Sobolev embedding implies regularity of measure in metric measure spaces

Abstract

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

4 pages, Funding agency's name has been updated

References in corpus (1)

• A note on metric-measure spaces supporting Poincaré inequalities