Green function estimates for subordinate Brownian motions: Stable and beyond

Abstract:

A subordinate Brownian motion $X$ is a Lévy process which can be obtained by replacing the time of the Brownian motion by an independent subordinator. In this paper, when the Laplace exponent $\phi$ of the corresponding subordinator satisfies some mild conditions, we first prove the scale invariant boundary Harnack inequality for $X$ on arbitrary open sets. Then we give an explicit form of sharp two-sided estimates of the Green functions of these subordinate Brownian motions in any bounded $C^{1,1}$ open set. As a consequence, we prove the boundary Harnack inequality for $X$ on any $C^{1,1}$ open set with explicit decay rate. Unlike previous work of Kim, Song and Vondraček, our results cover geometric stable processes and relativistic geometric stable process, i.e. the cases when the subordinator has the Laplace exponent \[ \phi (\lambda )=\log (1+\lambda ^{\alpha /2})\ \ \ \ (0<\alpha \leq 2, d > \alpha )\] and \[ \phi (\lambda )=\log (1+(\lambda +m^{2/\alpha })^{\alpha /2}-m)\ \ \ \ (0<\alpha <2, m>0, d >2) . \]