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We consider integrals in the sense of Choquet with respect to the$\delta$-dimensional Hausdorff content for continuously differentiablefunctions defined on open, connected sets in the Euclidean $n$-space, $n\geq2$, $0<\delta\le n$. In particular, for these functions we prove Sobolevinequalities 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 ofany given function. The results complement previously known Poincar\'e-Sobolevand Morrey inequalities.

On Choquet integrals and Sobolev type inequalities