Hardy and Hardy-Sobolev type inequalities involving mixed weights

We report about class of dilation-invariant inequalities for functions having compact support in cones $\mathcal C\subseteq\mathbb{R}^{d-k}\times\mathbb{R}^k$. The leading term has the form
$$ \int_\mathcal C\frac{|y|^a}{(|x|^2+|y|^2)^b}~\!|\nabla u|^p~\!dxdy~\!. $$
We include both classical spherical weights, initially examined by Il'in [Mat. Sb., 1961] and further discussed by Caffarelli-Kohn-Nirenberg [Compositio Math., 1984], as well as cylindrical weights, firstly investigated by Maz'ya in his monograph on Sobolev spaces.