Multipolar potentials and weighted Hardy inequalities

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 $\mu$. 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.