On dimension stable spaces of measures

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 $\beta$.