The Boundedness of the Ornstein-Uhlenbeck semigroup on variable Lebesgue spaces with respect to the Gaussian measure

The main result of this work is the proof of the boundedness of the Ornstein-Uhlenbeck semigroup $\{T_t{\}}_{t\ge 0}$ 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 $\beta >0$.