Spatial regularity of semigroups generated by L\'{e}vy type operators

We apply the probabilistic coupling approach to establish the spatial regularity of semigroups associated with L'{e}vy type operators, by assuming that the martingale problem of L'{e}vy type operators is well posed. In particular, we can prove the Lipschitz continuity of the semigroups under H"{o}lder continuity of coefficients, even when the L'evy kernel corresponding to L'{e}vy type operators is singular.