A Continuous Characterization of Triebel-Lizorkin Spaces Associated with Hermite Expansions

We study the properties of the Triebel-Lizorkin spaces associated with the multidimensional Hermite expansions on (). In addition to the endpoint estimates, we give the continuous characterizations of these spaces via harmonic and thermic extensions from into . Based on this result, we obtain the boundedness of Riesz transform associated with the Hermite expansions.

In the past few years, many authors have studied the Triebel-Lizorkin spaces associated with the Hermite operators.In [4], Epperson Using Mehler's formula, Epperson proved that the definition of the corresponding space ‖‖  ,  () is independent of the particular choice of the function .In [5], Dziubański continued this study and proved that the results in [4] also hold for the multidimensional Hermite expansions.Besides, Petrushev and Xu [6] showed that the Triebel-Lizorkin on R  induced by Hermite expansions can be characterized in terms of the needlet coefficients and that the Hermite-Triebel-Lizorkin space is, in general, different from the respective classical spaces.In addition, Olafsson and Zheng [7] study the Triebel-Lizorkin space associated with the Peetre type maximal function for Hermite operator , which is defined by They gave the maximal characterization of this space as follows.
where  * , () is the Peetre type maximal function for Hermite operator.
In [3], Thangavelu introduced Riesz transforms associated with Hermite expansions and proved that they are bounded operators on   (R  ), 1 <  < ∞.In [8], Harboure et al. gave a different proof to show that the  norms of these operators are bounded by a constant not depending on the dimension .Moreover, they also defined the Riesz transforms of higher order and obtained the free dimensional estimates of the   -bounds of these operators.After that, Stempak and Torrea [9] obtained the weighted   -inequalities for the gradient square function associated with the Poisson semigroup.They get the result proposed in [3] by using a slightly different proof and they also get the analogous result for the -function associated with the Poisson semigroup.
Although many results have been obtained about the Triebel-Lizorkin spaces associated with the Hermite operators, all of them only discuss the case of the discrete quasinorm of the Triebel-Lizorkin spaces associated with the Hermite operators.Up to now, as we know, there are few papers related to the continuous case due to the fact that the Hermite expansions do not satisfy the general convolution property.This makes the study of the continuous case a meaningful problem.We first present a general characterization of the Triebel-Lizorkin spaces associated with the multidimensional Hermite expansions satisfied continuous quasi-norm on R  .Furthermore, analogous to the classical Triebel-Lizorkin space, we construct the Littlewood-Paley theorem on these spaces.Finally, we obtain the endpoint estimate of these spaces and establish the boundedness of the Riesz transforms associated with Hermite functions expansions.

The Characterizations of 𝐹
where The results of Dziubański [5] show that the definition of  This theorem shows that any "discrete" quasi-norm of the Triebel-Lizorkin spaces associated with the Hermite expansions of type (13) has a "continuous" counterpart (15).
In order to prove our theorem, we need some necessary lemmas.
Lemma 2. Let {  }, {V  } be two sequences of complex numbers.Then, for any , we have where is the ( − 1)th linear mean of first  terms of {V  } and Δ  is the th difference operator.
Since the proof of Lemma 2 is easy, we omit the proof for the sake of compactness of the paper.
In [3], a basic estimate for the kernel of Riesz means for the Hermite expansions has been obtained.We describe the result in the following lemma.Lemma 3.For  > (3 − 2)/6, the following estimate is valid: where    (, ) is the kernel of the Riesz means     for the Hermite expansions, which is defined by where where It is obvious that () is locally bounded.Since (2  ) = (), it is bounded on (0, ∞) (see [3]).The boundedness of II can also be obtained by replacing  in the proof of the boundedness of I by − and following from the same estimate tactics as above.Hence, we have   * () ≤   ().We give the proof of Theorem 1 in what follows.
Proof.Consider the following.
Step 1.Let  ∈  ,  .We will prove that ‖‖  ,  in ( 13) can be estimated from above by the quasi-norm in (15).Let For the first term of the right-hand side of (26), taking the supremum with respect to  on 1 Afterwards, integrating the modified inequality with respect to  that appears now only in the maximal function yields where () is the Hardy-Littlewood maximal function.For 1 <  < ∞ and 1 <  < ∞, we first multiply (29) by 2  ; and then we apply the   -norm with respect to ; afterwards, we apply the   -norm with respect to . (30) Step 2. We will show that the quasi-norm in ( 15) can be deduced from above by We give the estimate of the first sum in (31).Let Then we have where  ∈ Z and  − 1 > (9 − 8)/6.In addition, following the same estimate procedure of (27), we have We use (), For the above inequality, we first apply the   -quasi-norm with respect to  and then apply   -quasi-norm with respect to .Note that  1 > .Then, we have In the proof of the above inequality, we use the following fact: Taking  −  = 2 >  − which is even stronger than the desires estimate.
The proof is complete.
The Poisson semigroup {  } >0 , associated with , is given by We refer the reader to Thangavelu's monograph [3] for a detailed description of the Poisson semigroup {  } >0 associated with .In [9], the authors studied the -function associated with the Poisson semigroup: where  ∈   , 1 ≤  < ∞, and (, ) is the Poisson integral of  that is given by the convergent series That is a consequent of a general result for symmetric contraction semigroups.This general result is a refinement done by Coifman et al. [10], of the Littlewood-Paley theory for symmetric contraction semigroups satisfying, also, positivity and conservation properties which was developed by Stein [11]; see also [12].Inspired by these results, we give the harmonic and thermic characterization of the Triebel-Lizorkin spaces associated with the Hermite expansions by the Poisson semigroup related to the Hermite expansion, which shows  0,2  =   , for 1 <  < ∞.
It means that ( 51) is an equivalent quasi-norm in  ,  .Here, we substituted  by  2 .

Endpoint Estimates
From Theorem 5, we know that  0,2  =   , for 1 <  < ∞.However, this conclusion does not hold when  = 1.So, the purpose of this section is to discuss the characterization of the space  0,2 1 .We first present the definition of the local Hardy space below.Definition 6.Let  ∈ (R  ), (0) = 1, and   () = ().The local Hardy space associated with the Hermite operator is defined by the quasi-norm         ℎ holds for all  ∈ ℎ 1  (R  ).
In order to prove Proposition 7, we recall some results from the theory of local Hardy spaces [13]; compare also [14].A function  is an atom for the local Hardy space ℎ 1  if there is a ball ( 0 , ),  ≤ 2 1−/2 , such that supp  ⊂  ( 0 , ) , The atomic norm in ℎ 1  is defined by where the infimum is taken over all decompositions  = ∑     ;   are ℎ 1  atoms.
Proof of Proposition 7. We only need to show that there exists a constant  > 0 such that where  is any ℎ 1  atom associated with a ball ( 0 , ).Let  * =  * ( 0 , 2).Clearly, this ball is concentric with .Then In the above inequality, we use the fact that sup If  ∉  * , we use the representation where and Φ  (, ) is the kernel of the projection operator   ; see [3] for detailed description.From Lemma 3, we conclude that We choose  − 1 > (3 − 2)/6; then we have Then we complete our proof of Proposition 7.
If  has the same properties as the function  (including In the sequel we omit the index  in (66).Then the  2 -valued counterpart of (54), (55) reads as follows.
which are equivalent quasi-norms.
Proof.Consider the following.

The Boundedness of the
it is reasonable to define the Riesz transforms  ±  ,  = 1, 2, . . ., , by This definition was suggested by Thangavelu [15]; compare also [3].The definition fits a general framework and, in dimension one, matches the notion of conjugacy for the Hermite function expansions investigated by Gosselin and Stempak [16].In [17], Stempak and Torrea have already proved that the kernels of the Riesz transforms satisfy the Calderón-Zygmund standard conditions.Their results are as below.
The results presented in Theorem 10 can be further enlarged to the boundedness of the -functions of Littlewood-Paley defined with either the heat semigroup or the Poisson semigroup.We use a theorem to describe this bounded result.Since the proof of Theorem 11 is very similar to the proof of Theorem 10, we omit the proof here in order to make the paper concise.

Theorem 11 .
If  is commutative with the operator  defined by (85) and if the kernel of  satisfies the Calderón-Zygmund size condition, then  is bounded on  ,  .