A trace inequality for solenoidal charges

Abstract

We prove that for $\alpha \in (d-1,d]$, one has the trace inequality \begin{align*} \int_{\mathbb{R}^d} |I_\alpha F| \;d\nu \leq C |F|(\mathbb{R}^d)\|\nu\|_{\mathcal{M}^{d-\alpha}(\mathbb{R}^d)} \end{align*} for all solenoidal vector measures $F$, i.e., $F\in M_b(\mathbb{R}^d,\mathbb{R}^d)$ and $\operatorname{div}F=0$. Here $I_\alpha$ denotes the Riesz potential of order $\alpha$ and $\mathcal M^{d-\alpha}(\mathbb{R}^d)$ the Morrey space of $(d-\alpha)$-dimensional measures on $\mathbb{R}^d$.