Integral comparisons of nonnegative positive definite functions on LCA groups

Abstract

In this paper we investigate the following questions. Let mu , nu be two regular Borel measures of finite total variation. When do we have a constant C satisfying ∫fdν≤C∫fdμ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\int f d\ u \\le C \\int f d\\mu \\end{aligned}$$\\end{document}whenever f is a continuous nonnegative positive definite function? How the admissible constants C can be characterized, and what is their optimal value? We first discuss the problem in locally compact abelian groups. Then we make further specializations when the Borel measures mu , nu are both either purely atomic or absolutely continuous with respect to a reference Haar measure. In addition, we prove a duality conjecture posed in our former paper.