Integral comparisons of nonnegative positive definite functions on LCA groups

Abstract

In this paper we investigate the following questions. Let $$\mu , \nu $$ μ , ν be two regular Borel measures of finite total variation. When do we have a constant C satisfying $$\begin{aligned} \int f d\nu \le C \int f d\mu \end{aligned}$$ ∫ f d ν ≤ C ∫ f d μ 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 $$\mu , \nu $$ μ , ν 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.