On Choquet integrals and Poincar?-Sobolev inequalities

We consider integral inequalities in the sense of Choquet with respect to the Hausdorff content 7-1 delta infinity. In particular, if Omega is a bounded John domain in Rn, n > 2, and 0 < delta < n, we prove that the corresponding (delta p/(delta - p), p)-Poincare-Sobolev inequalities hold for all continuously differentiable functions defined on Omega whenever delta/n < p < delta. We prove also that the (p, p)-Poincare inequality is valid for all p > delta/n.(c) 2023 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/).