A new version of Carleson measure associated with Hermite operator

Let L=−Δ+|x|2\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} be a Hermite operator, where Δ is the Laplacian on Rd\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}. In this paper we define a new version of Carleson measure associated with Hermite operator, which is adapted to the operator L. Then, we will use it to characterize the dual spaces and predual spaces of the Hardy spaces HLp(Rd)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H_{L}^{p}(\mathbb {R}^{d})$\end{document} associated with L.


Introduction
In recent years, the study of function spaces associated with Hermite operators has inspired great interest. Dziubański [7] introduced the Hardy space H p L (R d ), 0 < p ≤ 1, by using the heat maximal function and established its atomic characterization. Dziubański et al. [8] and Yang et al. [20] introduced and studied some BMO spaces and Morrey-Campanato spaces associated with operators. Deng et al. [5] introduced the space VMO L (R d ) and proved that (VMO L (R d )) * = H 1 L (R d ). Moreover, recently, Jiang et al. in [14] defined the predual spaces of Banach completions of Orlicz-Hardy spaces associated with operators. Bui et al. [3] considered the Besov and Triebel-Lizorkin spaces associated with Hermite operators.
One of the main purposes of studying the function spaces is to give the equivalent characterizations of them, for example, square functions characterizations for Hardy spaces [10], Carleson measure characterizations for BMO spaces [8] or Morry-Campanato spaces [6]. The aim of this paper is to give characterizations of the dual spaces and predual spaces of the Hardy spaces H p L (R d ) by a new version of Carleson measure. Now, let us review some known facts about the function spaces for L.
Let L be the basic Schrödinger operator in R d , d ≥ 1, the harmonic oscillator L = -+ |x| 2 . Let {T L t } t>0 be a semigroup of linear operators generated by -L and K L t (x, y) be their kernels. The Feynman-Kac formula implies that Dziubański [7] defined Hardy space H p L (R d ), 0 < p ≤ 1 as where The norm of Hardy space Remark 1 For simplicity, we just consider the case of d d+1 < p ≤ 1 in this paper. But all of our results hold for 0 < p ≤ 1.
Let ρ(x) = 1 1+|x| be the auxiliary function defined in [17]. This auxiliary function plays an important role in the estimates of the operators and in the description of the spaces associated with L. Then, for d d+1 < p ≤ 1 and 1 ≤ q ≤ ∞, a function a is an H p,q L -atom for the Hardy space H where the infimum is taken over all decompositions f = c j a j and a j are H p,q L -atoms. The atomic decomposition for H p L (R d ) is as follows (see [7] We define Campanato space associated with L as (cf. [1] or [20]). Definition 1 Let 0 ≤ α < 1, a locally integrable function g on R d belongs to L α if and only if g L α < ∞, where The duality of H p L (R d ) and L d (1/p-1) can be found in [12] or [20]. In order to give the Carleson measure characterization of L d(1/p-1) , we need some notations of the tent spaces (cf. [4]).
Let 0 < p < ∞ and 1 ≤ q ≤ ∞. Then the tent space T p q is defined as the space of functions where (x) is the standard cone whose vertex is x ∈ R d , i.e., is a Banach space. In fact, it is the completeness of T p 2 . Especially, T 1 2 = T 1 2 . In [19], the author proved the following result.
Then, for any f ∈ L 2 loc (R d ), we have: Therefore, in the harmonic analysis associated with L, the operators A j play the role of the classical partial derivatives ∂ x j in the Euclidean harmonic analysis (see [2,11,18]). Now, it is natural to consider the derivatives A i other than ∂ x j . In [13], the author defined the Lusin area integral operator by A j and characterized the Hardy space H 1 L (R d ). As a continuous study of the function spaces associated with L, in this paper we will define the Carleson measure by A j and characterize the dual spaces and predual spaces of Then the main results of this paper can be stated as follows.
Remark 2 In [8], the authors characterize the case p = 0, i.e., BMO L , by the heat semigroup with the classical derivatives. In [15], the authors characterize the space BMO L by the Poisson semigroup with the classical derivatives. In this paper, we will use the new derivatives A j of the Poisson semigroup to characterize the space L d(1/p-1) for d d+1 < p ≤ 1.
The paper is organized as follows. In Sect. 2, we give some estimates of the kernels. In Sect. 3, we give the proof of Theorem 1. The proofs of Theorem 2 will be given in Sect. 4.
Throughout the article, we will use A and C to denote the positive constants, which are independent of the main parameters and may be different at each occurrence. By B 1 ∼ B 2 , we mean that there exists a constant C > 1 such that 1

Estimates of the kernels
In this section, we give some estimates of the kernels, which we will use in the sequel. The proofs of these estimates can be found in [9].
( 2 ) ( By subordination formula, we can give the following estimates about the Poisson kernel. ( 4 ) Then, for any N > 0, there exist constants C > 0, C N > 0 such that Duong et al. [6] proved the following estimates about the kernel D L t (x, y).

Lemma 3
There exist constants C such that, for every N , there is a constant C N > 0, so that Let t = 1 2 ln 1+s 1-s , s ∈ (0, 1). Then The following estimations are very important for the proofs of the main result in this paper.

ρ(x)
and Lemma 1, we get Therefore, part (c) holds and this completes the proof of Proposition 4.
Lemma 4 and the subordination formula give the following.

Lemma 5
There is C > 0 for N ∈ N and |xx | ≤ |x-y| 2 , we can find C N > 0 such that

Carleson measure characterization of L α
Let s L denote the Littlewood-Paley g-function associated with L, i.e., , and A L denote the Lusin area integral associated with L, i.e., Then we can prove the following.

Lemma 6 The operators s L and A L are isometries on L 2 (R d ) up to constant factors. Exactly,
The proof of Lemma 6 is standard, we omit it.
. Then we have the following lemma.

Lemma 8 There exists C > 0 such that, for any H p,∞ L -atom a(x), we have A L a L p ≤ C.
The proofs of Lemmas 7 and 8 can be found in [8]. Now we can give the proof of Theorem 1.

Proof of Theorem
is absolutely convergent. To prove the assertion (a), we need to prove that, for any ball B = B(x 0 , r), Set B k = B(x 0 , 2 k r) and By Lemma 6, we have Note that For x ∈ B(x 0 , r), by Lemma 5(a) and (18), In the last step of the above, we use the facts d d+1 < p ≤ 1 to get d(1/p -1) -1 < 0. Thus we have It remains to estimate the constant term. Assume first that r < ρ(x 0 ). Taking k 0 such that .
. By Lemma 7, Lemma 8, and Proposition 3, we get This gives the proof of part (b) and then Theorem 1 is proved.

The predual space of Hardy space
In this section, we give a Carleson measure characterization of the space λ L d(1/p-1) (R d ).
Then, for any k ∈ N, we have By (22), we have .
We can take k 0 large enough such that 2 -k 0 /2 f 2 is small. This proves that Q L t f T p,∞ 2 < ∞ and η 1 (f ) = η 2 (f ) = η 3 (f ) = 0 follows from (23). Therefore Q L t f ∈ T p,∞ 2,0 . Now we give the proof of (22). Set B k = B(x 0 , 2 k r) and By Lemma 6, we have By For r ≥ ρ(x 0 ), we have Note that ρ(x) ≤ Cr for any x ∈ B(x 0 , r), again by Lemma 5(c), Then (22) follows from (24)-(27). For the reverse, by Theorem 1, we get f Let G(x) = (g(x)g(B))χ B . Then, by Lemma 7, we obtain By Hölder's inequality and Lemma 6, we have Now, we estimate E k . By Hölder's inequality again, we have that When r < ρ(x 0 ), then B g(x)g(B) dx = 0. Therefore, by Lemma 5(b), Therefore It follows that When r ≥ ρ(x 0 ), we have ρ(y) ≤ Cr for y ∈ B(x 0 , r). Then, by Lemma 5(a), Then we can get By (28) and (29), we know where Then we can get γ 1 (f ) = γ 2 (f ) = γ 3 (f ) = 0 as the proof of the first part of this theorem. Therefore f ∈ λ L α and the proof of Theorem 2 is completed.

Conclusions
This paper defines a new version of Carleson measure associated with Hermite operator, which is adapted to the operator L. Then, we characterize the dual spaces and predual spaces of the Hardy spaces H p L (R d ) associated with L. The main results of this paper are the central problems in harmonic analysis, which can be used in PED or geometry widely.

Funding
The research of the 1st author has been fully supported by the National Natural Science Foundation (Grant No. 11471018 and No. 11671031) of China.