Weighted Hardy's inequality in a limiting case and the perturbed Kolmogorov equation

TL;DR

This paper establishes a weighted Hardy inequality in a limiting case, proves the optimality of the constant, and applies these results to analyze the existence of solutions to a perturbed Kolmogorov equation with critical singular potential.

Contribution

It introduces a weighted Hardy inequality in a limiting case for the Kolmogorov operator and explores its implications for the existence of solutions to related parabolic problems.

Findings

Weighted Hardy inequality with optimal constant proved.

Existence and nonexistence results for solutions to perturbed Kolmogorov equations.

Extension of previous non-critical case results to the critical singular potential case.

Abstract

In this paper, we show a weighted Hardy inequality in a limiting case for functions in weighted Sobolev spaces with respect to an invariant measure. We also prove that the constant in the left-hand side of the inequality is optimal. As applications, we establish the existence and nonexistence of positive exponentially bounded weak solutions to a parabolic problem involving the Ornstein-Uhlenbeck operator perturbed by a critical singular potential in two dimensional case, according to the size of the coefficient of the critical potential. These results can be considered as counterparts in the limiting case of results which established in \cite{GGR(AA)} \cite{Hauer-Rhandi} in the non-critical cases, and are also considered as extensions of a result in \cite{Cabre-Martel} to the Kolmogorov operator case perturbed by a critical singular potential.