Integral comparisons of nonnegative positive definite functions on LCA groups

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 integral fd nu <= C integral fd 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.