Embeddings and characterizations of Lipschitz spaces

TL;DR

This paper provides a comprehensive analysis of Lipschitz spaces, introducing new sharp embeddings, characterizations via Fourier and wavelet methods, and extending classical results for Besov, Sobolev, and Lebesgue spaces.

Contribution

It presents novel sharp embeddings and characterizations of Lipschitz spaces, extending classical results and providing new tools for their analysis.

Findings

Sharp embeddings between Lipschitz and Besov spaces

New characterizations of Lipschitz norms via Fourier and wavelets

Extended embeddings into Lebesgue and Lorentz-Zygmund spaces

Abstract

In this paper we give a thorough study of Lipschitz spaces. We obtain the following new results: (1) Sharp Jawerth-Franke-type embeddings between the Besov and Lipschitz spaces extending the classical results for Besov and Sobolev spaces; (2) Sharp embeddings between Lipschitz spaces with different parameters extending the Br\'ezis-Wainger result; (3) Characterizations for Lipschitz spaces norms via Fourier transforms and wavelets; (4) Sharp embeddings from Lipschitz spaces into Lebesgue/Lorentz-Zygmund spaces.