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arXiv · 2604.13732

A note on Sobolev inequalities in the lower limit case

Petteri Harjulehto·Ritva Hurri-Syrjänen
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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.

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Petteri Harjulehto, Ritva Hurri-Syrjänen. 2026-04-15. A note on Sobolev inequalities in the lower limit case. https://arxiv.org/abs/2604.13732

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