On Choquet integrals and Poincar\'e-Sobolev inequalities

TL;DR

This paper establishes new Poincaré-Sobolev inequalities in the context of Choquet integrals with respect to Hausdorff content on bounded John domains, extending classical inequalities to this measure setting.

Contribution

It proves novel Poincaré-Sobolev inequalities involving Choquet integrals and Hausdorff content, broadening the scope of integral inequalities in geometric measure theory.

Findings

$(rac{ ext{delta} p}{ ext{delta} - p}, p)$-Poincaré-Sobolev inequalities hold for certain $p$

$(p,p)$-Poincaré inequality valid for all $p > ext{delta}/n$

Results extend classical inequalities to Hausdorff content measure setting

Abstract

We consider integral inequalities in the sense of Choquet with respect to the Hausdorff content $\mathcal{H}_\infty^{\delta}$. In particular, if $\Omega$ is a bounded John domain in $\mathbb{R}^n$, $n\geq 2$, and $0 <\delta \le n$, we prove that the corresponding $(\delta p/(\delta -p),p)$-Poincar\'e-Sobolev inequalities hold for all continuously differentiable functions defined on $\Omega$ whenever $\delta /n < p < \delta$. We prove also that the $(p,p)$-Poincar\'e inequality is valid for all $p>\delta /n$.