Weighted norm inequalities in a bounded domain by the sparse domination method

TL;DR

This paper establishes local two-weight Poincaré inequalities in bounded domains using sparse domination, extending harmonic analysis techniques to PDEs and weighted inequalities, with applications to supersolutions of the p-Laplace equation.

Contribution

It introduces a novel approach combining sparse domination with local-to-global techniques to prove weighted Poincaré inequalities in bounded domains.

Findings

Proves a local two-weight Poincaré inequality for cubes using sparse domination.

Establishes a local-to-global Poincaré inequality in domains satisfying a Boman chain condition.

Shows certain supersolutions support p-Poincaré inequalities, implying p-admissibility.

Abstract

We prove a local two-weight Poincar\'e inequality for cubes using the sparse domination method that has been influential in harmonic analysis. The proof involves a localized version of the Fefferman--Stein inequality for the sharp maximal function. By establishing a local-to-global result in a bounded domain satisfying a Boman chain condition, we show a two-weight $p$-Poincar\'e inequality in such domains. As an application we show that certain nonnegative supersolutions of the $p$-Laplace equation and distance weights are $p$-admissible in a bounded domain, in the sense that they support versions of the $p$-Poincar\'e inequality.