Estimates of Green Function for some perturbations of fractional Laplacian

Abstract

Suppose that Y(t) is a d-dimensional Levy symmetric process for which its Levy measure differs from the Levy measure of the isotropic alpha-stable process (0<alpha<2) by a finite signed measure. For a bounded Lipschitz set D we compare the Green functions of the process Y and its stable counterpart. We prove a few comparability results either one sided or two sided. Assuming an additional condition about the difference of the densities of the Levy measures, namely that it is of order of |x|^{-d+varrho} as x near 0, where varrho>0, we prove that the Green functions are comparable, provided D is connected. These results apply for example to alpha-stable relativistic process. This process was studied in recent years. In the paper we also considered one dimensional case for alpha<= 1 and proved that the Green functions for an open and bounded interval are comparable.