# Weighted multipolar Hardy inequalities and evolution problems with Kolmogorov operators perturbed by singular potentials

## TL;DR

This paper establishes weighted multipolar Hardy inequalities for Kolmogorov operators with singular potentials, providing conditions for the existence and nonexistence of positive solutions to related evolution problems.

## Contribution

It introduces new weighted Hardy inequalities for multipolar inverse square potentials and analyzes their implications for evolution equations with Kolmogorov operators.

## Key findings

- Weighted Hardy inequalities with optimal constants
- Criteria for existence of positive bounded solutions
- Nonexistence results based on inequality optimality

## Abstract

The main results in the paper are the weighted multipolar Hardy inequalities \begin{equation*} c\int_{\R^N}\sum_{i=1}^n\frac{u^2}{|x-a_i|^2}\,d\mu \leq\int_{\R^N}|\nabla u |^2d\mu+ K\int_{\R^N} u^2d\mu, \end{equation*} in $\R^N$ for any $u$ in a suitable weighted Sobolev space, with $0<c\le c_{o,\mu}$, $a_1,\dots,a_n\in \R^N$, $K$ constant. The weight functions $\mu$ are of a quite general type. The paper fits in the framework of the study of Kolmogorov operators \begin{equation*} Lu=\Delta u+\frac{\nabla \mu}{\mu}\cdot\nabla u, \end{equation*} perturbed by multipolar inverse square potentials, and of the related evolution problems. The necessary and sufficient conditions for the existence of positive exponentially bounded in time solutions to the associated initial value problem are based on weighted Hardy inequalities. The optimality of the constant constant $c_{o,\mu}$ allow us to state the nonexistence of positive solutions. We follow the Cabr\'e-Martel's approach. To this aim we state some properties of the operator $L$, of its corresponding $C_0$-semigroup and density results.