Singular parabolic operators in the half-space with boundary degeneracy: Dirichlet and oblique derivative boundary conditions

We study elliptic and parabolic problems governed by the singular elliptic operators $$\mathcal{L}=y^{{\alpha }_1}Tr\left(Q{D}_x^2\right)+2y^{\frac{{\alpha }_1+{\alpha }_2}{2}}q\cdot {\mathrm{\nabla }}_x{D}_y+\gamma y^{{\alpha }_2}{D}_{yy}+y^{\frac{{\alpha }_1+{\alpha }_2}{2}-1}(d,{\mathrm{\nabla }}_x)+c{y}^{{\alpha }_2-1}{D}_y-b{y}^{{\alpha }_2-2}$$ 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 $$\mathcal{L}=y^{\alpha }Tr\left(A{D}^2\right)+y^{\alpha -1}(v,\mathrm{\nabla })-b{y}^{\alpha -2},$$ where $v=(d,c)\in {\mathcal{R}}^{N+1}$, $b\in \mathcal{R}$ and $A=\left(\begin{array}{cc}Q& q^t\\ q& \gamma \end{array}\right)$ 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.