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Boundedness of the Maximal Function of the Ornstein-Uhlenbeck semigroup on variable Lebesgue spaces with respect to the Gaussian measure and consequences

Abstract:

The main result of this paper is the proof of the boundedness of the Maximal Function T∗ of the Ornstein-Uhlenbeck semigroup {Tt}t≥0 in Rd, on Gaussian variable Lebesgue spaces Lp(·)(γd), under a condition of regularity on p(·) following [5] and [8]. As an immediate consequence of that result, the Lp(·)(γd)-boundedness of the Ornstein-Uhlenbeck semigroup {Tt}t≥0 in Rd is obtained. Another consequence of that result is the Lp(·)(γd)-boundedness of the Poisson-Hermite semigroup and the Lp(·)(γd)-boundedness of the Gaussian Bessel potentials of order β > 0.