Regularity theory for parabolic operators in the half-space with boundary degeneracy

We study elliptic and parabolic problems governed by the singular elliptic operators \begin{align*} \mathcal L=y^{α_1}\mbox{Tr }\left(QD^2_xu\right)+2y^{\frac{α_1+α_2}{2}}q\cdot \nabla_xD_y+γy^{α_2} D_{yy}+Cy^{α_2-1}D_y \end{align*} under Neumann boundary condition, in the half-space ${\mathbb{R}}_{+}^{N+1}=\{(x,y):x\in {\mathbb{R}}^N,y>0\}$. 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.