  We study elliptic and parabolic problems governed by the singular elliptic
operators $$ \mathcal L=y^{\alpha_1}\mbox{Tr
}\left(QD^2_x\right)+2y^{\frac{\alpha_1+\alpha_2}{2}}q\cdot \nabla_xD_y+\gamma
y^{\alpha_2}
D_{yy}+y^{\frac{\alpha_1+\alpha_2}{2}-1}\left(d,\nabla_x\right)+cy^{\alpha_2-1}D_y-by^{\alpha_2-2}$$
in the half-space $\mathcal{R}^{N+1}_+=\{(x,y): x \in \mathcal{R}^N, y>0\}$,
under Dirichlet or oblique derivative boundary conditions. In the special case
$\alpha_1=\alpha_2=\alpha$ the operator $\mathcal L$ takes the form $$ \mathcal
L=y^{\alpha}\mbox{Tr
}\left(AD^2\right)+y^{\alpha-1}\left(v,\nabla\right)-by^{\alpha-2},$$ where
$v=(d,c)\in\mathcal{R}^{N+1}$, $b\in\mathcal{R}$ and $ A=\left(
\begin{array}{c|c}
  Q & { q}^t \\[1ex] \hline
  q& \gamma \end{array}\right)$ is an elliptic matrix. We prove elliptic and
parabolic $L^p$-estimates and solvability for the associated problems. In the
language of semigroup theory, we prove that $\mathcal L$ generates an analytic
semigroup, characterize its domain as a weighted Sobolev space and show that it
has maximal regularity.
