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_αF| \;dν\leq C |F|(\mathbb{R}^d)\|ν\|_{\mathcal{M}^{d-α}(\mathbb{R}^d)} \end{align*} for all solenoidal vector measures $F$, i.e., $F\in M_b({\mathbb{R}}^d,{\mathbb{R}}^d)$ and $\mathrm{div}F=0$. 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 ${\mathbb{R}}^d$.

arXiv comment8 pages

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Cited by in corpus (1)

• Failure of the local chain rule for the fractional variation