Weighted Sobolev Inequalities via the Meyers--Ziemer Framework: Measures, Isoperimetric Inequalities, and Endpoint Estimates

TL;DR

This paper develops a new global endpoint Sobolev inequality for measures, extending classical results with maximal functions, and explores its implications for weighted variation, capacity, isoperimetric inequalities, and fractional operators.

Contribution

It introduces a novel endpoint Sobolev inequality within the Meyers-Ziemer framework, incorporating maximal functions and extending to various inequalities and fractional operators.

Findings

Establishment of a new endpoint Sobolev inequality for measures.

Extension to weighted bounded variation functions and related inequalities.

Identification of a sharp bumped maximal function for non-endpoint cases.

Abstract

We establish a new global endpoint Sobolev inequality for measures that extends the classical theorem of Meyers-Ziemer by placing a maximal function on the right-hand side. This result has several significant consequences. It extends naturally to functions of weighted bounded variation and yields corresponding capacity and isoperimetric inequalities. The inequality is also closely connected to endpoint estimates for fractional operators, including bounds for fractional maximal functions and Hardy space endpoint estimates for the Riesz potential. Our main inequality yields a family of endpoint inequalities, characterized in terms of subrepresentation formulas, Lorentz space improvements, and isoperimetric inequalities for measures and bounded open sets. When one moves away from the endpoint to $p>1$, the analogous inequalities no longer hold in general; however, we identify a sharp bumped maximal function for which the corresponding non-endpoint inequality is valid. Finally, we show that this framework yields new $(p,p)$ two-weight Sobolev inequalities.