The Modulus of Continuity of a Measure with Finite Energy

A general class of energy integrals in ℝn, including standard Riesz and Bessel α-energy integrals, is considered and it is shown that the modulus of continuity of positive measures for which such integrals are finite satisfy certain weighted integrability conditions. These results are deduced from an equivalent formulation of the finite energy condition in terms of related positive harmonic functions in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$R^{n+1}_{+}$\end{document}. It is also shown that some of the results obtained are sharp.