On Choquet integrals and Sobolev type inequalities

Abstract

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