# Weighted Hardy's inequalities and Kolmogorov-type operators

## TL;DR

This paper establishes general conditions for weighted Hardy inequalities with respect to probability measures, determines the optimal constants, and explores their connection to Kolmogorov equations with singular potentials, including existence of solutions.

## Contribution

It provides new criteria for weighted Hardy inequalities involving probability measures and links these to Kolmogorov operators with singular potentials, including optimal constant determination.

## Key findings

- Derived conditions for weighted Hardy inequalities with probability measures.
- Identified the optimal constant for the inequality.
- Connected inequalities to the existence of solutions for perturbed Kolmogorov equations.

## Abstract

We give general conditions to state the weighted Hardy inequality \[ c\int_{\mathbb{R}^N}\frac{\varphi^2} {|x|^2}d\mu\leq\int_{\mathbb{R}^N}|\nabla \varphi |^2 d\mu+C\int_{\mathbb{R}^N} \varphi^2d\mu,\quad \varphi\in C_c^{\infty}(\mathbb{R}^N),\,c\leq c_{0,\mu}, \] with respect to a probability measure $d\mu$. Moreover, the optimality of the constant $c_{0,\mu}$ is given. The inequality is related to the following Kolmogorov equation perturbed by a singular potential \[ Lu+Vu=\left(\Delta u+\frac{\nabla \mu}{\mu}\cdot \nabla u\right)+\frac{c}{|x|^2}u \] for which the existence of positive solutions to the corresponding parabolic problem can be investigated. The hypotheses on $d\mu$ allow the drift term to be of type $\frac{\nabla \mu}{\mu}= -|x|^{m-2}x$ with $m> 0$.