Let $\alpha(x)$ be a measurable function such that $0<\alpha_1\le\alpha(x)\le \alpha_2<2$ for all $x\in \mathbb{R}^d$, and $\kappa(x,z)$ be a positive measurable function which is bounded from above and below and satisfies that $\kappa(x,z)=\kappa(x,-z)$ for all $x,z\in\mathbb{R}^d$. Under H\"{o}lder continuous assumptions on $\alpha(x)$ and $\kappa(x,z)$, we show existence, upper and lower bounds, and regularity properties of heat kernel associated with the following non-local operator with variable order
$$Lf(x):=\int_{\mathbb{R}^d}\Big(f(x+z)-f(x)-\nabla f(x)\cdot z \mathds{I}_{\{|z|\le 1\}}\Big) \frac{\kappa(x,z)}{|z|^{d+\alpha(x)}}\,dz.$$
This talk is based on a joint work with Zhen-Qing Chen and Jian Wang.