A Trace Inequality for Solenoidal Charges

Abstract We prove that for α ∈ ( d − 1, ), one has the trace inequality $$ {\int}_{\mathbb{R}^{d}} |I_{\alpha} F| d\nu \leq C |F|(\mathbb{R}^{d})\|\nu\|_{\mathcal{M}^{d-\alpha}(\mathbb{R}^{d})} ∫ ℝ d | I α F ν ≤ C ( ) ∥ M − all solenoidal vector measures F , i.e., $F\in M_{b}(\mathbb {R}^{d};\mathbb {R}^{d})$ ∈ b ; and div = 0. Here I denotes Riesz potential of order $\mathcal M^{d-\alpha }(\mathbb ℳ Morrey space )-dimensional on $\mathbb {R}^{d}$ .