Uniform Hölder Estimates on Semigroups Generated by Non-Local Operators of Variable Order

Abstract: 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.