Abstract:
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).
Contact us: