Hardy-Sobolev inequalities involving mixed radially and cylindrically symmetric weights

We deal with weighted Hardy-Sobolev type inequalities for functions on ${\mathbb{R}}^d$, $d\ge 2$. The weights involved are anisotropic, given by products of powers of the distance to the origin and to a nontrivial subspace. We establish necessary and sufficient conditions for validity of these inequalities, and investigate the existence/nonexistence of extremal functions.