Heat kernel for non-local operators with variable order

Let $\alpha (x)$ be a measurable function taking values in $[{\alpha }_1,{\alpha }_2]$ for $0<{\text{A}}_1\le {\text{A}}_2<2$, and $\kappa (x,z)$ be a positive measurable function that is symmetric in $z$ and bounded between two positive constants. Under a uniform H"older continuous assumptions on $\alpha (x)$ and $x\mapsto \kappa (x,z)$, we obtain existence, upper and lower bounds, and regularity properties of the heat kernel associated with the following non-local operator of variable order $$\text{LL}f(x)={\int }_{{\mathbb{R}}^d}(f(x+z)-f(x)-⟨\mathrm{\nabla }f(x),z⟩{\text{I}}_{\{\mathrm{\mid }z\mathrm{\mid }\le 1\}})\frac{\kappa (x,z)}{\mathrm{\mid }z{\mathrm{\mid }}^{d+\alpha (x)}}dz\mathrm{.}$$ In particular, we show that the operator $\text{LL}$ generates a conservative Feller process on ${\mathbb{R}}^d$ having the strong Feller property, which is usually assumed a priori in the literature to study analytic properties of $\text{LL}$ via probabilistic approaches. Our near-diagonal estimates and lower bound estimates of the heat kernel depend on the local behavior of index function $\alpha (x)$, when $\alpha(x)\equiv \A\in(0,2)$, our results recover some results by Chen and Kumagai (2003) and Chen and Zhang (2016).