A note on Sobolev inequalities in the lower limit case

Abstract

We study Poincare-Sobolev type inequalities for compactly supported smooth functions which are defined in the Euclidean $n$-space and whose absolute value of gradient are Choquet $\hat{I}´/n$-integrable with respect to the $\hat{I}´$-dimensional Hausdorff content, $n\ge 2$, $\hat{I}´\in (0,n]$. In particular, our results imply a new Sobolev inequality for quasicontinuous functions defined in the Sobolev space $W_0^{1,1}({\mathbb{R}}^n)$. As an application we extend a recently introduced superlevel Sobolev inequality into a context of the Hausdorff content.