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

TL;DR

This paper establishes conditions for the existence of solutions to a parabolic PDE involving Ornstein-Uhlenbeck operators perturbed by multipolar inverse square potentials, highlighting optimal constants and nonexistence results.

Contribution

It provides necessary and sufficient conditions for solutions and demonstrates the optimality of constants in the context of multipolar inverse square potentials.

Findings

Conditions for existence of weak solutions are derived.

Optimality of constants in inequalities is established.

Nonexistence of positive exponentially bounded solutions is proved.

Abstract

We give necessary and sufficient conditions for the existence of weak solutions of a parabolic problem corresponding to the Kolmogorov operators perturbed by a multipolar inverse square potential with respect to the Gaussian probability measure which is the unique invariant measure for Ornstein-Uhlenbeck type operators. We state the optimality of the constant and, then, the nonexistence of positive exponentially bounded solutions to the parabolic problem.