Integral comparisons of nonnegative positive definite functions on LCA groups

TL;DR

This paper studies inequalities involving nonnegative positive definite functions on locally compact abelian groups, characterizing the best constants for measure comparisons and proving a related duality conjecture.

Contribution

It provides a comprehensive analysis of measure inequalities for positive definite functions on LCA groups, including characterizations of optimal constants and a proof of a duality conjecture.

Findings

Characterization of constants C for measure inequalities

Results for atomic and absolutely continuous measures

Proof of a duality conjecture from previous work

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 $$\int f d\nu \le C \int f d\mu$$ 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.