Hardy–Sobolev inequalities involving mixed radially and cylindrically symmetric weights

We deal with weighted Hardy–Sobolev type inequalities for functions on [Formula: see text], [Formula: see text]. 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.