On Choquet integrals and Poincaré-Sobolev inequalities

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 ${\mathbb{R}}^n$, $n\ge 2$, and $0<\delta \le n$, we prove that the corresponding $(\delta p/(\delta -p),p)$-Poincaré-Sobolev inequalities hold for all continuously differentiable functions defined on $\Omega$ whenever $\delta /n<p<\delta$. We prove also that the $(p,p)$-Poincaré inequality is valid for all $p>\delta /n$.