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 $q>p\geq 1$ 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 $\mu.$