Multipolar potentials and weighted Hardy inequalities

Abstract

\begin{abstract} In this paper we state the following weighted Hardy type inequality for any functions $φ$ in a weighted Sobolev space and for weight functions $μ$ of a quite general type \begin{equation*} c_{N,μ} \int_{\R^N}V\,φ^2μ(x)dx\le \int_{\R^N}|\nabla φ|^2μ(x)dx +C_μ\int_{\R^N}W φ^2μ(x)dx, \end{equation*} where $V$ is a multipolar potential and $W$ is a bounded function from above depending on $μ$. 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 $c_{N,μ}$ in the estimate is optimal by building a suitable sequence of functions. \end{abstract}