Sharp kernel bounds for parabolic operators with first order degeneracy

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