Maximal Operators, Littlewood–Paley Functions and Variation Operators Associated with Nonsymmetric Ornstein–Uhlenbeck Operators

In this paper, we establish Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document} boundedness properties for maximal operators, Littlewood–Paley functions and variation operators involving Poisson semigroups and resolvent operators associated with nonsymmetric Ornstein–Uhlenbeck operators. We consider the Ornstein–Uhlenbeck operators defined by the identity as the covariance matrix and having a drift given by the matrix -λ(I+R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\lambda (I+R)$$\end{document}, being λ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda >0$$\end{document} and R a skew-adjoint matrix. The semigroups associated with these Ornstein–Uhlenbeck operators are the basic building blocks of all the normal Ornstein–Uhlenbeck semigroups.


Introduction
In this paper, we are concerned with maximal operators, Littlewood-Paley functions and variation operators defined by Poisson semigroups and resolvent operators for nonsymmetric Ornstein-Uhlenbeck operators.
We denote by Q a real, symmetric and positive definite d×d matrix and by B a nonzero real d × d matrix having eigenvalues with negative real parts, being d ∈ N, d ≥ 1. We now introduce the Ornstein-Uhlenbeck semigroup defined by Q, named the covariance matrix, and B, called the drift matrix. For every t ∈ (0, ∞] we consider the matrix Q t given by . In [29], it was proved that D(L Q,B p ) coincides with the Sobolev space W 2,p (γ ∞ ).
Harmonic analysis associated with the symmetric Ornstein-Uhlenbeck operator has been much investigated over the last twenty five years. When Q = −B = I, where I denotes the identity matrix, the operator L reduces to the classical Ornstein-Uhlenbeck operator L I,−I = 1 2 Δ − x∇ and the Hermite polynomials are an orthonormal basis in L 2 (R d , γ ∞ ) of eigenfunctions of L I,−I . Muckenhoupt ([30]) studied maximal operator and Riesz transforms in the one dimensional L I,−I -setting. Sjögren ([36]) extended to higher dimensions Muckenhoupt's results about the maximal operator defined by {H I,−I t } t>0 . Harmonic analysis operators associated with L I,−I were studied in [15] and [28] (maximal operators); in [13,32] and [34] (Littlewood-Paley functions); in [12,16,33] and [34] (Riesz transforms); in [14] and [17] (spectral multipliers) and in [20] (variation and oscillation operators). Gutiérrez, Segovia and Torrea ( [19]) and Gutiérrez ([18]) studied Riesz transforms defined by the operator L I,B when B is symmetric.
Our objective in this paper is to establish L p boundedness properties of some maximal operators, Littlewood-Paley functions and variation operators involving the Poisson semigroups and the resolvent operators associated with the nonsymmetric Ornstein-Uhlenbeck operator considered by Mauceri and Noselli ([26] and [27]).
Assume that Q = I and B = −λ(I + R) where λ > 0 and R is a skewadjoint matrix as in [26] and [27]. After making a change of variables in (1.1) we get where dγ ∞,λ (y) = π λ −d/2 e −λ|y| 2 dy, and for x, y ∈ R d and t > 0. By using the subordination formula, the Poisson semigroup {P Q,B t } t>0 is given by Let k ∈ N and j = 1, . . . , d. We consider the maximal operator P Q,B * ,k,j defined by The Littlewood-Paley function g Q,B k,j is given by 1 2 .
Let ρ > 2. If g is a complex valued function defined in (0, ∞), the ρ-variation V ρ (g) of g is defined by Variation inequalities have been studied in probability, ergodic theory and harmonic analysis in recent years. The first variation inequality was due to Léplinge ( [24]) in the martingales setting. Later, Bourgain ([2]) studied variation operators associated with ergodic averages of dynamic systems, a paper which has motivated a lot of researches in ergodic theory and harmonic analysis.
After a change of variable we can write By using (2.1) we deduce that, by denoting T Q,B every of the operators considered in Theorem 1.1, the following equality holds Thus, we show that it is sufficient to prove Theorem 1.1 when λ = 1. In the sequel we assume λ = 1.
We can write Let σ > 0. We define the sets In order to obtain a more manageable form of the kernel h I,B1 t in the symmetric case, that is, when R = 0, the following change of variable due to S. Meda was introduced in [14] τ (s) = log 1 + s 1 − s , s ∈ (0, 1).
After a careful reading of the proof of [27,Lemmas 5.5 and 5.6] we can see that with minor modifications in those ones the following properties can be proved.
(i) There exists C and t 0 > 0 such that (ii) Suppose that the one-parameter group of rotations {e tR } t∈R generated by the matrix R is periodic of period D. Then, there exists an interval J = (a, b), with 0 < a < b < ∞, and C > 0 such that Here C, t 0 , a and b depend on δ.
Note that (2.2) and (2.3) also hold when t 0 is replaced by t 1 ∈ (0, t 0 ) and J is replaced by an interval J 1 ⊂ J, respectively. In the sequel we consider t 0 > 0 and an interval J = (a, b), with 0 < a < b < ∞, satisfying (2.2) and (2.3), respectively, and such that there exist n, m ∈ N and β > 0 for which 4) for certain N ⊂ (0, ∞) of measure zero, and being a disjoint union. Let σ > 0. We choose a smooth function ϕ in R d × R d satisfying that and the global part T glob of T by The following results were proved in [15] and they will be useful to prove that the global parts of the operators in Theorem 1.1 are bounded from In the study of the local parts of the operators in Theorem 1.1 we will use the L p -boundedness properties of the operator S σ defined by where δ > 0. Operators of this type appear also when the symmetric case is considered (see, for instance, [20]).
Proof. We include a sketch of the proof of this property for the sake of completeness.
We have that |x By using interpolation we deduce that the operator S σ is bounded from We are going to explain the method we use to prove Theorem 1.1. We extend the procedure developed by Mauceri and Noselli ([26] and [27]).
Suppose that X is a Banach space of complex functions defined in (0, ∞). Let k ∈ N and j = 1, ..., d. We consider the operator T X k,j defined by It is clear that T X k,j reduces to P I,B1 * ,k,j and g I,B1 k,j when X = L ∞ ((0, ∞), dt) and X = L 2 ((0, ∞), dt t ), respectively. Furthermore, let ρ > 2. We consider on the space C(0, ∞) of continuous functions on (0, ∞) and the seminorm V ρ defined in (1.2). By identifying those functions in C(0, ∞) that differ by a constant, the space V ρ (0, ∞) consisting of all those g ∈ C(0, ∞) such that V ρ (g) < ∞ endowed with V ρ is a Banach space. We have that T t (x, y)dy, x ∈ R d and t > 0, (2.5) where P I,B1 t (x, y), x, y ∈ R d and t > 0, denotes the Poisson integral kernel and x, y ∈ R d and t > 0.
Differentiations under the integral sign are justified. Indeed, we have x, y ∈ R d and u > 0.
Since |e uB * | ≤ e −cu , u > 0, we get By using [1, Lemma 4] we obtain, for each x, y ∈ R d and t > 0, where x, y ∈ R d and t > 0.
Assume that X is one of the following Banach spaces: L ∞ ((0, ∞), dt), L 2 ((0, ∞), dt/t) and V ρ (0, ∞). We consider the operator We define δ p as follows We denote by E p the sets (0, t 0 ) or J # D in Lemma 2.2 associated to t 0 and satisfying the covering property (2.4).
Then, for every h ≥ 0, the operator S Suppose that the claim has been proved. Since x, y ∈ R d and t > 0.
By using the semigroup property of {H I,B1 t } t>0 we deduce that Our objective is to prove the Claim 1. In order to see the L p -boundedness properties of the operator S E,h k,j we study separately the local part and the global part of S E,h k,j . We analyze firstly the local part S E,h k,j,loc of S E,h k,j . We consider the operator where 3 2 , z ∈ R d and t > 0.
Here W u , u > 0 denotes the classical heat kernel given by We also define By using Minkowski inequality we deduce that By using [17, Lemma 3.6] from Claim 3 we deduce that the operator Minkowski's inequality leads to Thus, Claim 1 is proved when we establish Claims 2, 3 and 4. Let k ∈ N, j = 1, ..., d, and M ≥ 1. We consider the operator T X k,j,M defined by It is clear that According to [ In order to establish the L pboundedness properties for the maximal operators S I,B1 * ,k,j,M we need to work harder because, as far as we know, the L p -boundedness properties for the corresponding maximal operators involving the heat semigroup {H I,B1 t } t>0 have not been studied.
Let f ∈ C ∞ c (R d ). We can write Then, for every x ∈ R d and t > 0, Differentiation under the integral can be justified as above by considering that M > (d + 1)/2. We take E ⊂ (0, ∞) and h ≥ 0. As in the previous case, we define the operator S E,h k,j,M by x, y ∈ R d and t > 0.
In order to prove the L p -boundedness properties of the operator T X k,j,M where X = L ∞ ((0, ∞), dt), X = L 2 ((0, ∞), dt t ) and X = V ρ (0, ∞), we can proceed by following the same steps than in the previous case by considering the operator Here It is natural to ask if L p -boundedness properties of this operator when α = 1 can be proved by using the procedure in this paper. At this moment we can not apply our procedure because we do not know how to deal with the global parts of the operators when α = 1.
We now comment about some special cases. We consider α = 0 and X = L ∞ ((0, ∞), dt). By using the method in [25, §4] we can see that We can write We have that Then, the cases α = 0 and α = 2 are connected.
The arguments used in the symmetric Ornstein-Uhlenbeck setting in [20,32,33] and [34] do not work for the global parts of the operator T X k,α in the nonsymmetric context. Our objective in a next paper is to establish L p -boundedness properties of T X k,α -type operators for general nonsymmetric Ornstein-Uhlenbeck operators by using some of the ideas developed by Casarino, Ciatti and Sjögren ([5,6] and [4]).

Proof of Claims 2 and 4
Our objective in this section is to prove Claims 2 and 4 stated in the previous section for the operators in Theorem 1.1.

Proof of Claim 2
We consider firstly the operators P I,B1 * ,k,j , g I,B1 k,j and V I,B1 ρ,k,j . In the sequel X represents one of the following Banach spaces: We are going to study the operator where E ⊂ (0, ∞) and h ≥ 0. The definitions can be found in Sect. 2. We can write, for each x, y ∈ R d and t > 0, By using Minkowski inequality we get (3.1) According to [1,Lemma 4] it follows that We have that By using again (3.2) we obtain We first prove a technical result. (3.7) Proof. When B 1 = −I the operator L I,B1 is the symmetric Ornstein-Uhlenbeck and we have that Then, (3.7) will be proved when we see that (3.8) Since R + R * = 0 we get = |e −u x − e uR y| 2 , x,y ∈ R d and u > 0.
We have that x, y ∈ R d and u > 0, (3.9) and x, y ∈ R d and u > 0. (3.10) We can write, for every x, y ∈ R d and u > 0, We need to establish some estimations. We have that Then, according to [26,Lemma 3 We manipulate to get, for every x, y ∈ R d and u > 0, It follows that x, y ∈ R d and u ∈ (0, 1).
Thus we obtain that By using this estimation, we get, when (x, y) ∈ L 2 , On the other hand, by (3.9) and (3.10) and using Lemma 2.1 we deduce that We conclude that (3.8) holds. Thus we proved (3.7).
According to (3.6) and (3.7), by using Lemma 2.5 we conclude that the operator D E,h k,j,loc is bounded from L p (R d , γ ∞,1 ) into itself, for every 1 ≤ p ≤ ∞.
We now consider the operators S I,B1 * ,k,j,M , G I,B1 k,j,M and V I,B1 ρ,k,j,M . Let E ⊂ (0, ∞) and h ≥ 0. We define the operator where z ∈ R n and t > 0.
Minkowski inequality leads to Our objective is to see that the operator Z E,h k,j,M,loc defined by By using again Minkowski inequality we get We are going to prove a preliminary bound.
Proof. We firstly consider k = 0. We have that where c i ∈ R, i = 0, ..., k − 1. It follows that We have that when M > 1 2 . We also obtain By using (3.7) we get and then, by virtue of Lemma 2.5, the operator Z E,h k,j,M,loc is bounded from L p (R d , γ ∞,1 ) into itself, for every 1 ≤ p ≤ ∞.

Proof of Claim 4
Let E ⊂ (0, ∞) and h ≥ 0. By using Minkowski inequality we obtain

It was proved (see (3.3), (3.4) and (3.5)) that
Then, In a similar way by using (3.11) we can see that We denote now by E 1 the sets (0, t 0 ) and J # D associated with δ 1 = 1/10 given in Lemma 2.2. The arguments developed in the proof of [26, Proposition 4.7] allow us to prove that We define We observe that Then, we only have to establish that the operator Z For that, we first prove the following bounds. (3.14) Proof. Let η ∈ (0, 1). First we observe that from (3.9) we deduce that x,y ∈ R d and u > 0. Then, x, y ∈ R d and u > 0.
By using Lemma 2.2 we have that Qs(x,y)) x, y ∈ R d and s ∈ τ −1 (E p ). Since Thus we obtain x, y ∈ R d and u ∈ E p .

Proof of Claim 3
In this section we prove the Claim 3. Assume that E ⊂ (0, ∞) and h ≥ 0.
We recall that , z ∈ R d and u > 0.
We can write, for each x, y ∈ R d and t > 0, Then, by using Minkowski inequality and (3.3) we get that If h > 0, by taking into account that |x − y| ≤ C, when (x, y) ∈ L 2 , and that |z| On the other hand, we have that, for every x ∈ R d and t > 0, We conclude that From well-known L p -boundedness properties of the maximal operator defined by the classical heat semigroup we deduce that the local maximal operator U E,h * ,k,j,loc is bounded from L p (R d , dx) into itself, for every 1 < p < ∞ and from L 1 (R d , dx) into L 1,∞ (R d , dx).

4.1.2.
We now study the local maximal operator where According to (3.11) and Minkowski inequality we can write By proceeding as in section 4.1.1 we get that, if h > 0, On the other hand, by using (3.12), if k ≥ 1 we get Furthermore, we obtain Then, We conclude that and thus, we establish that U E,h * ,k,j,M,loc is bounded from L p (R d , dx) into itself, for every 1 < p < ∞, and from L 1 (R d , dx) into L 1,∞ (R d , dx).

4.2.1.
We consider the local Littlewood-Paley function g E,h k,j,loc defined by By using Minkowski inequality and (3.4) we get As in section 4.1.1 we deduce that if h > 0 the operator g E,h k,j,loc is bounded from L p (R d , dx) into itself, for every 1 < p < ∞, and from L 1 (R d , dx) into L 1,∞ (R d , dx).
We now study the operator g E,0 k,j,loc . We consider the Littlewood-Paley function g E,0 k,j defined by We are going to see that g E,0 k,j is bounded from we have that (see for instance, [10, p. 15 (11) By using Plancherel equality and (4.2) we get Minkowski inequality and (3.4) lead to We now use the Banach-valued Calderón-Zygmund theory for singular integrals (see [35]). We recall that the operator U E,0 k,j is defined by It is clear that

By using again Minkowski inequality and (3.4)
In a similar way we obtain, for every i = 1, ..., d, Let N ∈ N, N ≥ 2. The space F N = L 2 ((1/N, N ), dt/t) is a Banach and separable space.
Assume that f ∈ C ∞ c (R d ). Let x ∈ R d . We consider the mapping F x : We observe that F x (y), y ∈ R d , is continuous in [1/N, N ]. Thus, F x is continuous in R d . Indeed, let y 0 ∈ R d . We can write, by (3.2) and Minkowski inequality Since by using the dominated convergence theorem we deduce that Since F N is a separable Banach space, Pettis' Theorem ([37, Theorem p. 131]) implies that F x is F N -strongly measurable.
By (4.3) we get We define where the integral is understood in the F N -Bochner sense. Suppose that g ∈ F N . We have that We can write Hence, for every According to (4.3), (4.4) and (4.5) and by taking into account that the constant C in (4.5) does not depend on N , the vector-valued Calderón-Zygmund theory ( [35]) allows us to conclude that U E,0 k,j defines a bounded operator from for every 1 < p < ∞; By using monotone convergence theorem we deduce that g E,0 k,j is bounded from L p (R d , dx) into itself, for every 1 < p < ∞, and from

4.2.2.
We are going to study the local Littlewood-Paley function G E,h k,j,M,loc defined by Suppose that h > 0. Minkowski inequality and (3.11) lead to It follows that G E,h k,j,M,loc is bounded from L p (R d , dx) into itself, for every 1 < p < ∞, and from L 1 (R d , dx) into L 1,∞ (R d , dx).
In order to study the operator G E,0 k,j,M,loc we use the vector-valued Calderón-Zygmund theory ( [35]).
We define By using Plancherel equality, Minkowski inequality and (3.11) we obtain We consider the operator U E,0 k,j,M defined by We have that Minkowski inequality and (3.11) lead to We also obtain, for every i = 1, .., d, By proceeding as in the previous case we can conclude that the operator G E,0 k,j,M,loc is bounded from L p (R d , dx) into itself, for every 1 < p < ∞, and from By using (3.5) we can proceed as in section 4.1.1 to prove that, when We now study the variation operator v E,0 ρ,k,j,loc . We consider first, E = (0, ∞). The classical Poisson kernel is given by , z ∈ R d and t > 0.
We have that , z ∈ R d and t > 0.
We get = C(φ t * R j f )(x), x ∈ R d and t > 0, for a certain C ∈ R. Here, R j denotes the j-th Euclidean Riesz transform. We define It is clear that φ(z) = ψ(|z|), z ∈ R d . We have that ψ(u) → 0 as u → ∞, and ∞ 0 |ψ (u)|u d du < ∞. According to Lemma 2.4 in [3], the variation operator associated with {T t } t>0 where T t f = φ t * f , t > 0, is bounded from L p (R d , dx) into itself, for every 1 < p < ∞, and from L 1 (R d , dx) into L 1,∞ (R d , dx). Since R j is bounded from L p (R d , dx) into itself, for every 1 < p < ∞ we derive the same boundedness property for v (0,∞),0 ρ,k,j .
According to (3.5) and Minkoswki inequality we obtain and, in a similar way, for every i = 1, ..., d, (4.7) Let N ∈ N, N ≥ 2. We consider the space V ρ 1 N , N consisting in all g ∈ C 1 N , N such that By identifying those functions that differ by a constant V ρ 1 N , N , V N ρ is a Banach space. Assume that f ∈ C ∞ c (R d ). Let x ∈ R d . We define, for every y ∈ R d , the function F x (y) ∈ C 1 N , N defined by
Since F x is continuous, it is V ρ 1 N , N -strongly measurable by Pettis' Theorem. Indeed, F x is weakly measurable. Furthermore, if Q represents the set of rational numbers, we have that F x (R d ) = F x (Q d ) Vρ 1 N ,N . By using (4.6) we get where the integral is understood in the V ρ -Bochner sense. Let 0 < a = 1. We define the functional T a in V ρ 1 N , N as follows T a (g) = g(a) − g (1). licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creativecommons.org/licenses/by/4.0/.
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