Coupling by reflection and Hölder regularity for non-local operators of variable order

Abstract:

We consider the non-local operator of variable order as follows: \begin{equation*} Lf(x)= \int _{{\mathbb {R}}^d\setminus \{0\}}\big (f(x+z)-f(x)-\langle \nabla f(x),z\rangle {\mathbb {1}}_{\{|z|\le 1\}}\big )\frac {n(x,z)}{|z|^{d+\alpha (x)}} dz, \end{equation*} where $\alpha (x)\in [\alpha _0,\alpha _2]$ for any $x\in {\mathbb {R}}^d$ and some constants $0<\alpha _0\le \alpha _2<2$ and $n(x,z)$ is a positive Borel measurable function bounded from above and below. Under mild continuity conditions on $\alpha (x)$ and $n(x,z)$, we establish the Hölder regularity for the associated semigroups. The proof is based on a new construction of the coupling by reflection for non-local operator $L$, and the results successfully apply to both stable-like processes in the sense of Bass and time-change of symmetric stable processes.