Open Access

New real-variable characterizations of Hardy spaces associated with twisted convolution

Journal of Inequalities and Applications20152015:170

https://doi.org/10.1186/s13660-015-0687-3

Received: 2 February 2015

Accepted: 8 May 2015

Published: 28 May 2015

Abstract

In this paper, we give some new real-variables characterizations of the Hardy space associated with twisted convolution, including Poisson maximal function, area integral, and Littlewood-Paley g-function.

Keywords

twisted convolution Hardy space Poisson maximal function area integral Littlewood-Paley g-function

MSC

42B30 42B25 42B35

1 Introduction

In this paper, we consider the 2n linear differential operators
$$ Z_{j}=\frac{\partial}{\partial{z_{j}}}+\frac{1}{4} \bar{z}_{j},\qquad \bar{Z}_{j}=\frac{\partial}{\partial {\bar{z}_{j}}}- \frac{1}{4}z_{j} \quad \text{on } \mathbb{C}^{n}, j=1,2,\ldots,n. $$
(1)
Together with the identity they generate a Lie algebra \(h^{n}\) which is isomorphic to the \(2n+1\) dimensional Heisenberg algebra. The only nontrivial commutation relations are
$$ [Z_{j}, \bar{Z}_{j} ]=-\frac{1}{2}I, \quad j=1,2,\ldots,n. $$
(2)
The operator L defined by
$$L=-\frac{1}{2}\sum_{j=1}^{n} (Z_{j}\bar{Z}_{j}+\bar{Z}_{j}Z_{j} ) $$
is nonnegative, self-adjoint, and elliptic. Therefore it generates a diffusion semigroup \(\{T_{t}^{L}\}_{t>0}=\{e^{-tL}\}_{t>0}\). The operators in (1) generate a family of ‘twisted translations’ \(\tau_{w}\) on \(\mathbb{C}^{n}\) defined on measurable functions by
$$\begin{aligned} (\tau_{w}f ) (z) =&\exp \Biggl(\frac{1}{2}\sum _{j=1}^{n}(w_{j}z_{j}+ \bar{w}_{j}\bar{z}_{j}) \Biggr)f(z) \\ =&f(z+w)\exp \biggl(\frac{i}{2}\operatorname{Im}(z\cdot\bar{w}) \biggr). \end{aligned}$$
The ‘twisted convolution’ of two functions f and g on \(\mathbb{C}^{n}\) can now be defined as
$$\begin{aligned} (f\times g) (z) =&\int_{\mathbb{C}^{n}}f(w)\tau_{-w}g(z)\,dw \\ =&\int_{\mathbb{C}^{n}}f(z-w)g(w)\bar{\omega}(z,w)\,dw, \end{aligned}$$
where \(\omega(z,w)=\exp (\frac{i}{2}\operatorname{Im}(z\cdot\bar{w}) )\). More about twisted convolution can be found in [13].

In [4], the authors defined the Hardy space \(H_{L}^{1}(\mathbb{C}^{n})\) associated with a twisted convolution. They gave several characterizations of \(H_{L}^{1}(\mathbb{C}^{n})\) via maximal functions, the atomic decomposition, and the behavior of the local Riesz transform. As applications, the boundedness of Hömander multipliers on Hardy spaces is considered in [5]. The ‘twisted cancelation’ and Weyl multipliers were introduced for the first time in [6]. Recently, Huang and Wang [7] defined the Hardy space \(H_{L}^{p}(\mathbb{C}^{n})\) associated with a twisted convolution for \(\frac{2n}{2n+1}< p\leq1\). Huang gave the characterizations of the Hardy space associated with twisted convolution by the Lusin area integral function and the Littlewood-Paley function defined by the heat kernel in [8] and established the boundedness of the Weyl multiplier by these characterizations in [9]. Recently, Huang and Liu gave the molecular characterization of Hardy space associated with twisted convolution in [10]. The purpose of this paper is to give some new real-variable characterizations for \(H_{L}^{p}(\mathbb {C}^{n})\), including the Poisson maximal function, the Lusin area integral, and the Littlewood-Paley g-function defined by the Poisson kernel.

We first give some basic notations concerning \(H_{L}^{p}(\mathbb{C}^{n})\). Let \(\mathcal{B}\) denote the class of \(C^{\infty}\)-functions φ on \(\mathbb{C}^{n}\), supported on the ball \(B(0,1)\) such that \(\|\varphi\|_{\infty}\leq1\) and \(\|\nabla\varphi\|_{\infty}\leq 2\). For \(t>0\), let \(\varphi_{t}(z)=t^{-2n}\varphi(z/t)\). Given \(\sigma>0\), \(0<\sigma\leq+\infty\), and a tempered distribution f, define the grand maximal function
$$M_{\sigma}f(z)=\sup_{\varphi\in \mathcal{B}}\sup_{0< t< \sigma} \bigl\vert \varphi_{t}\times f(z)\bigr\vert . $$
Then the Hardy space \(H_{L}^{p}(\mathbb{C}^{n})\) can be defined by
$$H_{L}^{p}\bigl(\mathbb{C}^{n}\bigr)= \bigl\{ f\in \mathcal{S}'\bigl(\mathbb{C}^{n}\bigr): M_{\infty}f \in L^{p}\bigl(\mathbb{C}^{n}\bigr) \bigr\} . $$
For any \(f\in H_{L}^{p}(\mathbb{C}^{n})\), define \(\| f\|_{ H_{L}^{p}(\mathbb{C}^{n})}=\| M_{\infty}f\|_{L^{p}}\).

Definition 1

Let \(\frac{2n}{2n+1}< p\le1\le q\le\infty\) and \(p\neq q\). A function \(a(z)\) is a \(H_{L}^{p,q}\)-atom for the Hardy space \(H_{L}^{p}(\mathbb{C}^{n})\) associated to a ball \(B(z_{0},r)\) if
$$\begin{aligned} (1)&\quad \operatorname{supp} a\subset B(z_{0},r); \\ (2)& \quad \|a\|_{q}\leq\bigl\vert B(z_{0},r)\bigr\vert ^{1/q-1/p}; \\ (3)&\quad \int_{\mathbb{C}^{n}} a(w)\bar{\omega}(z_{0},w) \,dw=0. \end{aligned}$$

We define the atomic Hardy space \(H^{p,q}_{L}(\mathbb{C}^{n})\) to be the set of all tempered distributions of the form \(\sum_{j} \lambda_{j}a_{j}\) (the sum converges in the topology of \(\mathcal{S}'(\mathbb{C}^{n})\)), where \(a_{j}\) are \(H_{L}^{p,q}\)-atoms and \(\sum_{j}|\lambda_{j}|^{p}<+\infty\).

The atomic quasi-norm in \(H_{L}^{p,q}(\mathbb{C}^{n})\) is defined by
$$\|f\|_{L\text{-atom}}=\inf \biggl\{ \biggl(\sum_{j}| \lambda_{j}|^{p} \biggr)^{1/p} \biggr\} , $$
where the infimum is taken over all decompositions \(f=\sum_{j} \lambda_{j}a_{j}\) and \(a_{j}\) are \(H_{L}^{p,q}\)-atoms.

The following result has been proved in [4] and [7].

Proposition 1

Let \(\frac{2n}{2n+1}< p\leq1\). Then for a tempered distribution f on \(\mathbb{C}^{n}\), the following are equivalent:
  1. (i)

    \(M_{\infty}f \in L^{p}(\mathbb{C}^{n})\).

     
  2. (ii)

    For some σ, \(0<\sigma<+\infty\), \(M_{\sigma}f \in L^{p}(\mathbb{C}^{n})\).

     
  3. (iii)
    For some radial function \(\varphi\in\mathcal{S}\), such that \(\int_{\mathbb{C}^{n}}\varphi(z)\, dz\neq0\), we have
    $$\sup_{0< t< 1}\bigl\vert \varphi_{t}\times f(z)\bigr\vert \in L^{p}\bigl(\mathbb{C}^{n}\bigr). $$
     
  4. (iv)

    f can be decomposed as \(f=\sum_{j} \lambda_{j}a_{j}\), where \(a_{j}\) are \(H_{L}^{p,q}\)-atoms and \(\sum_{j}|\lambda_{j}|^{p}<+\infty\).

     

Corollary 1

Let \(\frac{2n}{2n+1}< p\leq1\) and \(1< q\le\infty\). Then \(H_{L}^{p,q}(\mathbb{C}^{n})=H^{p}_{L}(\mathbb{C}^{n})\) with equivalent norms.

Let \(\{P_{t}^{L}\}_{t>0}\) be the Poisson semigroup generated by the operator L. Then, for \(f\in L^{2}(\mathbb{C}^{n})\), the function \(e^{-t\sqrt{L}}f\) has the special Hermite expansion (cf. [11])
$$e^{-t\sqrt{L}}f(z)= (2\pi )^{-n}\sum_{k=0}^{\infty}e^{-\sqrt {2k+n}t}f\times \varphi_{k}(z), $$
where \(\varphi_{k}\) are Laguerre functions. Therefore \(e^{-t\sqrt{L}}f\) is given by the twisted convolution with the kernel
$$ P_{t}(z)= (2\pi )^{-n}\sum _{k=0}^{\infty}e^{-\sqrt{2k+n}t} \varphi_{k}(z). $$
(3)
The Poisson maximal function is defined by
$$M_{P}(f) (z)=\sup_{t>0}\bigl\vert P_{t}\times f(z)\bigr\vert . $$
We can characterize the Hardy space \(H_{L}^{1}(\mathbb{C}^{n})\) as follows.

Theorem 1

\(f\in H_{L}^{1}(\mathbb{C}^{n})\) if and only if \(f\in L^{1}(\mathbb{C}^{n})\) and \(M_{P}(f)\in L^{1}(\mathbb{C}^{n})\). Moreover, we have
$$\|f\|_{H_{L}^{1}}\sim\bigl\Vert M_{P}(f)\bigr\Vert _{L^{1}}. $$
We define the area integral associated to \(\{P_{t}^{L}\}_{t>0}\) by
$$\bigl(S_{L}^{k}f\bigr) (z)= \biggl(\int _{0}^{+\infty}\int_{|z-w|< t} \bigl\vert D_{t}^{k}f(w)\bigr\vert ^{2} \frac{dw\,dt}{t^{2n+1}} \biggr)^{1/2}, $$
the Littlewood-Paley g-function by
$$\mathcal{G}_{L}^{k}(f) (z)= \biggl(\int_{0}^{\infty} \bigl\vert D_{t}^{k}f(z)\bigr\vert ^{2} \frac{dt}{t} \biggr)^{1/2}, $$
and we consider the \(g_{\lambda}^{*}\)-function associated with L defined by
$$g_{\lambda,k}^{*}f(z)= \biggl(\int_{0}^{\infty}\int _{\mathbb {C}^{n}} \biggl(\frac{t}{t+|z-w|} \biggr)^{2\lambda n}\bigl\vert D_{t}^{k}f(w)\bigr\vert ^{2} \frac{dw\,dt}{t^{2n+1}} \biggr)^{1/2}, $$
where \(D_{t}^{k}f(z)=t^{k} (\partial_{t}^{k}P_{t}^{L}f )(z)\).

Now we can prove the main result of this paper.

Theorem 2

  1. (a)
    A function \(f\in H_{L}^{1}(\mathbb{C}^{n})\) if and only if its Lusin area integral \(S_{L}^{k}f\in L^{1}(\mathbb{C}^{n})\) and \(f\in L^{1}(\mathbb{C}^{n})\). Moreover, we have
    $$\|f\|_{H_{L}^{1}}\sim\bigl\Vert S_{L}^{k}f\bigr\Vert _{L^{1}}. $$
     
  2. (b)
    A function \(f\in H_{L}^{1}(\mathbb{C}^{n})\) if and only if its Littlewood-Paley g-function \(\mathcal{G}_{L}^{k}f\in L^{1}(\mathbb{C}^{n})\) and \(f\in L^{1}(\mathbb{C}^{n})\). Moreover, we have
    $$\|f\|_{H_{L}^{1}}\sim\bigl\Vert \mathcal{G}_{L}^{k}f \bigr\Vert _{L^{1}}. $$
     
  3. (c)
    A function \(f\in H_{L}^{1}(\mathbb{C}^{n})\) if and only if its \(g_{\lambda}^{*}\)-function \(g_{\lambda,k}^{*}f\in L^{1}(\mathbb{C}^{n})\) and \(f\in L^{1}(\mathbb{C}^{n})\), where \(\lambda>3\). Moreover, we have
    $$\|f\|_{H_{L}^{1}}\sim\bigl\Vert g_{\lambda,k}^{*}f\bigr\Vert _{L^{1}}. $$
     

Remark 1

In this paper, we just give the proofs of our results for \(p=1\). In fact, we can prove the case \(\frac{2n}{2n+1}< p<1\) under more conditions (such as that f vanishes weakly at infinity). The proofs of the case \(\frac{2n}{2n+1}< p<1\) are quite similar to the case \(p=1\), so we omit them.

Throughout the article, we will use C to denote a positive constant, which is independent of the main parameters and may be different at each occurrence. By \(B_{1} \sim B_{2}\), we mean that there exists a constant \(C>1\) such that \(\frac{1}{C} \leq \frac{B_{1}}{B_{2}}\leq C\).

2 Preliminaries

In this section, we give some preliminaries that we will use in the sequel.

Let \(K_{t}(z)\) be the heat kernel of \(\{T_{t}^{L}\}_{t>0}\). Then we can get (cf. [11])
$$ K_{t}(z)= (4\pi )^{-n} (\sinh t )^{-n}e^{-\frac{1}{4}|z|^{2}(\coth t)}. $$
(4)
It is easy to prove that the heat kernel \(K_{t}(z)\) has the following estimates (cf. [8]).

Lemma 1

There exists a positive constant \(C>0\) such that
  1. (i)

    \(|K_{t}(z) |\leq C t^{-n}e^{-C\frac{|z|^{2}}{t}}\);

     
  2. (ii)

    \(|\nabla K_{t}(z) |\leq C t^{-n-\frac{1}{2}}e^{-C\frac{|z|^{2}}{t}}\).

     
Let \(Q_{t}^{k}(z)\) be the twisted convolution kernel of \(Q_{t}^{k}=t^{2k}\partial_{s}^{k}T_{s}^{L}|_{s=t^{2}}\). Then
$$Q_{t}^{k}(z)=t^{2k}\partial_{s}^{k}K_{s}(z)|_{s=t^{2}}. $$
We have the following estimates [8].

Lemma 2

There exist constants \(C, C_{k}>0\) such that
  1. (i)

    \(|Q_{t}^{k}(z) |\leq C_{k} t^{-2n}e^{-C t^{-2}|z|^{2}}\);

     
  2. (ii)

    \(| \nabla Q_{t}^{k}(z) |\leq C_{k} t^{-2n-1}e^{-C t^{-2}|z|^{2}}\).

     

By the subordination formula, we can give the following estimates as regards the Poisson kernel.

Lemma 3

There exist constants \(C_{k}>0\), \(A>0\) such that
  1. (a)
    $$ 0< P_{t}(z)\leq C_{k}\frac{t}{(t^{2}+A|z|^{2})^{(2n+1)/2}}; $$
    (5)
     
  2. (b)
    $$ \bigl\vert \nabla P_{t}(z)\bigr\vert \leq C_{k}\frac{\sqrt{t}}{(t^{2}+A|z|^{2})^{(2n+1)/2}}. $$
    (6)
     

Lemma 4

Let \(D_{t}^{k}(z)\) be the integral kernel of the operator \(D_{t}^{k}\). Then there exist constants \(C_{k}>0\), \(A>0\), such that
  1. (a)
    $$ \bigl|D_{t}^{k}(z) \bigr|\leq C_{k}\frac {t}{(t^{2}+A|z|^{2})^{(2n+1)/2}}; $$
     
  2. (b)
    $$ \bigl\vert \nabla D_{t}^{k}(z)\bigr\vert \leq C_{k}\frac{\sqrt{t}}{(t^{2}+A|z|^{2})^{(2n+1)/2}}. $$
     

We also need some basic properties about the tent space (cf. [12]).

Let \(0< p<\infty\), and \(1\leq q\leq\infty\). Then the tent space \(T^{p}_{q}\) is defined as the space of functions f on \(\mathbb{C}^{n}\times\mathbb{R}^{+}\), so that
$$\biggl(\int_{\Gamma(z)}\bigl\vert f(w,t)\bigr\vert ^{q}\frac {dw\,dt}{t^{2n+1}} \biggr)^{1/q}\in L^{p}\bigl( \mathbb{C}^{n}\bigr), \quad \text{when } 1\leq q< \infty $$
and
$$\sup_{(w,t)\in\Gamma(z)}\bigl\vert f(w,t)\bigr\vert \in L^{p} \bigl(\mathbb{C}^{n}\bigr),\quad \text{when } q=\infty, $$
where \(\Gamma(z)\) is the standard cone whose vertex is \(z\in\mathbb{C}^{n}\), i.e.,
$$\Gamma(z)=\bigl\{ (w,t):|w-z|< t\bigr\} . $$
Assume \(B(z_{0},r)\) is a ball in \(\mathbb{C}^{n}\), its tent \(\hat{B}\) is defined by \(\hat{B}=\{(w,t):|w-z_{0}|\leq r-t\}\). A function \(a(z,t)\) supported in a tent \(\hat{B}\), B a ball in \(\mathbb{C}^{n}\), is said to be an atom in the tent space \(T^{p}_{q}\) if and only if it satisfies
$$\biggl(\int_{\hat{B}}\bigl\vert a(z,t)\bigr\vert ^{2}\frac{dz\,dt}{t} \biggr)^{1/2}\leq |B|^{1/2-1/p}. $$
The atomic decomposition of \(T^{p}_{q}\) is stated as follows.

Proposition 2

When \(0< p\leq1\), then for any \(f\in T^{p}_{2}\) can be written as \(f=\sum\lambda_{k}a_{k}\), where \(a_{k}\) are atoms and \(\sum|\lambda_{k}|^{p}\leq C\|f\|_{T^{p}_{2}}^{p}\).

3 The proofs of the main results

Let
$$M_{H}f(z)=\sup_{t>0}\bigl\vert K_{t} \times f(z)\bigr\vert ,\quad f\in L^{1}\bigl(\mathbb{C}^{n} \bigr) $$
be the heat maximal function. Then we can characterize \(H_{L}^{1}(\mathbb{C}^{n})\) by the maximal function \(M_{H}f\) as follows (cf. [4] or [8]).

Lemma 5

\(f\in H_{L}^{1}(\mathbb{C}^{n})\) if and only if \(M_{H}f\in L^{1}(\mathbb{C}^{n})\) and \(f\in L^{1}(\mathbb{C}^{n})\).

Now, we give the proof of Theorem 1.

Proof of Theorem 1

If \(f\in H_{L}^{1}(\mathbb{C}^{n})\), then, by Lemma 5, we get \(M_{H}f\in L^{1}(\mathbb{C}^{n})\). Since
$$P_{t}(z)=\frac{1}{\sqrt{\pi}}\int_{0}^{\infty}K_{t^{2}/4\mu }(z)e^{-\mu} \mu^{-1/2}\, d\mu, $$
we have \(\|M_{P}(f)\|_{L^{1}}\leq C \|M_{H}(f)\|_{L^{1}}\), i.e., \(M_{P}f\in L^{1}(\mathbb{C}^{n})\).
For the reverse, there exists a function η defined on \((1,\infty )\) that is rapidly decreasing at ∞ and satisfies the moment conditions (cf. [13])
$$\int_{1}^{\infty}\eta(t)\, dt=1, \qquad \int _{1}^{\infty}t^{k}\eta(t)\, dt=0,\quad k=1,2, \ldots. $$
Let
$$ \Phi(z)=\int_{1}^{\infty}\eta(t)P_{t}(z)\, dt. $$
(7)
Since
$$\bigl(1+s^{2}\bigr)^{-(2n+1)/2}=\sum_{k< R}a_{k}s^{k}+O \bigl(s^{R}\bigr),\quad 0\leq s< \infty $$
for appropriate binomial coefficients \(a_{k}\), we have
$$ \frac{t}{(t^{2}+A|z|^{2})^{(2n+1)/2}}=\sum_{k< R}a_{k}t|z|^{-1-2n} \biggl(\frac{t}{|z|} \biggr)^{k}+O\bigl(t^{R+1}|z|^{-2n-1-R} \bigr). $$
(8)
By (8) and Lemma 3, we know that Φ and any derivative of Φ are rapidly decreasing. Thus \(\Phi\in\mathcal{S}\) and
$$\int_{\mathbb{C}^{n}}\Phi(z)\, dz=\int_{1}^{\infty}\eta(t)\, dt=1. $$
Therefore,
$$M_{\Phi}(f) (z)\leq M_{P}(f) (z)\dot{\int}_{1}^{\infty}\bigl\vert \eta(t)\bigr\vert \, dt\leq C M_{P}(f) (z). $$
This proves that \(M_{P}(f)\in L^{1}(\mathbb{C}^{n})\) implies \(f\in H_{L}^{1}(\mathbb{C}^{n})\) and the proof of Theorem 1 is complete. □

In order to get our results, we need the following lemma (cf. Lemma 5 in [8]).

Lemma 6

  1. (i)
    The operators \(S_{L}^{k}\) and \(\mathcal{G}_{L}^{k}\) are isometries on \(L^{2}(\mathbb{C}^{n})\) up to constant factors. Exactly,
    $$ \bigl\Vert \mathcal{G}_{L}^{k} f \bigr\Vert _{L^{2}}\sim \| f \|_{L^{2}},\qquad \bigl\Vert S_{L}^{k} f \bigr\Vert _{L^{2}}\sim \| f \|_{L^{2}}. $$
     
  2. (ii)
    When \(\lambda>1\), there exists a constant \(C>0\), such that
    $$C^{-1}\|f\|_{L^{2}}\leq\bigl\Vert g_{\lambda,k}^{*}f\bigr\Vert _{L^{2}}\leq C\|f\|_{L^{2}}. $$
     
We define the new Lusin type area integral operator by
$$\bigl(S_{L,\alpha}^{k}f\bigr) (z)= \biggl(\int _{0}^{+\infty}\int_{|z-w|< \alpha t}\bigl\vert D_{t}^{k}f(w)\bigr\vert ^{2}\frac{dw\,dt}{t^{2n+1}} \biggr)^{1/2}, $$
where \(\alpha>0\).

Lemma 7

It is easy to see that the above definition of the area integral operator is independent of α in the sense of \(\|(S_{L}^{\alpha}f)\|_{L^{p}}\sim \|(S_{L}^{\beta}f)\|_{L^{p}}\), for \(0<\alpha<\beta<\infty\) and \(0< p<\infty\) (cf. [12]). In the following, we use \(S_{L}^{k}\) to denote \(S_{L,1}^{k}\).

Proof of Theorem 2

(a) By Lemma 4, we can prove that there exists a constant \(C>0\) such that for any atom \(a(z)\) of \(H_{L}^{1}(\mathbb{C}^{n})\), we have
$$ \bigl\Vert S_{L}^{k}a\bigr\Vert _{L^{1}}\leq C. $$
(9)
In the following, we will show that \(f\in H_{L}^{1}(\mathbb{C}^{n})\) when \(S_{L}^{k}f\in L^{1}(\mathbb{C}^{n})\) and \(f\in L^{1}(\mathbb{C}^{n})\).
We first assume that \(f\in L^{1}(\mathbb{C}^{n})\cap L^{2}(\mathbb{C}^{n})\). When \(S_{L}^{k}f\in L^{1}(\mathbb{C}^{n})\), we know \(D_{t}^{k}f\in T_{2}^{1}\). By Proposition 2, we get
$$ D_{t}^{k}f(z)=\sum _{j}\lambda_{j}a_{j}(z,t), $$
(10)
where \(a_{j}(z,t)\) are atoms of \(T^{1}_{2}\) and \(\sum_{j}|\lambda_{j}|<\infty\). By the spectrum theorem (cf. [14]), we can prove
$$ f(z)=4\int_{0}^{\infty}D_{t}^{k} \bigl(D_{t}^{k}f(z) \bigr)\frac{dt}{t}. $$
(11)
By (10) and (11), we get
$$f(z)=4\int_{0}^{+\infty} D_{t}^{k} \biggl( \sum_{j}\lambda_{j}a_{j}(z,t) \biggr)\frac{dt}{t} =C\sum_{j} \lambda_{j}\int_{0}^{+\infty}D_{t}^{k}a_{j}(z,t) \frac{dt}{t}. $$
Therefore, it is sufficient to prove \(\alpha_{j}=\int_{0}^{+\infty}D_{t}^{k}a_{j}(z,t)\frac{dt}{t}\), \(i=1,2, \ldots\) , are bounded in \(H_{L}^{1}(\mathbb{C}^{n})\) uniformly, i.e., there exists a constant \(C>0\) such that for any atom \(a(z,t)\) in \(T^{1}_{2}\),
$$\|\alpha\|_{H_{L}^{1}}= \biggl\Vert \int_{0}^{+\infty }D_{t}^{k}a(z,t) \frac{dt}{t}\biggr\Vert _{H_{L}^{1}}\leq C. $$
We assume that \(a(z,t)\) is supported in \(\hat{B}(z_{0},r)\), where \(\hat{B}(z_{0},r)\) denotes the tent of the ball \(B(z_{0},r)\), then
$$\Bigl\Vert \sup_{t>0}\bigl\vert e^{-t\sqrt{L}}\alpha(z) \bigr\vert \Bigr\Vert _{L^{1}} \leq\Bigl\Vert \Bigl(\sup _{t>0}\bigl\vert e^{-t\sqrt{L}}\alpha(z)\bigr\vert \Bigr) \chi_{B^{*}}\Bigr\Vert _{L^{1}} +\Bigl\Vert \Bigl(\sup _{t>0}\bigl\vert e^{-t\sqrt{L}}\alpha(z)\bigr\vert \Bigr) \chi_{(B^{*})^{c}}\Bigr\Vert _{L^{1}} =I_{1}+I_{2}, $$
where \(B^{*}=B(z_{0}, 2r)\).
By the Hölder inequality, we get
$$I_{1}\leq\bigl\vert B^{*}\bigr\vert ^{1/2} \biggl(\int_{\mathbb{C}^{n}}\Bigl(\sup_{t>0}\bigl\vert e^{-t\sqrt{L}}\alpha(z)\bigr\vert \Bigr)^{2}\, dz \biggr)^{1/2} \leq\bigl\vert B^{*}\bigr\vert ^{1/2}\|\alpha\|_{L^{2}}. $$
By the self-adjointness of \(D_{t}^{k}\) and Lemma 5, we can get
$$\begin{aligned} \|\alpha\|_{L^{2}} =&\sup_{\|\beta\|_{L^{2}}\leq1}\int _{\mathbb{C}^{n}}\alpha(z)\bar{\beta}(z)\, dz \\ =&\sup_{\|\beta\|_{L^{2}}\leq1}\int_{\mathbb{C}^{n}} \biggl(\int _{0}^{+\infty}D_{t}^{k}a(z,t) \frac{dt}{t} \biggr)\bar {\beta}(z)\, dz \\ =&\sup_{\|\beta\|_{L^{2}}\leq1}\int_{0}^{+\infty}\int _{\mathbb{C}^{n}}D_{t}^{k}a(z,t)\bar{\beta}(z)\, dz \frac{dt}{t} \\ =&\sup_{\|\beta\|_{L^{2}}\leq1}\int_{0}^{+\infty}\int _{\mathbb{C}^{n}}a(z,t)D_{t}^{k}\bar{\beta}(z)\, dz \frac{dt}{t} \\ \leq&\sup_{\|\beta\|_{L^{2}}\leq1} \biggl(\int_{\mathbb{C}^{n}}\int _{0}^{+\infty}\bigl\vert a(z,t)\bigr\vert ^{2}\frac{dz\, dt}{t} \biggr)^{1/2} \\ &{} \times \biggl(\int_{\mathbb{C}^{n}} \int_{0}^{+\infty} \bigl\vert D_{t}^{k}\bar{\beta}(z)\bigr\vert ^{2}\frac{dz\, dt}{t} \biggr)^{1/2} \\ \leq&|B|^{-1/2}\|\beta\|_{L^{2}}\leq|B|^{-1/2}. \end{aligned}$$
This gives the proof of \(I_{1}\leq C\).
By Lemma 2, we can prove
$$\begin{aligned}& \sup_{s>0} \biggl\vert e^{-s\sqrt{L}}\int _{0}^{+\infty }D_{t}^{k}a(z,t) \frac{dt}{t} \biggr\vert \\& \quad = \sup_{s>0} \biggl\vert e^{-s\sqrt{L}}\int _{0}^{+\infty}(-t\sqrt {L})^{k}e^{-t\sqrt{L}}a(z,t) \frac{dt}{t} \biggr\vert \\& \quad = \sup_{s>0} \biggl\vert \int_{0}^{+\infty}(-t \sqrt{L})^{k}e^{-(s+t)\sqrt{L}}a(z,t)\frac{dt}{t} \biggr\vert \\& \quad = \sup_{s>0} \biggl\vert \int_{0}^{+\infty} \biggl(\frac{t}{s+t} \biggr)^{k} \bigl(-(s+t) \sqrt{L} \bigr)^{k}e^{-(s+t)\sqrt{L}}a(z,t)\frac{dt}{t} \biggr\vert \\& \quad = \sup_{s>0} \biggl\vert \int_{0}^{+\infty} \biggl(\frac{t}{s+t} \biggr)^{k} \int_{\mathbb{C}^{n}}D_{s+t}^{k}(z-w)a(w,t) \frac{dw\, dt}{t} \biggr\vert \\& \quad \leq \sup_{s>0}\int_{0}^{+\infty} \frac{t}{s+t} \int_{\mathbb{C}^{n}}\frac{s+t}{((s+t)^{2}+A|z-w|^{2})^{(2n+1)/2}}\bigl\vert a(w,t)\bigr\vert \frac{dw \, dt}{t} \\& \quad \leq \sup_{s>0} \biggl(\int_{0}^{r} \int_{B}(s+t)^{-4n} \biggl(1+A\frac{|z-w|^{2}}{(s+t)^{2}} \biggr)^{-(2n+1)} \biggl(\frac{t}{s+t} \biggr)^{2} \frac{dw \, dt}{t} \biggr)^{1/2} \\& \qquad {} \times \biggl(\int_{0}^{r}\int _{B}\bigl\vert a(w,t)\bigr\vert ^{2} \frac{dw\, dt}{t} \biggr)^{1/2} \\& \quad \leq |B|^{-1/2}|z-z_{0}|^{-(2n+1)} \biggl(\int _{0}^{r}\int_{B}t\, dw\, dt \biggr)^{1/2} \\& \quad \leq Cr|z-z_{0}|^{-(2n+1)}. \end{aligned}$$
Then we get
$$I_{2}\leq C r\int_{(B^{*})^{c}}|z-z_{0}|^{-(2n+1)} \, dz\leq C. $$
When \(f\in L^{1}(\mathbb{C}^{n})\), we can proceed similarly to Proposition 14 in [15]. In fact, we let \(f_{s}=T_{2^{-s}}^{L}f\), \(s\geq0\). Then, by \(f\in L^{1}(\mathbb{C}^{n})\) and Lemma 3, we know \(f_{s}\in L^{2}(\mathbb{C}^{n})\) and \(\|S_{L}^{k}f_{s}\|_{1}\leq\|S_{L}^{k}f\|_{1}\). By the above proof, we get
$$\|f_{s} \|_{H_{L}^{1}(\mathbb{C}^{n})}\lesssim \bigl\Vert S_{L}^{k}f_{s} \bigr\Vert _{L^{1}}\leq \bigl\Vert S_{L}^{k}f \bigr\Vert _{L^{1}}. $$
By the monotone convergence theorem, we have
$$\|f_{s}-f_{n} \|_{H_{L}^{1}}\leq \bigl\Vert S_{L}^{k} (f_{s}-f_{n}) \bigr\Vert _{L^{1}}\rightarrow0,\quad \text{when } s,n\rightarrow +\infty. $$
Therefore, \(\{f_{s}\}\) is a Cauchy sequence in \(H^{1}_{L}(\mathbb{C}^{n})\) and there exists \(g\in H^{1}_{L}(\mathbb{C}^{n})\) such that
$$\lim_{s\rightarrow+\infty}f_{s}=g \quad \text{in } H^{1}_{L}\bigl(\mathbb{C}^{n}\bigr). $$
As
$$\lim_{s\rightarrow+\infty}f_{s}=f \quad \text{in } ( \mathit{BMO}_{L})^{*}, $$
we know \(f=g\in H^{1}_{L}(\mathbb{C}^{n})\) and \(\|f\|_{H_{L}^{1}(\mathbb{C}^{n})}\lesssim\|S_{L}^{k}f\|_{L^{1}}\).

This gives the proof of Theorem 2(a).

(b) Firstly, by Lemma 4, we can prove that there exists a positive constant C such that for any atom \(a(z)\) of \(H_{L}^{1}(\mathbb{C}^{n})\), we have
$$\bigl\Vert \mathcal{G}_{L}^{k}a\bigr\Vert _{L^{1}}\leq C. $$
For the reverse, by (a), it is sufficient to prove
$$ \bigl\Vert S_{L}^{k+1}f\bigr\Vert _{L^{1}}\leq C \bigl\Vert \mathcal{G}_{L}^{k}f\bigr\Vert _{L^{1}}. $$
(12)
Our proof is motivated by [16]. Let
$$F(z) (t)= \bigl(\partial_{t}^{k}e^{-t\sqrt{L}}f \bigr) (z),\qquad V(z,s)=e^{-s\sqrt{L}}F(z). $$
Then
$$V(z,s) (t)=e^{-s\sqrt{L}} \bigl(\partial_{t}^{k}e^{-t\sqrt{L}}f \bigr) (z)= \bigl(\partial_{t}^{k}e^{-(s+t)\sqrt{L}}f \bigr) (z). $$
Therefore
$$\begin{aligned} \int_{0}^{+\infty} \bigl\vert V(z,s) (t) \bigr\vert ^{2}t^{2k-1} \, dt =&\int_{0}^{+\infty} \bigl\vert \bigl(\partial_{t}^{k}e^{-(s+t)\sqrt {L}}f \bigr) (z) \bigr\vert ^{2}t^{2k-1}\, dt \\ =&\int_{s}^{+\infty} \bigl\vert \bigl( \partial_{t}^{k}e^{-t\sqrt{L}}f \bigr) (z) \bigr\vert ^{2}(t-s)^{2k-1} \, dt. \end{aligned}$$
Hence
$$\sup_{s>0} \int_{0}^{+\infty} \bigl\vert V(z,s) (t) \bigr\vert ^{2}t^{2k-1}\, dt \leq\int _{0}^{+\infty} \bigl\vert \bigl(t^{k} \partial_{t}^{k}e^{-t\sqrt {L}}f \bigr) (z) \bigr\vert ^{2}\frac{dt}{t} = \bigl(\mathcal{G}_{L}^{k}f(z) \bigr)^{2}. $$
Let \(\mathbf{X}=L^{2} ((0,\infty), t^{2k-1}\, dt )\). Then
$$\sup_{s>0} \bigl\Vert e^{-s\sqrt{L}}F(z) \bigr\Vert _{\mathbf{X}}=\mathcal {G}_{L}^{k}f(z)\in L^{1} \bigl(\mathbb{C}^{n}\bigr). $$
Therefore \(F\in H_{\mathbf{X}}^{1}(\mathbb{C}^{n})\), here \(H_{\mathbf{X}}^{1}(\mathbb{C}^{n})\) can be seen as a vector-valued Hardy space. This shows that \(\widetilde{S_{L}^{1}}F(z)\in L^{1}(\mathbb{C}^{n})\), where
$$\widetilde{ S_{L}^{1}}F(z)= \biggl(\int _{0}^{+\infty}\int_{|z-w|< 2t} \bigl\Vert D_{t}^{1}F(w) \bigr\Vert _{\mathbf{X}}^{2} \frac{dw\, dt}{t^{2n+1}} \biggr)^{1/2}. $$
By
$$\begin{aligned} \bigl(S_{L}^{1}F(z) \bigr)^{2} =&\int _{0}^{+\infty}\int_{|z-w|< 2t} \bigl\Vert D_{t}^{1}(z) \bigr\Vert _{\mathbf{X}}^{2} \frac{dw\, dt}{t^{2n+1}} \\ =&\int_{0}^{+\infty}\int_{\vert z-w\vert < 2t} \int_{0}^{+\infty} \bigl\vert (-t\sqrt{L})e^{-t\sqrt{L}}F(w) (s) \bigr\vert ^{2}s^{2k-1}\, ds\frac{dw\, dt}{t^{2n+1}} \\ =&\int_{0}^{+\infty} \int_{0}^{+\infty} \int_{\vert z-w\vert < 2t} \bigl\vert (-\sqrt{L})^{k+1}e^{-(s+t)\sqrt{L}}f(w) \bigr\vert ^{2}t^{1-2n}s^{2k-1}\, dw\, dt\, ds \\ =&\int_{0}^{+\infty}\int_{s}^{+\infty} \int_{\vert z-w\vert < 2(t-s)} \bigl\vert (-\sqrt{L})^{k+1}e^{-t\sqrt{L}}f(w) \bigr\vert ^{2}(t-s)^{1-2n}s^{2k-1}\, dw\,dt\,ds \\ =&\int_{0}^{+\infty}\int_{0}^{t} \int_{\vert z-w\vert < 2(t-s)} \bigl\vert (-\sqrt{L})^{k+1}e^{-t\sqrt{L}}f(w) \bigr\vert ^{2}(t-s)^{1-2n}s^{2k-1}\, dw\,ds\,dt \\ \geq&\int_{0}^{+\infty}\int_{0}^{t/2} \int_{\vert z-w\vert < 2(t-s)} \bigl\vert (-\sqrt{L})^{k+1}e^{-t\sqrt{L}}f(w) \bigr\vert ^{2}(t-s)^{1-2n}s^{2k-1}\, dw\,ds\,dt \\ \geq&\int_{0}^{+\infty}\int_{0}^{t/2} \int_{\vert z-w\vert < t} \bigl\vert (-\sqrt{L})^{k+1}e^{-t\sqrt{L}}f(w) \bigr\vert ^{2}t^{1-2n}s^{2k-1}\, dw\,ds\,dt \\ =&\frac{1}{2k2^{2k}}\int_{0}^{+\infty}\int _{\vert z-w\vert < t} \bigl\vert (-t\sqrt {L})^{k+1}e^{-t\sqrt{L}}f(w) \bigr\vert ^{2} t^{-1-2n}\, dw\,dt \\ =&\frac{1}{2k2^{2k}}\int_{0}^{+\infty}\int _{\vert z-w\vert < t} \bigl\vert D_{t}^{k+1}f(w) \bigr\vert ^{2}\frac{dw\, dt}{t^{2n+1}}=\frac{1}{2k2^{2k}} \bigl(S_{L}^{k+1}f(z) \bigr)^{2}, \end{aligned}$$
we get \(S_{L}^{k+1}f\in L^{1}(\mathbb{C}^{n})\). Then \(f\in H_{L}^{1}(\mathbb{C}^{n})\) follows from (a).

This completes the proof of Theorem 2(b).

(c) By \(S_{L}^{k}f(z)\leq (\frac{1}{2} )^{2\lambda n}g_{\lambda,k}^{*}f(z)\), we know \(f\in H_{L}^{1}(\mathbb{C}^{n})\) when \(g_{\lambda,k}^{*}f\in L^{1}(\mathbb{C}^{n})\) and \(f\in L^{1}(\mathbb{C}^{n})\). In the following, we show there exists a constant \(C>0\) such that for any atom \(a(z)\) of \(H^{1}_{L}(\mathbb{C}^{n})\), we have
$$\bigl\Vert g_{\lambda,k}^{*}a \bigr\Vert _{L^{1}}\leq C. $$
Without loss of generality, we may assume \(a(z)\) is supported in \(B(0,r)\), then
$$\begin{aligned} g_{\lambda,k}^{*}a(z)^{2} =&\int_{0}^{\infty} \int_{\mathbb{C}^{n}} \biggl(\frac{t}{t+|z-w|} \biggr)^{2\lambda n} \bigl\vert D_{t}^{k}a(w) \bigr\vert ^{2} \frac{dw \, dt}{t^{2n+1}} \\ =&\int_{0}^{\infty} \int_{|z-w|< t} \biggl(\frac{t}{t+|z-w|} \biggr)^{2\lambda n} \bigl\vert D_{t}^{k}a(w) \bigr\vert ^{2}\frac{dw\, dt}{t^{2n+1}} \\ &{}+\sum_{i=1}^{\infty}\int _{0}^{\infty} \int_{2^{i-1}t\leq|z-w|< 2^{i}t} \biggl( \frac{t}{t+|z-w|} \biggr)^{2\lambda n} \bigl\vert D_{t}^{k}a(w) \bigr\vert ^{2}\frac{dw\, dt}{t^{2n+1}} \\ \leq& C S^{1}_{L}a(z)^{2}+\sum _{i=1}^{\infty}2^{-2i\lambda n}S^{k}_{L,2^{i}}a(z)^{2}. \end{aligned}$$
Therefore,
$$\bigl\Vert g_{\lambda,k}^{*}a \bigr\Vert _{L^{1}}\leq C \bigl\Vert S_{L}^{1}a \bigr\Vert _{L^{1}}+ \sum _{i=1}^{\infty}2^{-i\lambda n} \bigl\Vert S^{k}_{L,2^{i}}a \bigr\Vert _{L^{1}}. $$
By part (a), we have \(\|S_{L}^{k}a\|_{L^{1}}\leq C\). In the following, we will prove that
$$ \bigl\Vert S^{k}_{L,2^{i}}a\bigr\Vert _{L^{1}}\leq C 2^{3in}. $$
(13)
First, by Lemma 5, we can obtain
$$ \bigl\Vert S^{k}_{L,2^{i}}a \bigr\Vert _{L^{1}(B(0, 2^{i+2}r))}\leq \bigl\vert B\bigl(0, 2^{i+2}r\bigr) \bigr\vert ^{1/2} \bigl\Vert S^{k}_{L,2^{i}}a \bigr\Vert _{L^{2}} \leq C 2^{2i n}. $$
(14)
Let \(z\notin B(0, 2^{i+2}r)\). We have
$$\begin{aligned} S^{k}_{L,2^{i}} a(z)^{2} \leq& \int _{0}^{\infty}\int_{|z-w|< 2^{i}t} \biggl( \int_{B(0,r)} \bigl\vert D^{k}_{t}(w-v)-D^{k}_{t}(w) \bigr\vert \bigl\vert a(v) \bigr\vert \, dv \biggr)^{2} \frac{dw \, dt}{t^{2n+1}} \\ \leq& \int^{\frac{|z|}{2^{i+1}}}_{0} \int_{|z-w|< 2^{i}t} ( \cdots )^{2} \frac{dw\, dt}{t^{2n+1}} + \int^{\infty}_{\frac{|z|}{2^{i+1}}} \int_{|z-w|< 2^{i}t} ( \cdots )^{2} \frac{dw\, dt}{t^{2n+1}} \\ =&I_{1}+I_{2}. \end{aligned}$$
For \(z \notin B(0,2^{i+2}r)\), when \(|z-w|< 2^{i}t \leq\frac{|z|}{2}\), we have \(|w| \sim|z|\). By Lemma 4, we get
$$\begin{aligned} I_{1} \leq& C \int^{\frac{|z|}{2^{i+1}}}_{0} \int _{|z-w|< 2^{i} t} \biggl( \int_{B(0,r)} \frac{\sqrt{t}}{(t^{2}+A|w|^{2})^{(2n+1)/2}}|v| \bigl\vert a(v) \bigr\vert \, dv \biggr)^{2} \frac{dw\, dt}{t^{2n+1}} \\ \leq& C 2^{2i n} \int^{\frac{|z|}{2^{i+1}}}_{0} t^{-4n} \biggl( \frac{|z|}{t} \biggr)^{-(4n+3)} \biggl( \frac{r}{t} \biggr)^{2} \frac{dt}{t} \leq C 2^{2i n-i}\frac{ r^{2}}{|z|^{4n+2}}. \end{aligned}$$
By Lemma 4 again, we get
$$\begin{aligned} I_{2} \leq& C \int^{\infty}_{\frac{|z|}{2^{i+1}}} \int _{|z-w|< 2^{i}t} \biggl( \int_{B(0,r)} t^{-2n} \biggl( \frac{r}{t} \biggr) \bigl\vert a(v) \bigr\vert \, dv \biggr)^{2} \frac{dw\, dt}{t^{2n+1}} \\ \leq& C 2^{2i n}\int^{\infty}_{\frac{|z|}{2^{i+1}}} t^{-4n} \biggl( \frac{r}{t} \biggr)^{2} \frac{dt}{t} \leq C 2^{i(6n+2)} \frac{ r^{2}}{|z|^{2(2n+1)}}. \end{aligned}$$
Thus,
$$ \int_{|z| \geq2^{i+2}r} \bigl\vert S^{k}_{L,2^{i}} a(z)\bigr\vert \, dz \leq C 2^{3i n+i} \int_{|z| \geq 2^{i+2}r} \frac{r}{|z|^{2n+1}}\, dz\leq C 2^{3i n}. $$
(15)
Therefore, when \(\lambda>3\), we prove \(\|g_{\lambda,k}^{*}a \| _{L^{1}}\leq C\). Then Theorem 2(c) is proved. □

Declarations

Acknowledgements

This paper is supported by National Natural Science Foundation of China (11471018), the Beijing Natural Science Foundation (1142005).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
College of Sciences, North China University of Technology

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