We study elliptic and parabolic problems governed by the singular elliptic operators \(\begin{aligned}&{\mathcal {L}}=y^{\alpha _1}\text{ Tr } \left( QD^2_x\right) +2y^{\frac{\alpha _1+\alpha _2}{2}}q\cdot \nabla _xD_y+\gamma y^{\alpha _2} D_{yy}\\&\qquad +y^{\frac{\alpha _1+\alpha _2}{2}-1}\left( d,\nabla _x\right) +cy^{\alpha _2-1}D_y-by^{\alpha _2-2} \end{aligned}\) in the half-space \(\mathbb {R}^{N+1}_+=\{(x,y): x \in \mathbb {R}^N, y>0\}\) , under Dirichlet or oblique derivative boundary conditions. In the special case \(\alpha _1=\alpha _2=\alpha \) the operator \({\mathcal {L}}\) takes the form \(\begin{aligned} {\mathcal {L}}&=y^{\alpha }\text{ Tr } \left( AD^2\right) +y^{\alpha -1}\left( v,\nabla \right) -by^{\alpha -2}, \end{aligned}\) where \(v=(d,c)\in \mathbb {R}^{N+1}\) , \(b\in \mathbb {R}\) 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.
