  A comprehensive analysis of Sobolev-type inequalities for the
Ornstein-Uhlenbeck operator in the Gauss space is offered. A unified approach
is proposed, providing one with criteria for their validity in the class of
rearrangement-invariant function norms. Optimal target and domain norms in the
relevant inequalities are characterized via a reduction principle to
one-dimensional inequalities for a Calder\'on type integral operator patterned
on the Gaussian isoperimetric function. Consequently, the best possible norms
in a variety of specific families of spaces, including Lebesgue, Lorentz,
Lorentz-Zygmund, Orlicz and Marcinkiewicz spaces, are detected. The reduction
principle hinges on a preliminary discussion of the existence and uniqueness of
generalized solutions to equations, in the Gauss space, for the
Ornstein-Uhlenbeck operator, with a just integrable right-hand side. A decisive
role is also played by a pointwise estimate, in rearrangement form, for these
solutions.
