On More General Lipschitz Spaces

Abstract

The present paper deals with (logarithmic) Lipschitz spaces of type ${\mathrm{Lip}}_{p,q}^{(1,–\alpha )}(1\le p\le \infty ,0<q\le \infty ,\alpha >\frac{1}{q})$. We study their properties and derive some (sharp) embedding results. In that sense this paper can be regarded as some continuation and extension of our papers [8, 9], 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 Brézis-Wainger result [6] about the 'almost' Lipschitz continuity of elements of the Sobolev spaces $H_p^{1+\frac{n}{p}}({\mathbb{R}}^n)$ when $1<p<\infty$.