Weighted Hardy inequalities and Ornstein-Uhlenbeck type operators perturbed by multipolar inverse square potentials

In this paper our main results are the multipolar weighted Hardy inequality c Sigma(n)(i=1) integral phi(2)/vertical bar x-a(i)vertical bar(2) d mu <= integral(RN) vertical bar del phi vertical bar(2)d mu+K integral(RN) phi(2)d mu, c <= c(o), where the functions phi belong to a weighted Sobolev space H-mu(1), and the proof of the optimality of the constant c(o) = c(o)(N) := (N-2/2)(2). The Gaussian probability measure d mu is the unique invariant measure for Ornstein-Uhlenbeck type operators. This estimate allows us to get necessary and sufficient conditions for the existence of positive solutions to a parabolic problem corresponding to the Kolmogorov operators defined on smooth functions and perturbed by a multipolar inverse square potential Lu+Vu = (Delta u+del mu/mu del u) + Sigma(n)(i=1) c/vertical bar x-a(i)vertical bar(2)u, x is an element of R-N, c > 0, a(1),..., a(n) is an element of R-N. (C) 2018 Elsevier Inc. All rights reserved.