Small time characterization of heat kernels in a weighted setting

We provide a small time characterization for the Gaussian bounds on the heat kernel of the \(C_0\) $C_0$ -semigroup generated by a second-order degenerate elliptic operator on the weighted Lebesgue space \(L^2_w(\mathbb {R}^d)\) $L_w^2({\mathbb{R}}^d)$ , where w belongs to the class \(\mathcal {A}_1\) ${\mathcal{A}}_1$ of Muckenhoupt weights. The result extends (Elst in Proc Am Math Soc 134:707–714, 2005, Thereom 1) to a weighted setting.