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

Abstract In this paper our main results are the multipolar weighted Hardy inequality c ∑ i = 1 n ∫ R N φ 2 | x − a i | 2 d μ ≤ ∫ R N | ∇ φ | 2 d μ + K ∫ R N φ 2 d μ , c ≤ c o , where the functions φ belong to a weighted Sobolev space H μ 1 , and the proof of the optimality of the constant c o = c o (N) : = (N − 2 2) 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 potential L u + V u = (Δ u + ∇ μ μ ⋅ ∇ u) + ∑ i = 1 n c | x − a i | 2 u , x ∈ R N , c > 0 , a 1 , … , a n ∈ R N. [ABSTRACT FROM AUTHOR]