A note on Sobolev inequalities in the lower limit case

TL;DR

This paper investigates Poincare-Sobolev inequalities involving Hausdorff content, leading to a new Sobolev inequality for quasicontinuous functions and extending superlevel Sobolev inequalities.

Contribution

It introduces novel Poincare-Sobolev inequalities related to Hausdorff content and extends superlevel Sobolev inequalities to this context.

Findings

Derived a new Sobolev inequality for quasicontinuous functions.

Extended superlevel Sobolev inequalities using Hausdorff content.

Established inequalities for functions with gradient in Choquet $rac{ ext{delta}}{n}$-integrable class.

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 $\delta /n$-integrable with respect to the $\delta$-dimensional Hausdorff content, $n\geq 2$, $\delta\in (0,n]$. In particular, our results imply a new Sobolev inequality for quasicontinuous functions defined in the Sobolev space $W^{1,1}_0(\mathbb{R}^n)$. As an application we extend a recently introduced superlevel Sobolev inequality into a context of the Hausdorff content.