A unified approach to degenerate problems in the half-space

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 $\mathbb R^{N+1}_+=\{(x,y): x \in \mathbb R^N, y>0\}$.