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arXiv · 2203.15623

On Choquet integrals and Poincaré-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 $\mathcal{H}_\infty^δ$. In particular, if $Ω$ is a bounded John domain in $\mathbb{R}^n$, $n\geq 2$, and $0 <δ\le n$, we prove that the corresponding $(δp/(δ-p),p)$-Poincaré-Sobolev inequalities hold for all continuously differentiable functions defined on $Ω$ whenever $δ/n < p < δ$. We prove also that the $(p,p)$-Poincaré inequality is valid for all $p>δ/n$.

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P. Harjulehto, R. Hurri-Syrjänen. 2022-12-22. On Choquet integrals and Poincaré-Sobolev inequalities. https://arxiv.org/abs/2203.15623

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