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

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


Introduction and Preliminaries
The Ornstein-Uhlenbeck semigroup {T t } t≥0 is the semigroup of operators generated in L 2 (γ d ) by the Ornstein-Uhlenbeck operator as infinitesimal generator, i.e., formally T t = e −tL . In view of the spectral theorem, for f = ∞ k=0 J k f ∈ L 2 (R d , γ d ) and t ≥ 0, T t is defined as where { h ν } ν are the normalized Hermite polynomials in d variables, and is the orthogonal projection of L 2 (R d , γ d ) onto .
Using Mehler's formula, it can be proved that the Ornstein-Uhlenbeck semigroup has an integral representation as for all f ∈ L 1 (R d , γ d ). Taking the change of variable s = 1 − e −2t , we obtain that T t f (x) = 1 (πs) d/2 The maximal function of the Ornstein-Uhlenbeck semigroup is defined by It is well know that the Ornstein-Uhlenbeck semigroup {T t } t≥0 in R d is a Markov operator semigroup in L p (R d , γ d ), 1 ≤ p ≤ ∞, i.e. a positive conservative symmetric diffusion semigroup, strongly L p -continuous in L p (R d , γ d ), 1 ≤ p ≤ ∞; with the Ornstein-Uhlenbeck operator L as its infinitesimal generator, see [2], [1] or [12]. Its properties can be obtained directly from the general theory of Markov semigroups, see [1] or [11]. It is also well known that the maximal function Even thought there are some known results about boundedness of operators on Gaussian variable Lebesgue spaces L p(·) (γ d ), see for instance [5], as far as we know, there is not proof in the literature of boundedness of the Ornstein-Uhlenbeck semigroup {T t } 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 {T t } t≥0 on R d is bounded for Gaussian variable Lebesgue spaces L p(·) (R d , γ d ), under certain conditions of regularity on p(·), that will be determined later (see Definitions 1.1, 1.2, 1.5 and 2.1) As a consequence of Theorems 1.1 we obtain, An important remark is needed here. Observe that from Theorem 1.1 we can not conclude, as in the classical case, that the semigroup {T t } is a contraction semigroup is L p(·) (R d , γ d ). We do not know if that is actually true for this case. Therefore questions like some form of hypercontractivity for the semigroup in this context are unknown.
Additionally, let us consider the Poisson-Hermite semigroup as the subordinated semigroup to the Ornstein-Uhlenbeck semigroup, using the Bochner's subordination formula, see E. Stein [10], defined then as, It is also well known that the Poisson-Hermite semigroup {P t } t≥0 is a strongly continuous, symmetric, conservative semigroup of positive contractions in L p (γ d ), 1 ≤ p ≤ ∞, with infinitesimal generator (−L) 1/2 . Additionally, the maximal function of the Poisson-Hermite semigroup is defined by As a consequence of the boundedness of {T t }, we will prove that {P t } t≥0 is also bounded for Gaussian variable Lebesgue spaces L p(·) (R d , γ d ) under the same conditions of regularity on p(·).
Finally, the Gaussian Bessel potential of order β > 0, J β is defined as for all x ∈ R d .
It can be proved, using P. A. Meyer's multiplier theorem, that the Gaussian Bessel potentials J β are L p (γ d )-bounded 1 < p < ∞. Moreover we will see that as consequence of Theorem 1.3 we obtain the boundedness of Gaussian Bessel potential on L p(·) (R d , γ d ).
Then there exists a constant C > 0 such that Now, for completeness, let us introduce some basic background on variable Lebesgue spaces with respect to a Borel measure µ.
Any µ-measurable function p(·) : R d → [1, ∞] is an exponent function; the set of all the exponent functions will be denoted by P(R d , µ). For E ⊂ R d we set p − (E) = ess inf x∈E p(x) and p + (E) = ess sup x∈E p(x).
We use the abbreviations p + = p + (R d ) and p − = p − (R d ).
We will need the following technical result; for its proof see Lemma 3.26 in [4].
Then there exists a constant C depending on d, N and the LH ∞ constant of ρ(·) such that given any set E and any function F with 0 ≤ F (y) ≤ 1 for y ∈ E, and the norm words, there are at most N balls (resp. cubes) that intersect at the same time.
The following definition was introduced for the first time by Berezhnoǐ in [3], defined for a family of disjoint balls or cubes. In the context of variable spaces, it has been considered in [6], allowing the family to have bounded overlappings. Definition 1.10. Given an exponent p(·) ∈ P(R d ), we will say that p(·) ∈ G, if for every family of balls (or cubes) for all functions f ∈ L p(·) (R d ) and g ∈ L p (·) (R d ). The constant only depends on N. Lemma 1.11 (Teorema 7.3.22 in [6]). If p(·) ∈ LH(R d ), then p(·) ∈ G.
As usual, in what follows C represents a constant that is not necessarily the same in each occurrence; also we will use the following notation: given two functions f , g, the symbols and denote, that there is a constant c such that f ≤ cg and cf ≥ g, respectively. When both inequalities are satisfied, that is, f g f , we will denote f ≈ g.

Proofs of the main results.
In this section we are going to consider Lebesgue variable spaces with respect to the Gaussian measure γ d , L p(·) (R d , γ d ). The next condition was introduced by E. Dalmasso and R. Scotto in [5].
Definition 2.1 with Observation 2.2 and Lemma 2.3 end up strengthening the regularity conditions on the exponent functions p(·) to obtain the boundedness of the maximal function T * . As a consequence of Lemma 1.11, we have Proof. Let We will prove that inf(A) inf(B) and inf(B) inf(A). In fact, taking λ ∈ A then On the other hand, taking λ ∈ B then Hence, we get

Boundedness of the maximal function of Ornstein
For x ∈ R d let us consider admissible (or hyperbolic) balls, It is well known that the Gaussian measure is essentially constant on B h (x), see [12, Chapter 1].
As it is nowadays a standard technique in Gaussian harmonic analysis, we split T t into its local part and its global part, , for x ∈ R d , where, using the integral representation (4) and the change of variables s = 1 − e −2t we will write Therefore the maximal function of the Ornstein-Uhlenbeck semigroup will be bounded by the sum of the operators, and which we call the local and global maximal operators respectively.
Next, we will need the following technical lemma to handle the proof of boundedness of the local part, for the proof see [12], for an earlier formulation see also [7]. Lemma 2.6. Let us define the secuence x k = √ k for k ∈ N. For this strictly increasing secuence, we obtain a family of disjoint balls B k j , for k ∈ N and 1 ≤ j ≤ N k with the following properties: Now, we present the boundedness of the local maximal operator T * 0 .
There exists a constant C > 0 such that Proof. We follow the proof of Theorem 3.3. in [5]. Without lost of generality let us assume that f ≥ 0.
Following [8] we obtain that if y ∈ B h (x) then e −u(s) ≤ C d e − |x−y| 2 s and therefore Now, given x ∈ R d , by Lemma 2.6, there exists B ∈ F such that x ∈ B and B h (x) ⊂B, so we get, Set φ s (z) = e − |z| 2 s s d/2 and since {φ s } s>0 is an approximation of identity, we have Therefore, Thus, for all x ∈ R d . Since, the right hand side is independent of s we immediately get for all x ∈ R d . Let f ∈ L p(·) (R d , γ d ). Using the characterization of the norm by duality, from (17) and following again [5] we obtain that where c B is the center of the balls B andB. Using Hölder's inequality and the boundedness of the maximal operator M H−L on L p(·) (R d ), we get By Lemma 2.5, we have that and therefore, Since the family of ballsF has bounded overlaps; applying Corollary 2.4, to the functions f e −|·| 2 /p(·) ∈ L p(·) (R d ) and ge −|·| 2 /p (·) ∈ L p (·) (R d ) and again applying Lemma 2.5, we get Taking supremum on all the functions g ∈ L p (·) (R d , γ d ) with g p (·),γ d ≤ 1, we obtain that Finally, we will obtain the boundedness of the global maximal operator Then there exists a constant C > 0 such that Proof. Suppose that f ≥ 0. Again, we follow the proof of Theorem 3.5 in [5].
On the other hand, using Lemma 1.7 with G(x) = 1 C Cx∩Ex P (x, y)g 2 (y)dy ≤ 1 and applying the inequality (7), we obtain that Now, in order to estimate the last two integrals, we apply Hölder's inequality.
Then, by Fubbini's theorem we get, Now, we need to estimate the integral R d Cx∩Ex P (x, y)g 2 (y)dy p∞ dx.
We proceed in an analogous way, but applying the Hölder's inequality to the exponent p ∞ , and applying the inequality (8) in Lemma 1.7. Thus, it follows With this we obtain that T * 1 (f χ E (·) ) p(·),γ d ≤ C d,p , then by homogenity of the norm the result holds for all function f ∈ L p(·) (R d , γ d ). Hence Now, the proof of the L p(·) (γ d )-boundedness of the maximal function T * of the Ornstein-Uhlenbeck semigroup, Theorem 1.1, is a immediate consequence of Theorems 2.7 and 2.8, since we have On the other hand, the L p(·) (γ d )-boundedness of the Additionally, from the L p(·) (R d , γ d )-boundedness of T * we obtain , from the pointwise convergence of the Ornstein-Uhlenbeck semigroup (see [9]), we have, On the other hand, Applying Lebesgue's dominated convergence theorem, we have Thus,

Consequences of the Boundedness of the Ornstein-Uhlenbeck semigroup
Another consequence of Theorem 1.1 is the boundedness of Poisson-Hermite semigroup in L p(·) (R d , γ d ), Theorem 1.3: Proof. of Theorem 1.3. Let f ∈ L p(·) (R d , γ d ) with f p(·),γ d ≤ 1, then by Corollary 1.2, we have for every s > 0 and therefore T s f C p(·),γ d ≤ 1.
For fixed t > 0, since the measure µ 1/2 t (ds) is a probability measure, using Jensen's inequality, and Fubini's theorem we get that the modular is less or equal to 1. In fact,

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Thus, P t f ∈ L p(·) (R d , γ d ) and P t f p(·),γ d ≤ C, for all t > 0. Now, by homogeneity of the norm and the linearity of P t we obtain the general result. P t f p(·),γ d ≤ C f p(·),γ d for any function f ∈ L p(·) (R d , γ d ) and t > 0.
We want to thank the reviewer for his/her commentaries and corrections which improved greatly the presentation as well as the clarity of the results of the paper.