On dimension stable spaces of measures

Abstract

In this paper, we define spaces of measures $D{S}_{\hat{I}}^2({\mathbb{R}}^d)$ with dimensional stability ${\hat{I}}^2\in (0,d)$. These spaces bridge between $M_b({\mathbb{R}}^d)$, the space of finite Radon measures, and $D{S}_d({\mathbb{R}}^d)={\mathrm{H}}^1({\mathbb{R}}^d)$, the real Hardy space. We show the spaces $D{S}_{\hat{I}}^2({\mathbb{R}}^d)$ support Sobolev inequalities for ${\hat{I}}^2\in (0,d]$, while for any ${\hat{I}}^2\in [0,d]$ we show that the lower Hausdorff dimension of an element of $D{S}_{\hat{I}}^2({\mathbb{R}}^d)$ is at least ${\hat{I}}^2$.

30 pages