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arXiv · 2405.10728

On dimension stable spaces of measures

Daniel Spector·Dmitriy Stolyarov
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Abstract

In this paper, we define spaces of measures $DS_\beta(\mathbb{R}^d)$ with dimensional stability $\beta \in (0,d)$. These spaces bridge between $M_b(\mathbb{R}^d)$, the space of finite Radon measures, and $DS_d(\mathbb{R}^d)= \mathrm{H}^1(\mathbb{R}^d)$, the real Hardy space. We show the spaces $DS_\beta(\mathbb{R}^d)$ support Sobolev inequalities for $\beta \in (0,d]$, while for any $\beta \in [0,d]$ we show that the lower Hausdorff dimension of an element of $DS_\beta(\mathbb{R}^d)$ is at least $\beta$.

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Daniel Spector, Dmitriy Stolyarov. 2024-05-17. On dimension stable spaces of measures. https://arxiv.org/abs/2405.10728

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