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

TL;DR

This paper proves the boundedness of the Ornstein-Uhlenbeck semigroup on Gaussian variable Lebesgue spaces, extending to Poisson-Hermite semigroup and Gaussian Bessel potentials, under regularity conditions on the variable exponent.

Contribution

It establishes the boundedness of the Ornstein-Uhlenbeck semigroup on variable Lebesgue spaces with Gaussian measure, under specific regularity conditions on the exponent function.

Findings

Boundedness of Ornstein-Uhlenbeck semigroup on variable Lebesgue spaces

Extension to Poisson-Hermite semigroup boundedness

Boundedness of Gaussian Bessel potentials of order β

Abstract

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 $\beta> 0$.