Multipolar potentials and weighted Hardy inequalities

TL;DR

This paper establishes a weighted Hardy inequality involving multipolar potentials and general weight functions, introducing a new method with vector functions and integral identities, and proves the optimality of the constant involved.

Contribution

It presents a new weighted Hardy inequality for multipolar potentials with a general class of weights, using a novel approach involving vector functions and integral identities, and confirms the optimality of the inequality's constant.

Findings

Derived a weighted Hardy inequality for multipolar potentials.

Introduced a new method using vector functions and integral identities.

Proved the optimality of the inequality constant.

Abstract

\begin{abstract} In this paper we state the following weighted Hardy type inequality for any functions $\varphi$ in a weighted Sobolev space and for weight functions $\mu$ of a quite general type \begin{equation*} c_{N,\mu} \int_{\R^N}V\,\varphi^2\mu(x)dx\le \int_{\R^N}|\nabla \varphi|^2\mu(x)dx +C_\mu \int_{\R^N}W \varphi^2\mu(x)dx, \end{equation*} where $V$ is a multipolar potential and $W$ is a bounded function from above depending on $\mu$. The method to get the result is based on the introduction of a suitable vector value function and on an integral identity that we state in the paper. We prove that the constant $c_{N,\mu}$ in the estimate is optimal by building a suitable sequence of functions. \end{abstract}