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.
Highlights
Introduction and PreliminariesThe Ornstein-Uhlenbeck semigroup {Tt}t≥0 is the semigroup of operators generated in L2(γd) by the Ornstein-Uhlenbeck operator d 1 ∂2 ∂ L = 2 ∆x − x, ∇x = i=12 ∂x2i − xi ∂xi (1)as infinitesimal generator, i.e., formally Tt = e−tL
Using Mehler’s formula, it can be proved that the Ornstein-Uhlenbeck semigroup has an integral representation as
The main result of this paper is the proof that the maximal function T ∗ of the Ornstein-Uhlenbeck semigroup {Tt}t≥0 on Rd is bounded for Gaussian variable Lebesgue spaces Lp(·)(Rd, γd), under certain conditions of regularity on p(·), that will be determined later
Summary
|ν|=k k=0 where {hν}ν are the normalized Hermite polynomials in d variables, and. Using Mehler’s formula, it can be proved that the Ornstein-Uhlenbeck semigroup has an integral representation as. Taking the change of variable s = 1 − e−2t, we obtain that. It is well known that the maximal function T ∗ is bounded in Lp(Rd, γd), 1 < p ≤ ∞. Even thought there are some known results about boundedness of operators on Gaussian variable Lebesgue spaces Lp(·)(γd), see for instance [5], as far as we know, there is not proof in the literature of boundedness of the Ornstein-. Uhlenbeck semigroup {Tt}t≥0, nor of the boundedness of the maximal function of the semigroup. The main result of this paper is the proof that the maximal function T ∗ of the Ornstein-Uhlenbeck semigroup {Tt}t≥0 on Rd is bounded for Gaussian variable Lebesgue spaces Lp(·)(Rd, γd), under certain conditions of regularity on p(·), that will be determined later (see Definitions 1.1, 1.2, 1.5 and 2.1). Questions like some form of hypercontractivity for the semigroup in this context are unknown
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