On Hausdorff measure and an inequality due to Maz'ya

TL;DR

This paper provides an elementary proof of Maz'ya's inequality, explores its connection with Hausdorff measure, and discusses its relation to isoperimetric and Sobolev inequalities.

Contribution

It introduces an elementary proof of Maz'ya's inequality and examines its links with Hausdorff measure, isoperimetric, and Sobolev inequalities.

Findings

Elementary proof of Maz'ya's inequality

Approximation property for Hausdorff measure established

Relations between Maz'ya's inequality, isoperimetric, and Sobolev inequalities discussed

Abstract

We give an "elementary" proof of an inequality due to Maz'ya. As a prerequisite we prove an approximation property for the Hausdorff measure. We also comment on the relations between Maz'ya's inequality, the isoperimetric inequality and the Sobolev inequality.