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arXiv · 2106.05595

Hardy-Sobolev inequalities and weighted capacities in metric spaces

Lizaveta Ihnatsyeva·Juha Lehrbäck·Antti V. Vähäkangas
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Abstract

Let $\Omega$ be an open set in a metric measure space $X$. Our main result gives an equivalence between the validity of a weighted Hardy-Sobolev inequality in $\Omega$ and quasiadditivity of a weighted capacity with respect to Whitney covers of $\Omega$. Important ingredients in the proof include the use of a discrete convolution as a capacity test function and a Maz'ya type characterization of weighted Hardy-Sobolev inequalities.

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Lizaveta Ihnatsyeva, Juha Lehrbäck, Antti V. Vähäkangas. 2021-06-10. Hardy-Sobolev inequalities and weighted capacities in metric spaces. https://arxiv.org/abs/2106.05595

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