Hardy-type inequalities with mixed weights

We discuss the best constant $m_{p,a,{b}}(\mathcal C)$ in inequalities of the type
$$\begin{equation*} m_{p,a,{b}}(\mathcal C)\int\limits_{\mathcal C}|y|^{a}|z|^{-{{b}}-p}|u|^p~\!dz\le \int\limits_{\mathcal C}|y|^{a}|z|^{-{{b}}}|\nabla u|^p~\!dz~, \quad u\in C^\infty_c({\mathcal C})~\!, \end{equation*}$$
where ${\mathcal C}\subseteq \mathbb R^d$ is a cone, $p>1$, and $a, {b}\in\mathbb R$. Here $z=(x,y)$ is the variable in $\mathbb R^d\equiv\mathbb R^{d-k}\times\mathbb R^k$.
The talk is based on the joint work with G. Cora and R. Musina (Italy).