Multipolar potentials and weighted Hardy inequalities

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