Green function estimates for subordinate Brownian motions : stable and beyond

A subordinate Brownian motion $X$ is a L'evy process which can be obtained by replacing the time of the Brownian motion by an independent subordinator. In this paper, when the Laplace exponent $\varphi$ 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 on 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 {KSV2, KSV4}, our results cover geometric stable processes and relativistic geometric stable process, i.e. the cases when the subordinator has the Laplace exponent $$\varphi (\lambda )=\log (1+{\lambda }^{\alpha \mathrm{/}2})(0<\alpha \le 2,d>\alpha )$$ and $$\varphi (\lambda )=\log (1+(\lambda +m^{\alpha \mathrm{/}2}{)}^{2\mathrm{/}\alpha }-m)(0<\alpha <2,m>0,d>2)\mathrm{.}$$