz spaces of type $Lip^{(1,-\alpha)}_{p,q}$, $1\leq p\leq\infty$, $0<q\leq\infty$, $\alpha>1/q$.
We study their properties and derive some (sharp) embedding results. In that sense this paper can be regarded as some continuation and extension of some of our earlier papers, but there are also connections with some recent work of Triebel concerning Hardy inequalities and sharp embeddings.
Recall that the nowadays almost classical' forerunner of investigations of this type is the Br\'ezis-Wainger result about the almost' Lipschitz continuity of elements of the Sobolev spaces $H^{1+n/p}_p(R^n)$ when $1<p<\infty$.

Keywords: limiting embeddings, Lipschitz spaces, function