Gradient Estimates for Elliptic Operators with Second-Order Discontinuous Coefficients

We consider the second-order elliptic operator $$\begin{aligned} L =\Delta +(a-1)\sum _{i,j=1}^N\frac{x_ix_j}{|x|^2}D_{ij}+c\frac{x}{|x|^2}\cdot \nabla -\frac{b}{|x|^{2}}, \end{aligned}$$ \(a>0\), \(b,\ c \in \mathbb {R}\) and we prove gradient estimates for the heat kernel.