On Choquet integrals and Poincar\'e-Sobolev inequalities

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 $p>\delta /n$.