  We study elliptic and parabolic problems governed by 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), \qquad\alpha_1,
\alpha_2 \in\mathbb R \end{equation*} in the half-space $\mathbb
R^{N+1}_+=\{(x,y): x \in \mathbb R^N, y>0\}$.
