Sobolev embedding implies regularity of measure in metric measure spaces

TL;DR

This paper demonstrates that Sobolev embeddings in metric measure spaces imply a lower bound on measure growth, establishing regularity without requiring the doubling condition on the measure.

Contribution

It proves that Sobolev embedding implies measure regularity in metric spaces without assuming doubling conditions, extending previous results.

Findings

Measure satisfies a polynomial growth condition

Sobolev embedding implies measure regularity

Results hold without doubling measure assumption

Abstract

We prove that if the Sobolev embedding $M^{1,p}(X)\hookrightarrow L^q(X)$ holds for some $q>p\geq 1$ in a metric measure space $(X,d,\mu),$ then a constant $C$ exists such that $\mu(B(x,r))\geq Cr^n$ for all $x\in X$ and all $0<r\leq 1,$ where $\frac{1}{p}-\frac{1}{q}=\frac{1}{n}.$ This was proved in \cite{Gor17} assuming a doubling condition on the measure $\mu.$