Estimates of Green functions for some perturbations of fractional Laplacian

Abstract

Suppose that $Y_t$ is a $d$-dimensional symmetric Lévy process such that its Lévy measure differs from the Lévy measure of the isotropic $\alpha$-stable process ($0 < \alpha < 2$) by a finite signed measure. For a bounded Lipschitz open set $D$ we compare the Green functions of the process $Y$ with those of its stable counterpart, and we prove several comparability results, both one-sided and two-sided. In particular, assuming an additional condition about the difference between the densities of the Lévy measures, namely that it is of the order of $|x|^{-d+\varrho}$ as $|x|\to 0$, where $\varrho > 0$, we prove that the Green functions are comparable, provided $D$ is connected. These results apply, for example, to the relativistic $\alpha$-stable process. The bounds for its Green functions were previously known for $d > \alpha$ and smooth sets. Here we consider also the one-dimensional case for $\alpha \ge 1$, and we prove that the Green functions for a bounded open interval are comparable, a case that, to the best of our knowledge, had not been treated in the literature.