On Choquet integrals and Poincaré-Sobolev inequalities

Abstract

We consider integral inequalities in the sense of Choquet with respect to the Hausdorff content H∞δ. In particular, if Ω is a bounded John domain in Rn, n≥2, and 0<δ≤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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