  In this paper, we consider the Cauchy problem for Klein-Gordon equation with
a cubic convolution nonlinearity in $\R^3$. By making use of Bourgain's method
in conjunction with a precise Strichartz estimate of S.Klainerman and D.Tataru,
we establish the $H^s (s<1)$ global well-posedness of the Cauchy problem for
the cubic convolution defocusing Klein-Gordon-Hartree equation. Before arriving
at the previously discussed conclusion, we obtain global solution for this
non-scaling equation with small initial data in $H^{s_0}\times H^{s_0-1}$ where
$s_0=\frac\gamma 6$ but not $\frac\gamma2-1$, for this equation that we
consider is a subconformal equation in some sense. In doing so a number of
nonlinear prior estimates are already established by using Bony's
decomposition, flexibility of Klein-Gordon admissible pairs which are slightly
different from that of wave equation and a commutator estimate. We establish
this commutator estimate by exploiting cancellation property and utilizing
Coifman and Meyer multilinear multiplier theorem. As far as we know, it seems
that this is the first result on low regularity for this Klein-Gordon-Hartree
equation.
