A note on self improvement of Poincar\'e-Sobolev inequalities via Garsia-Rodemich spaces

TL;DR

This paper demonstrates how Garsia-Rodemich spaces can be used to show self-improving properties of Poincaré-Sobolev inequalities, broadening understanding of their behavior in various contexts.

Contribution

It introduces a novel approach using Garsia-Rodemich conditions to analyze the self-improvement of Poincaré-Sobolev inequalities.

Findings

Garsia-Rodemich conditions characterize weak type inequalities.

Poincaré-Sobolev inequalities exhibit self-improvement properties.

The approach applies in very general contexts.

Abstract

We use the characterization of weak type inequalities via Garsia-Rodemich conditions to show self improving properties of Poincar\'e-Sobolev inequalities in a very general context.