On Sobolev inequalities with Choquet integrals

Petteri Harjulehto, Ritva Hurri-Syrjänen

Published 2023-11-16Version 1

We consider integrals in the sense of Choquet with respect to the Hausdorff content ${\mathcal{H}}^{\delta}_\infty$ for continuously differentiable functions defined on open, connected sets $\Omega$ in ${\mathbb{R}}^n$, $n\geq 2$, $0<\delta\le n$. In particular, for these functions we prove Sobolev inequalities in the limiting case $p=\delta /n$ and the case $p>n$, here $p$ being the integrability exponent of the gradient of the given function. These results complement previous results for Poincar\'e-Sobolev and Trudinger inequalities.