A note on Sobolev inequalities in the lower limit case

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 $\delta \mathrm{/}n$-integrable with respect to the $\delta$-dimensional Hausdorff content, $n\ge 2$, $\delta \in (0,n]$. 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.