On Choquet integrals and Sobolev type inequalities

Abstract

We consider integrals in the sense of Choquet with respect to the $δ$-dimensional Hausdorff content for continuously differentiable functions defined on open, connected sets in the Euclidean $n$-space, $n\geq 2$, $0<δ\le n$. In particular, for these functions we prove Sobolev inequalities in the limiting case $p=δ/n$ and in the case $p>δ$, here $p$ is the integrability exponent of the absolute value of the gradient of any given function. The results complement previously known Poincaré-Sobolev and Morrey inequalities.