Sobolev embedding for $M^{1,p}$ spaces is equivalent to a lower bound of the measure

TL;DR

This paper proves that the Sobolev embedding theorem for $M^{1,p}$ spaces on metric-measure spaces is equivalent to having a lower measure bound, establishing a fundamental geometric-analytic connection.

Contribution

It demonstrates the equivalence between Sobolev embeddings and measure lower bounds for $M^{1,p}$ spaces, extending known implications to a full equivalence.

Findings

Sobolev embeddings are equivalent to measure lower bounds

Lower measure bounds imply Sobolev embeddings

Sobolev embeddings characterize measure growth conditions

Abstract

It has been known since 1996 that a lower bound for the measure, $\mu(B(x,r))\geq br^s$, implies Sobolev embedding theorems for Sobolev spaces $M^{1,p}$ defined on metric-measure spaces. We prove that, in fact Sobolev embeddings for $M^{1,p}$ spaces are equivalent to the lower bound of the measure.