Multipolar potentials and weighted Hardy inequalities

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{document}$ \begin{align*} c_{N, \mu} \int_{ \mathbb{R}^N}V\, \varphi^2\mu(x)dx\le \int_{ \mathbb{R}^N}|\nabla \varphi|^2\mu(x)dx +C_\mu \int_{ \mathbb{R}^N}W \varphi^2\mu(x)dx, \end{align*} $\end{document} where $ V $ is a multipolar potential and $ W $ is a bounded function from above depending on $ \mu $. Our method is based on introducing a suitable vector-valued function and 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.