  The main result of this work is the proof of the boundedness of the
Ornstein-Uhlenbeck semigroup $ \{T_t \}_{t\geq 0} $ in $ {\mathbb R}^d $ on
Gaussian variable Lebesgue spaces under a condition of regularity on $p(\cdot)$
following previous papers by E. Dalmaso R. Scotto and S. P\'erez. As a
consequence of this result, we obtain the boundedness of Poisson-Hermite
semigroup and the boundedness of the Gaussian Bessel potentials of order
$\beta> 0$.
