Sharp kernel bounds for parabolic operators with first order degeneracy

Abstract

We prove sharp upper and lower estimates for the parabolic kernel of the singular elliptic operator \begin{align*} \mathcal L&=\mbox{Tr }\left(AD^2\right)+\frac{\left(v,\nabla\right)}y, \end{align*} in the half-space ${\mathbb{R}}_{+}^{N+1}=\{(x,y):x\in {\mathbb{R}}^N,y>0\}$ under Neumann or oblique derivative boundary conditions at $y=0$.