Higher order Riesz transforms for Hermite expansions

In this paper, we consider the Riesz transform of higher order associated with the harmonic oscillator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L=-\Delta+|x|^{2}$\end{document}L=−Δ+|x|2, where Δ is the Laplacian on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {R}^{d}$\end{document}Rd. Moreover, the boundedness of Riesz transforms of higher order associated with Hermite functions on the Hardy space is proved.

for f ∈ L  (R d ) and Then the Poisson semigroup is defined as The relation between the heat kernel and the Poisson kernel is Let A j = ∂ ∂x j + x j and A -j = A * j = -∂ ∂x j + x j , j = , , . . . , d. Then we can denote L as We define operators R L ±j , j = , , . . . , d R j and R -j are called the Riesz transforms associated with L. The definition was first suggested by Thangavelu in []. Let e j be the coordinate vectors in R d , then Therefore, for f ∈ L  (R d ), and In [], the author proved that R L j were bounded on the local Hardy spaces h  (R d ) which were defined by Goldberg in []. Thangavelu asked one question: whether it was possible to characterize h  (R d ) by R L j , i.e., whether the equality Remark  When we consider the boundedness of Riesz transforms for L on Hardy spaces, the main tool is Littlewood-Paley characterizations of Hardy spaces. In fact, we have the following equality (cf. []): , the authors proved the boundedness of R L ±j on L p (R d ), where they considered the semigroup generated by L + b for b <  on L p (R d ).
In this paper, we prove that the higher ordered Riesz transforms are bounded on the Hardy spaces associated with Hermite functions. More precisely, let and define the m-ordered Riesz transforms as : where we say a(x) is an atom for the space can be defined as In [], the authors proved the following result.
Proposition  There exists C >  satisfying Then . This semigroup is generated by the operator -(L + b).
The subordination formula is The Poisson integral of f (x) can be defined as The main results of this paper are as follows.
The organization of this paper is as follows. In Section , we give some estimations of the heat kernel and the Poisson kernel associated with L + b. In Section , Theorem  is proved. In Section , we prove Theorem .
Throughout the article, we use A and C to denote the positive constants, which are independent of the main parameters and may be different at each occurrence. By

Estimations of the kernels
Then the following inequality can be proved by the Feynman-Kac formula: () By the subordination formula, we get the following.
Proof (a) By subordination formula and Lemma , we have By () and (b) By subordination formula again, we know We also have Then (b) follows from () and (). y). Then, by Lemma , we can prove (cf. [] or []) the following.
Let t =   ln +s -s , s ∈ (, ). Then The proof of the following proposition is motivated by [].
Proposition  There is A > , for N ∈ N and |xx | ≤ |x-y|  , we can find C N >  such that and t =   ln +s -s ∼ s, s →  + , for s ∈ (,   ], we have If x · y ≥ , then |x| ≤ |x + y|. So Therefore, When s ∈ [   , ), Since t =   ln +s -s > s for s ∈ [   , ), we get Therefore, and Therefore, when s ∈ (,   ], we have When s ∈ [   , ), we have By ()-(), we get Similar to the proof of (), for any N > , we can prove Since ρ(x) =  +|x| , we get Since x and y are symmetric, we also have Then (a) follows from ()-().
(b) Note that For J  , let The subordination formula gives the following lemma.

Square function characterizations of H 1 L (R d )
We define square functions for k = , , . . . . The proof of the following lemma can be found in [].
Then, by Lemma , we can prove (cf. Section  in [] or []) the following.
Motivated by [], we can prove the following.
Lemma  There is C >  satisfying Hence By Lemma , we can prove the following.
Similar to the proof of Lemma  in [], we have the following.

Lemma  Let a be an H ,∞ L -atom. Then we can find a constant C >  satisfying
As pointed out in [], we cannot get that an operator is bounded on H p L (R d ) by just proving that it is uniformly bounded on atoms. But we have the following lemma (cf. p., Theorem . in []).

Lemma  Let T be an integral operator with the kernel in the Campanato space d(/p-) and satisfy Ta L p ≤ C for all the H p,q L -atom a(x), then T is a bounded operator from H
In the following, we prove D b t (x, y) = tA j P b t (x, y) belongs to BMO L , which is defined in [].

Lemma  For every t >  and x
Proof For any ball B(y  , r), if r < ρ(y  ) and r < t, then by Lemma (b) we have If t ≤ r < ρ(y  ), then by Lemma (a) If r ≥ ρ(y  ), then by Lemma (a) we have Then Lemma  follows from ()-().
Now, let us prove Theorem .
Proof of Theorem  When f ∈ L  (R d ) and G b (f ) ∈ L  (R d ), by Proposition  and Lemma , we have The reverse can be proved by Lemmas ,  and . Theorem  is proved.

Riesz transform associated with L
We introduce the following version of Riesz transform: We can prove the following.
This proves Theorem .
The proof of the following lemma can be found in [].
Now, we can prove Theorem .

Proof of Theorem
. We prove Theorem  by an inductive argument. When m = , Theorem  has been proved in []. We assume that Theorem  holds for m -, by Lemma  and Theorem , Therefore Theorem  holds.

Conclusions
In this paper, we consider the Riesz transforms of higher order associated with a harmonic oscillator and prove the boundedness of them on the Hardy space. It is well known that the Riesz transforms play an important role in the study of harmonic analysis and partial differential equations. These results are very good progress on the harmonic analysis of Hermite operators.