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arXiv · 2108.09077

The Hajłasz capacity density condition is self-improving

Javier Canto·Antti V. Vähäkangas
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Abstract

We prove a self-improvement property of a capacity density condition for a nonlocal Hajlasz gradient in complete geodesic spaces. The proof relates the capacity density condition with boundary Poincaré inequalities, adapts Keith-Zhong techniques for establishing local Hardy inequalities and applies Koskela-Zhong arguments for proving self-improvement properties of local Hardy inequalities. This leads to a characterization of the Hajlasz capacity density condition in terms of a strict upper bound on the upper Assouad codimension of the underlying set, which shows the self-improvement property of the Hajlasz capacity density condition.

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Javier Canto, Antti V. Vähäkangas. 2021-11-23. The Hajłasz capacity density condition is self-improving. https://arxiv.org/abs/2108.09077

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