%0 Journal Article
%T Uniform H£żlder Estimates on Semigroups Generated by Non-Local Operators of Variable Order
%A Dejun Luo
%A Jian Wang
%J Mathematics
%D 2014
%I arXiv
%X 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.
%U http://arxiv.org/abs/1403.3163v2