  Let G(x,y) and G_D(x,y) be the Green functions of rotationally invariant
symmetric \alpha-stable process in R^d and in an open set D respectively, where
0<\alpha < 2. The inequality G_D(x,y)G_D(y,z)/G_D(x,z) \le c(G(x,y)+G(y,z)) is
a very useful tool in studying (local) Schrodinger operators. When the above
inequality is true with a constant c=c(D)>0, then we say that the 3G theorem
holds in D.
  In this paper, we establish a generalized version of 3G theorem when D is a
bounded \kappa-fat open set, which includes a bounded John domain. The 3G we
consider is of the form G_D(x,y)G_D(z,w)/G_D(x,w), where y may be different
from z. When y=z, we recover the usual 3G.
  The 3G form G_D(x,y)G_D(z,w)/G_D(x,w) appears in non-local Schrodinger
operator theory. Using our generalized 3G theorem, we give a concrete class of
functions belonging to the non-local Kato class, introduced by Chen and Song,
on \kappa-fat open sets.
  As an application, we discuss relativistic \alpha-stable processes
(relativistic Hamiltonian when \alpha=1) in \kappa-fat open sets. We identify
the Martin boundary and the minimal Martin boundary with the Euclidean boundary
for relativistic \alpha-stable processes in \kappa-fat open sets. Furthermore,
we show that relative Fatou type theorem is true for relativistic stable
processes in \kappa-fat open sets.
  The main results of this paper hold for a large class of symmetric Markov
processes, as are illustrated in the last section of this paper. We also
discuss the generalized 3G theorem for a large class of symmetric stable Levy
processes.
