  We study, in $L^{1}(\R^n;\gamma)$ with respect to the gaussian measure,
non-tangential maximal functions and conical square functions associated with
the Ornstein-Uhlenbeck operator by developing a set of techniques which allow
us, to some extent, to compensate for the non-doubling character of the
gaussian measure. The main result asserts that conical square functions can be
controlled in $L^1$-norm by non-tangential maximal functions. Along the way we
prove a change of aperture result for the latter. This complements recent
results on gaussian Hardy spaces due to Mauceri and Meda.
