  We study Sobolev spaces with weights in the half-space
$\mathbb{R}^{N+1}_+=\{(x,y): x \in \mathbb{R}^N, y>0\}$, adapted to the
singular elliptic operators \begin{equation*} \mathcal L
=y^{\alpha_1}\Delta_{x} +y^{\alpha_2}\left(D_{yy}+\frac{c}{y}D_y
-\frac{b}{y^2}\right). \end{equation*}
