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arXiv · 2404.05222

Fractional Hardy inequalities and capacity density

Lizaveta Ihnatsyeva·Kaushik Mohanta·Antti V. Vähäkangas
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Abstract

We prove that a pointwise fractional Hardy inequality implies a fractional Hardy inequality, defined via a Gagliardo-type seminorm. The proof consists of two main parts. The first one is to characterize the pointwise fractional Hardy inequality in terms of a fractional capacity density condition. The second part is to show the deep open-endedness or self-improvement property of the fractional capacity density, which we accomplish in the setting of a complete geodesic space equipped with a doubling measure. These results are new already in the standard Euclidean setting.

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Lizaveta Ihnatsyeva, Kaushik Mohanta, Antti V. Vähäkangas. 2024-04-08. Fractional Hardy inequalities and capacity density. https://arxiv.org/abs/2404.05222

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