A Trace Inequality for Solenoidal Charges

Abstract

We prove that for α ∈ (d − 1,d), one has the trace inequality∫ℝd|IαF|dν≤C|F|(ℝd)∥ν∥Md−α(ℝd)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\int}_{\\mathbb{R}^{d}} |I_{\\alpha} F| d\ u \\leq C |F|(\\mathbb{R}^{d})\\|\ u\\|_{\\mathcal{M}^{d-\\alpha}(\\mathbb{R}^{d})} $$\\end{document} for all solenoidal vector measures F, i.e., Fin M_{b}(mathbb {R}^{d};mathbb {R}^{d}) and divF = 0. Here Iα denotes the Riesz potential of order α and mathcal M^{d-alpha }(mathbb {R}^{d}) the Morrey space of (d − α)-dimensional measures on mathbb {R}^{d}.