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 inequalityc∑i=1n∫RNφ2|x−ai|2dμ≤∫RN|∇φ|2dμ+K∫RNφ2dμ,c≤co, where the functions φ belong to a weighted Sobolev space Hμ1, and the proof of the optimality of the constant co=co(N):=(N−22)2. The Gaussian probability measure dμ 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 potentialLu+Vu=(Δu+∇μμ⋅∇u)+∑i=1nc|x−ai|2u,x∈RN,c>0, a1,…,an∈RN.