Multipolar Hardy inequalities and mutual interaction of the poles

TL;DR

This paper establishes a weighted Hardy inequality involving multiple poles in Euclidean space, analyzing the optimal constant and the influence of pole proximity, with implications for weighted Sobolev spaces.

Contribution

It introduces a multipolar Hardy inequality with optimal constants and explores how the interaction between poles affects the inequality's validity.

Findings

Derived the weighted Hardy inequality with multiple poles.

Identified the optimal constant depending on the weight and dimension.

Analyzed the effect of pole proximity on the inequality's constants.

Abstract

In this paper we state the weighted Hardy inequality \begin{equation*} c\int_{{\mathbb R}^N}\sum_{i=1}^n \frac{\varphi^2 }{|x-a_i|^2}\, \mu(x)dx\le \int_{{\mathbb R}^N} |\nabla\varphi|^2 \, \mu(x)dx +k \int_{\mathbb{R}^N}\varphi^2 \, \mu(x)dx \end{equation*} for any $ \varphi$ in a weighted Sobolev spaces, with $c\in]0,c_o[$ where $c_o=c_o(N,\mu)$ is the optimal constant, $a_1,\dots,a_n\in \mathbb{R}^N$, $k$ is a constant depending on $\mu$. We show the relation between $c$ and the closeness to the single pole. To this aim we analyze in detail the difficulties to be overcome to get the inequality.