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arXiv · 2203.15623

On Choquet integrals and Poincar\'e-Sobolev inequalities

P. Harjulehto·R. Hurri-Syrjänen
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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 $p>\delta /n$.

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P. Harjulehto, R. Hurri-Syrjänen. 2022-03-29. On Choquet integrals and Poincar\'e-Sobolev inequalities. https://arxiv.org/abs/2203.15623

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