Uniform H\"{o}lder Estimates on Semigroups Generated by Non-Local   Operators of Variable Order
  We consider the non-local operator of variable order as follows   $$Lf(x)= \int_{\R^d\setminus\{0\}}\big(f(x+z)-f(x)-\<\nabla f(x),z\>   \I_{\{|z|\le 1\}}\big)\frac{n(x,z)}{|z|^{d+\alpha(x)}}\,dz.$$ Under mild conditions on $\alpha(x)$ and $n(x,z)$, we establish the H\"{o}lder regularity for the associated semigroups. The proof is based on the probabilistic coupling method, and it successfully applies to both stable-like processes in the sense of Bass and time-change of symmetric stable processes. 
Dejun Luo, Jian Wang
math.PR