On embedding of Besov spaces of zero smoothness into Lorentz spaces

TL;DR

This paper investigates the embedding properties of zero smoothness Besov spaces into Lorentz spaces, establishing that such embeddings only occur when the parameters are equal, thus answering a question posed by Besov.

Contribution

The paper provides a negative result on the embedding of zero smoothness Besov spaces into Lorentz spaces, clarifying the conditions under which these embeddings are possible.

Findings

Embedding occurs only when p=q.

Answer to Besov's posed question.

Clarifies the structure of zero smoothness Besov spaces.

Abstract

We show that the zero smoothness Besov space $B_{p,q}^{0,1}$ does not embed into the Lorentz space $L_{p,q}$ unless $p=q$; here $p,q\in (1,\infty)$. This answers negatively a question proposed by O. V. Besov.