Non-collapsing condition and Sobolev embeddings for Haj{\l}asz-Besov spaces

TL;DR

This paper investigates the connection between Sobolev embeddings for Haj{}asz-Besov spaces on metric measure spaces and the non-collapsing measure condition, providing new embedding results involving rearrangement invariant quasi-norms.

Contribution

It establishes the relationship between Sobolev embeddings and the non-collapsing measure condition, and introduces new embeddings for Haj{}asz-Besov spaces with specific smoothness moduli.

Findings

Sobolev embeddings are characterized by the non-collapsing measure condition.

New embedding results for Haj{}asz-Besov spaces with rearrangement invariant quasi-norms.

The measure's non-collapsing condition is essential for certain Sobolev embeddings.

Abstract

In this paper we will focus on understanding the relation between Sobolev embedding theorems for Haj{\l}asz-Besov spaces defined on a doubling metric measure space $(\Omega,d,\mu)$ and the non-collapsing condition of the measure, i.e. \[ \inf_{x\in\Omega}\mu(B(x,1))>0. \] We will also obtain embedding results for Haj{\l}asz-Besov spaces whose modulus of smoothness is generated by a rearrangement invariant quasi-norm.