A Trace Inequality for Solenoidal Charges

Abstract

AbstractWe prove that for α ∈ (d − 1,d), 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 | d ν ≤ C | F | ( ℝ d ) ∥ ν ∥ M d − α ( ℝ d ) for all solenoidal vector measures F, i.e., $F\in M_{b}(\mathbb {R}^{d};\mathbb {R}^{d})$ F ∈ M b ( ℝ d ; ℝ d ) and divF = 0. Here Iα denotes the Riesz potential of order α and $\mathcal M^{d-\alpha }(\mathbb {R}^{d})$ ℳ d − α ( ℝ d ) the Morrey space of (d − α)-dimensional measures on $\mathbb {R}^{d}$ ℝ d .