The multi-parameter Hausdorff operators on $H^1$ and $L^p$

TL;DR

This paper characterizes functions for which multi-parameter Hausdorff operators are bounded on Hardy and Lebesgue spaces, providing operator norms and improving previous results while answering an open question.

Contribution

It offers a complete characterization of boundedness for multi-parameter Hausdorff operators on Hardy and Lebesgue spaces, advancing the theoretical understanding.

Findings

Characterization of functions for boundedness of Hausdorff operators

Explicit operator norms derived

Improvement over previous results and resolution of an open question

Abstract

In the present paper, we characterize the nonnegative functions $\varphi$ for which the multi-parameter Hausdorff operator $\mathcal H_\varphi$ generated by $\varphi$ is bounded on the multi-parameter Hardy space $H^1(\mathbb R\times\cdots\times\mathbb R)$ or $L^p(\mathbb R^n)$, $p\in [1,\infty]$. The corresponding operator norms are also obtained. Our results improve some recent results in \cite{FZ, LM2, LM3, We} and give an answer to an open question posted by Liflyand \cite{Li}.

Equations

$${\mathcal{H}}_{\phi }f(x)={\int }_0^{\infty }f\left(\frac{x}{t}\right)\frac{\phi (t)}{t}dt.$$

$${\mathcal{H}}_{\phi }f(x_1,\dots ,x_n)={\int }_0^{\infty }\cdots {\int }_0^{\infty }f\left(\frac{x_1}{t_1},\dots ,\frac{x_n}{t_n}\right)\frac{\phi (t_1,\dots ,t_n)}{t_1\dots t_n}d{t}_1\cdots d{t}_n.$$

$${\otimes }_{j=1}^n{\Phi }_{t_j}^{(j)}(\mathbf{x}):=\prod _{j=1}^n\frac{1}{t_j}{\Phi }^{(j)}\left(\frac{x_j}{t_j}\right),\mathbf{x}=(x_1,\dots ,x_n)\in {\mathbb{R}}^n.$$

$$\Vert f{\Vert }_{H^1(\mathbb{R}\times \cdots \times \mathbb{R})}:={\Vert M_{\Phi }f\Vert }_{L^1(\mathbb{R})}<\infty ,$$

$$M_{\Phi }f(\mathbf{x})=\underset{(t_1,\dots ,t_n)\in (0,\infty {)}^n}{\sup }|f\ast ({\otimes }_{j=1}^n{\Phi }_{t_j}^{(j)})(\mathbf{x})|,\mathbf{x}\in {\mathbb{R}}^n.$$

$$f\otimes \cdots \otimes f(\mathbf{x})=\prod _{j=1}^{n}f(x_j),\mathbf{x}=(x_1,\dots ,x_n)\in {\mathbb{R}}^n,$$

$$C_1\Vert f{\Vert }_{H^1(\mathbb{R})}^n\le {\Vert f\otimes \cdots \otimes f\Vert }_{H^1(\mathbb{R}\times \cdots \times \mathbb{R})}\le C_2\Vert f{\Vert }_{H^1(\mathbb{R})}^n.$$

$${\int }_0^{\infty }\cdots {\int }_0^{\infty }\phi (t_1,\dots ,t_n)d{t}_1\cdots d{t}_n<\infty .$$

$$\Vert {\mathcal{H}}_{\phi }{\Vert }_{H^1(\mathbb{R}\times \cdots \times \mathbb{R})\to H^1(\mathbb{R}\times \cdots \times \mathbb{R})}={\int }_0^{\infty }\cdots {\int }_0^{\infty }\phi (t_1,\dots ,t_n)d{t}_1\cdots d{t}_n.$$

$$\Vert f{\Vert }_{H^1(\mathbb{R}\times \cdots \times \mathbb{R})}:=\sum _{\mathbf{e}\in \{0,1{\}}^n}\Vert {\mathbf{H}}_{\mathbf{e}}f{\Vert }_{L^1({\mathbb{R}}^n)},$$

$${\int }_0^{\infty }\cdots {\int }_0^{\infty }\frac{\phi (t_1,\dots ,t_n)}{t_1^{1-1/p}\dots t_n^{1-1/p}}d{t}_1\cdots d{t}_n<\infty .$$

$$\Vert {\mathcal{H}}_{\phi }{\Vert }_{L^p({\mathbb{R}}^n)\to L^p({\mathbb{R}}^n)}={\int }_0^{\infty }\cdots {\int }_0^{\infty }\frac{\phi (t_1,\dots ,t_n)}{t_1^{1-1/p}\dots t_n^{1-1/p}}d{t}_1\cdots d{t}_n.$$

$${\mathcal{H}}_{\phi }^{\ast }f(x_1,\dots ,x_n)={\int }_0^{\infty }\cdots {\int }_0^{\infty }f(t_1{x}_1,\dots ,t_n{x}_n)\phi (t_1,\dots ,t_n)d{t}_1\dots d{t}_n.$$

$${\int }_0^{\infty }\cdots {\int }_0^{\infty }\frac{\phi (t_1,\dots ,t_n)}{t_1\dots t_n}d{t}_1\cdots d{t}_n<\infty .$$

$$\Vert {\mathcal{H}}_{\phi }^{\ast }{\Vert }_{H^1(\mathbb{R}\times \cdots \times \mathbb{R})\to H^1(\mathbb{R}\times \cdots \times \mathbb{R})}={\int }_0^{\infty }\cdots {\int }_0^{\infty }\frac{\phi (t_1,\dots ,t_n)}{t_1\dots t_n}d{t}_1\cdots d{t}_n.$$

$${\int }_0^{\infty }\cdots {\int }_0^{\infty }\frac{\phi (t_1,\dots ,t_n)}{t_1^{1/p}\dots t_n^{1/p}}d{t}_1\cdots d{t}_n<\infty .$$

$$\Vert {\mathcal{H}}_{\phi }^{\ast }{\Vert }_{L^p({\mathbb{R}}^n)\to L^p({\mathbb{R}}^n)}={\int }_0^{\infty }\cdots {\int }_0^{\infty }\frac{\phi (t_1,\dots ,t_n)}{t_1^{1/p}\dots t_n^{1/p}}d{t}_1\cdots d{t}_n.$$

$${\int }_{{\mathbb{R}}^n}{\mathcal{H}}_{\phi }f(\mathbf{x})g(\mathbf{x})d\mathbf{x}={\int }_{{\mathbb{R}}^n}f(\mathbf{x}){\mathcal{H}}_{\phi }^{\ast }g(\mathbf{x})d\mathbf{x}.$$

$${\int }_{{\mathbb{R}}^n}{\mathcal{H}}_{\phi }^{\ast }f(\mathbf{x})g(\mathbf{x})d\mathbf{x}={\int }_{{\mathbb{R}}^n}f(\mathbf{x}){\mathcal{H}}_{\phi }g(\mathbf{x})d\mathbf{x}.$$

$$\widehat{{\mathcal{H}}_{\phi }f}={\mathcal{H}}_{\phi }^{\ast }\hat{f}.$$

$$\widehat{{\mathcal{H}}_{\phi }^{\ast }f}={\mathcal{H}}_{\phi }\hat{f}.$$

$$\widehat{{\mathcal{H}}_{\phi }f}(\mathbf{y})$$

$$\Vert {\mathcal{H}}_{\phi }f{\Vert }_{L^p({\mathbb{R}}^n)}$$

$$\Vert {\mathcal{H}}_{\phi }{\Vert }_{L^p({\mathbb{R}}^n)\to L^p({\mathbb{R}}^n)}\le {\int }_0^{\infty }\cdots {\int }_0^{\infty }\frac{\phi (t_1,\dots ,t_n)}{t_1^{1-1/p}\dots t_n^{1-1/p}}d{t}_1\cdots d{t}_n.$$

$$f_{\epsilon }(\mathbf{x})=\prod _{j=1}^n|x_j{|}^{-1/p-\epsilon }{\chi }_{\{y_j\in \mathbb{R}:|y_j|\ge 1\}}(x_j)$$

$${\mathcal{H}}_{\phi }{f}_{\epsilon }(\mathbf{x})=\prod _{j=1}^n|x_j{|}^{-1/p-\epsilon }{\int }_0^{|x_1|}d{t}_1\cdots {\int }_0^{|x_{n-1}|}d{t}_{n-1}{\int }_0^{|x_n|}\frac{\phi (t_1,\dots ,t_n)}{t_1^{1-1/p-\epsilon }\dots t_n^{1-1/p-\epsilon }}d{t}_n$$

$$\Vert {\mathcal{H}}_{\phi }{f}_{\epsilon }{\Vert }_{L^p({\mathbb{R}}^n)}$$

$$\Vert {\mathcal{H}}_{\phi }{\Vert }_{L^p({\mathbb{R}}^n)\to L^p({\mathbb{R}}^n)}$$

$$\Vert {\mathcal{H}}_{\phi }{\Vert }_{L^p({\mathbb{R}}^n)\to L^p({\mathbb{R}}^n)}\ge {\int }_0^{\infty }\cdots {\int }_0^{\infty }\frac{\phi (t_1,\dots ,t_n)}{t_1^{1-1/p}\dots t_n^{1-1/p}}d{t}_1\cdots d{t}_n.$$

$$\Vert {\mathcal{H}}_{\phi }{\Vert }_{L^p({\mathbb{R}}^n)\to L^p({\mathbb{R}}^n)}={\int }_0^{\infty }\cdots {\int }_0^{\infty }\frac{\phi (t_1,\dots ,t_n)}{t_1^{1-1/p}\dots t_n^{1-1/p}}d{t}_1\cdots d{t}_n,$$

Full text

This work is supported by Vietnam National Foundation for Science and Technology Development (Grant No. 101.02-2017.304)

∗Corresponding author

1. Introduction and main result

Let $\varphi$ be a locally integrable function on $(0,\infty)$. The classical one-parameter Hausdorff operator ${\mathcal{H}}_{\varphi}$ is defined for suitable functions $f$ on ${\mathbb{R}}$ by

[TABLE]

The Hausdorff operator ${\mathcal{H}}_{\varphi}$ is an interesting operator in harmonic analysis. There are many classical operators in analysis which are special cases of the Hausdorff operator if one chooses suitable kernel functions $\varphi$, such as the classical Hardy operator, its adjoint operator, the Cesàro type operators, the Riemann-Liouville fractional integral operator. See the survey article [13] and the references therein. In the recent years, there is an increasing interest in the study of boundedness of the Hausdorff operator on some function spaces, see for example [1, 2, 4, 7, 8, 12, 13, 14, 15, 16, 17, 18, 19].

When $\varphi$ is a locally integrable function on $(0,\infty)^{n}$, there are several high-dimensional extensions of ${\mathcal{H}}_{\varphi}$. One of them is the multi-parameter Hausdorff operator ${\mathcal{H}}_{\varphi}$ defined for suitable functions $f$ on ${\mathbb{R}}^{n}$ by

[TABLE]

Let $\Phi^{(1)},\ldots,\Phi^{(n)}$ be $C^{\infty}$-functions with compact support satisfying $\int_{{\mathbb{R}}}\Phi^{(1)}(x)dx=\cdots=\int_{{\mathbb{R}}}\Phi^{(n)}(x)dx=1$. Then, for any $(t_{1},\ldots,t_{n})\in(0,\infty)^{n}$, we denote

[TABLE]

Following Gundy and Stein [6], we define the multi-parameter Hardy space $H^{1}({\mathbb{R}}\times\cdots\times{\mathbb{R}})$ as the set of all functions $f\in L^{1}({\mathbb{R}}^{n})$ such that

[TABLE]

where $M_{\Phi}f$ is the multi-parameter smooth maximal function of $f$ defined by

[TABLE]

Remark 1.1**.**

1. (i)

$\|\cdot\|_{H^{1}({\mathbb{R}}\times\cdots\times{\mathbb{R}})}$* defines a norm on $H^{1}({\mathbb{R}}\times\cdots\times{\mathbb{R}})$, whose size depends on the choice of $\{\Phi^{(j)}\}_{j=1}^{n}$, but the space $H^{1}({\mathbb{R}}\times\cdots\times{\mathbb{R}})$ does not depend on this choice.* 2. (ii)

If $f$ is in $H^{1}({\mathbb{R}})$, then the function

[TABLE]

is in $H^{1}({\mathbb{R}}\times\cdots\times{\mathbb{R}})$. Moreover, there exist two positive constants $C_{1},C_{2}$ independent of $f$ such that

[TABLE]

In the setting of two-parameter, Liflyand and Móricz showed in [15] that ${\mathcal{H}}_{\varphi}$ is bounded on $H^{1}({\mathbb{R}}\times{\mathbb{R}})$ provided $\varphi\in L^{1}((0,\infty)^{2})$. In the setting of $n$-parameter, one of Weisz’s important results (see [18, Theorem 7]) showed that ${\mathcal{H}}_{\varphi}$ is bounded on $H^{1}({\mathbb{R}}\times\cdots\times{\mathbb{R}})$ provided $\varphi(t_{1},\ldots,t_{n})=\prod_{i=1}^{n}\varphi_{i}(t_{i})$ with $\varphi_{i}\in L^{1}({\mathbb{R}})$ for all $1\leq i\leq n$. Recently, in the setting of two-parameter, Fan and Zhao showed in [4] that the condition $\varphi\in L^{1}((0,\infty)^{2})$ is also a necessary condition for $H^{1}({\mathbb{R}}\times{\mathbb{R}})$-boundedness of ${\mathcal{H}}_{\varphi}$ if $\varphi$ is nonnegative valued. However, it seems that Fan-Zhao’s method can not be used to obtain the exact norm of ${\mathcal{H}}_{\varphi}$ on $H^{1}({\mathbb{R}}\times{\mathbb{R}})$. So, in the setting of $n$-parameter, a natural question arises: Can one find the exact norm of ${\mathcal{H}}_{\varphi}$ on $H^{1}({\mathbb{R}}\times\cdots\times{\mathbb{R}})$? Very recently, in the setting of one-parameter, this question was solved by Hung, Ky and Quang [7].

Motivated by the above question and an open question posted by Liflyand [12, Problem 5], we characterize the nonnegative functions $\varphi$ for which ${\mathcal{H}}_{\varphi}$ is bounded on $H^{1}({\mathbb{R}}\times\cdots\times{\mathbb{R}})$. More precisely, our main result is the following:

Theorem 1.1**.**

Let $\varphi$ be a nonnegative function in $L^{1}_{\rm loc}((0,\infty)^{n})$. Then ${\mathcal{H}}_{\varphi}$ is bounded on $H^{1}({\mathbb{R}}\times\cdots\times{\mathbb{R}})$ if and only if

[TABLE]

Moreover, in that case,

[TABLE]

Theorem 1.1 not only gives an affirmative answer to the above question, but also gives an answer to [12, Problem 5]. It should be pointed out that the norm of the Hausdorff operator ${\mathcal{H}}_{\varphi}$ (${\int_{0}^{\infty}}\cdots{\int_{0}^{\infty}}\varphi(t_{1},\ldots,t_{n})dt_{1}\ldots dt_{n}$) does not depend on the choice of the above functions $\{\Phi^{(j)}\}_{j=1}^{n}$, moreover, it still holds when the above norm $\|\cdot\|_{H^{1}({\mathbb{R}}\times\cdots\times{\mathbb{R}})}$ is replaced by

[TABLE]

where ${\bf H_{e}}f$’s are the multi-parameter Hilbert transforms of $f$. See Theorem 3.3 for details.

Also we characterize the nonnegative functions $\varphi$ for which ${\mathcal{H}}_{\varphi}$ is bounded on $L^{p}({\mathbb{R}}^{n})$, $p\in[1,\infty]$. Our next result can be stated as follows.

Theorem 1.2**.**

Let $p\in[1,\infty]$ and let $\varphi$ be a nonnegative function in $L^{1}_{\rm loc}((0,\infty)^{n})$. Then ${\mathcal{H}}_{\varphi}$ is bounded on $L^{p}({\mathbb{R}}^{n})$ if and only if

[TABLE]

Moreover, in that case,

[TABLE]

Throughout the whole article, we always assume that $\varphi$ is a nonnegative function in $L^{1}_{\rm loc}((0,\infty)^{n})$ and denote by $C$ a positive constant which is independent of the main parameters, but it may vary from line to line. The symbol $A\lesssim B$ means that $A\leq CB$. If $A\lesssim B$ and $B\lesssim A$, then we write $A\sim B$.

2. Norm of ${\mathcal{H}}_{\varphi}$ on $L^{p}({\mathbb{R}}^{n})$

The main purpose of this section is to give the proof of Theorem 1.2. Let us first consider the operator ${\mathcal{H}}^{*}_{\varphi}$ defined by

[TABLE]

Studying this operator on the spaces $L^{p}({\mathbb{R}}^{n})$ is useful in proving the main theorem (Theorem 1.1) in the next section.

Remark that ${\mathcal{H}}^{*}_{\varphi}={\mathcal{H}}_{\overline{\varphi}}$ with $\overline{\varphi}({\bf t})=\frac{\varphi(1/t_{1},\ldots,1/t_{n})}{t_{1}\ldots t_{n}}$ for all ${\bf t}=(t_{1},\ldots,t_{n})\in(0,\infty)^{n}$. Hence, by Theorems 1.1 and 1.2, we obtain:

Theorem 2.1**.**

${\mathcal{H}}^{*}_{\varphi}$* is bounded on $H^{1}({\mathbb{R}}\times\cdots\times{\mathbb{R}})$ if and only if*

[TABLE]

Moreover, in that case,

[TABLE]

Theorem 2.2**.**

Let $p\in[1,\infty]$. Then ${\mathcal{H}}^{*}_{\varphi}$ is bounded on $L^{p}({\mathbb{R}}^{n})$ if and only if

[TABLE]

Moreover, in that case,

[TABLE]

By Theorems 1.2, 2.2 and the Fubini theorem, ${\mathcal{H}}^{*}_{\varphi}$ can be viewed as the Banach space adjoint of ${\mathcal{H}}_{\varphi}$ and vice versa. More precisely, we have:

Theorem 2.3**.**

Let $p\in[1,\infty]$ and $1/p^{\prime}+1/p=1$.

1. (i)

If (1.2) holds, then, for all $f\in L^{p}({\mathbb{R}}^{n})$ and all $g\in L^{p^{\prime}}({\mathbb{R}}^{n})$,

[TABLE] 2. (ii)

If (2.2) holds, then, for all $f\in L^{p}({\mathbb{R}}^{n})$ and all $g\in L^{p^{\prime}}({\mathbb{R}}^{n})$,

[TABLE]

As a consequence of the above theorem, we get the following.

Corollary 2.1**.**

Let $p\in[1,2]$.

1. (i)

If (1.2) holds, then, for all $f\in L^{p}({\mathbb{R}}^{n})$,

[TABLE] 2. (ii)

If (2.2) holds, then, for all $f\in L^{p}({\mathbb{R}}^{n})$,

[TABLE]

Proof.

We prove only (i) since the proof of (ii) is similar. Moreover, from the Hausdorff-Young theorem and the fact that $L^{1}({\mathbb{R}}^{n})\cap L^{p}({\mathbb{R}}^{n})$ is dense in $L^{p}({\mathbb{R}}^{n})$, we consider only the case $p=1$. For all ${\bf y}=(y_{1},\ldots,y_{n})\in{\mathbb{R}}^{n}$, by Theorem 2.3(i) and the Fubini theorem, we get

[TABLE]

This completes the proof of Corollary 2.1.

∎

Proof of Theorem 1.2.

Since the case $p=\infty$ is trivial, we consider only the case $p\in[1,\infty)$. Suppose that (1.2) holds. For any $f\in L^{p}({\mathbb{R}}^{n})$, by the Minkowski inequality, we obtain

[TABLE]

This proves that ${\mathcal{H}}_{\varphi}$ is bounded on $L^{p}({\mathbb{R}}^{n})$, moreover,

[TABLE]

Conversely, suppose that ${\mathcal{H}}_{\varphi}$ is bounded on $L^{p}({\mathbb{R}}^{n})$. For any $\varepsilon>0$, take

[TABLE]

for all ${\bf x}=(x_{1},\ldots,x_{n})\in{\mathbb{R}}^{n}.$ Then, it is easy to see that $f_{\varepsilon}\in L^{p}({\mathbb{R}}^{n})$ and

[TABLE]

for all ${\bf x}=(x_{1},\ldots,x_{n})\in{\mathbb{R}}^{n}.$ Some simple computations give

[TABLE]

Therefore,

[TABLE]

Letting $\varepsilon\to 0$, we obtain

[TABLE]

This, together (2.3), implies that

[TABLE]

and thus ends the proof of Theorem 1.2.

∎

3. Norm of ${\mathcal{H}}_{\varphi}$ on $H^{1}({\mathbb{R}}\times\cdots\times{\mathbb{R}})$

The main purpose of this section is to give the proof of Theorem 1.1 and to show that the norm of the Hausdorff operator ${\mathcal{H}}_{\varphi}$ in Theorem 1.1 still holds when one replaces the norm $\|\cdot\|_{H^{1}({\mathbb{R}}\times\cdots\times{\mathbb{R}})}$ by the norm $\|\cdot\|_{*}$ (see (3.7) below).

Let $\mathbb{C}_{+}^{n}$ be the upper half-plan in $\mathbb{C}^{n}$, that is,

[TABLE]

Following Gundy-Stein [6] and Lacey [9], a function $F:\mathbb{C}_{+}^{n}\to\mathbb{C}$ is said to be in the Hardy space $\mathcal{H}^{1}_{a}(\mathbb{C}_{+}^{n})$ if it is holomorphic in each variable separately and

[TABLE]

Let $j\in\{1,\ldots,n\}$. For any $f\in L^{1}({\mathbb{R}}^{n})$, the Hilbert transform $H_{j}f$ computed in the $j^{\rm th}$ variable is defined by

[TABLE]

For any ${\bf e}=(e_{1},\ldots,e_{n})\in\mathbb{E}:=\{0,1\}^{n}$, denote

[TABLE]

with $H_{j}^{e_{j}}=I$ for $e_{j}=0$ while $H_{j}^{e_{j}}=H_{j}$ for $e_{j}=1$.

The following two theorems are well-known, see for example [6, 9, 10, 18].

Theorem 3.1**.**

A function $f$ is in $H^{1}({\mathbb{R}}\times\cdots\times{\mathbb{R}})$ if and only if ${\bf H_{e}}f$ is in $L^{1}({\mathbb{R}}^{n})$ for all ${\bf e}\in\mathbb{E}$. Moreover, in that case,

[TABLE]

Theorem 3.2**.**

Let $F\in\mathcal{H}^{1}_{a}(\mathbb{C}_{+}^{n})$. Then the boundary value function $f$ of $F$, which is defined by

[TABLE]

a. e. ${\bf x}=(x_{1},\ldots,x_{n})\in{\mathbb{R}}^{n}$, is in $H^{1}({\mathbb{R}}\times\cdots\times{\mathbb{R}})$. Moreover,

[TABLE]

and, for all ${\bf x}+i{\bf y}=(x_{1}+iy_{1},\ldots,x_{n}+iy_{n})\in\mathbb{C}_{+}^{n}$,

[TABLE]

where $P(u)=\frac{1}{1+u^{2}},u\in{\mathbb{R}}$, is the Poisson kernel on ${\mathbb{R}}$.

In order to prove Theorem 1.1, we also need the following two lemmas.

Lemma 3.1**.**

Let $\varphi$ be such that ${\mathcal{H}}_{\varphi}$ is bounded from $H^{1}({\mathbb{R}}\times\cdots\times{\mathbb{R}})$ into $L^{1}({\mathbb{R}}^{n})$. Then (1.1) holds.

Lemma 3.2**.**

Let $\varphi$ be such that (1.1) holds. Then:

1. (i)

${\mathcal{H}}_{\varphi}$* is bounded on $H^{1}({\mathbb{R}}\times\cdots\times{\mathbb{R}})$, moreover,*

[TABLE] 2. (ii)

If supp $\varphi\subset[0,1]^{n}$, then

[TABLE]

Proof of Lemma 3.1.

Since the function

[TABLE]

is in $H^{1}({\mathbb{R}})$ (see [7, Theorem 3.3]), Remark 1.1(ii) yields that

[TABLE]

is in $H^{1}({\mathbb{R}}\times\cdots\times{\mathbb{R}})$. Hence, the function

[TABLE]

${\bf x}=(x_{1},\ldots,x_{n})\in{\mathbb{R}}^{n}$, is in $L^{1}({\mathbb{R}}^{n})$ since ${\mathcal{H}}_{\varphi}$ is bounded from $H^{1}({\mathbb{R}}\times\cdots\times{\mathbb{R}})$ into $L^{1}({\mathbb{R}}^{n})$. As a consequence,

[TABLE]

which proves (1.1), and thus ends the proof of Lemma 3.1.

∎

Proof of Lemma 3.2.

(i) For any $f\in H^{1}({\mathbb{R}}\times\cdots\times{\mathbb{R}})$, by the Fubini theorem,

[TABLE]

for all ${\bf x}=(x_{1},\ldots,x_{n})\in{\mathbb{R}}^{n}$. Hence, by Theorem 1.2,

[TABLE]

This proves that ${\mathcal{H}}_{\varphi}$ is bounded on $H^{1}({\mathbb{R}}\times\cdots\times{\mathbb{R}})$, moreover,

[TABLE]

(ii) Let $\delta\in(0,1)$ be arbitrary. Set $\varphi_{\delta}({\bf t}):=\varphi({\bf t})\chi_{[\delta,1]^{n}}({\bf t})$ for all ${\bf t}\in(0,\infty)^{n}$. Then, by (3.1), we see that

[TABLE]

and

[TABLE]

For any $\varepsilon>0$, we define the function $F_{\varepsilon}:\mathbb{C}_{+}^{n}\to\mathbb{C}$ by

[TABLE]

where $\zeta^{1+\varepsilon}=|\zeta|^{1+\varepsilon}e^{i(1+\varepsilon)\arg\zeta}$ for all $\zeta\in\mathbb{C}$. Denote by $f_{\varepsilon}$ the boundary value function of $F_{\varepsilon}$, that is, $f_{\varepsilon}({\bf x})=\lim_{{\bf y}\to 0}F_{\varepsilon}({\bf x}+i{\bf y})$. Then, by Theorem 3.2,

[TABLE]

where the constants are independent of $\varepsilon$.

For all ${\bf z}={\bf x}+i{\bf y}=(x_{1}+iy_{1},\ldots,x_{n}+iy_{n})=(z_{1},\ldots,z_{n})\in\mathbb{C}^{n}_{+}$, by the Fubini theorem and Theorem 3.2, we get

[TABLE]

where $\phi_{\varepsilon,{\bf z}}(t_{1},\ldots,t_{n}):=\prod_{j=1}^{n}\frac{t_{j}^{\varepsilon}}{(z_{j}+it_{j})^{1+\varepsilon}}$. For any ${\bf t}=(t_{1},\ldots,t_{n})\in[\delta,1]^{n}$, a simple calculus gives

[TABLE]

Therefore, by Theorem 3.2 again,

[TABLE]

This, together with (3.3), yields

[TABLE]

as $\varepsilon\to 0$. As a consequence,

[TABLE]

This, together with (3), allows us to conclude that

[TABLE]

since $\lim_{\delta\to 0}\int_{(0,1]^{n}\setminus[\delta,1]^{n}}\varphi({\bf t})d{\bf t}=0$. Hence, by (3.1),

[TABLE]

This completes the proof of Lemma 3.2.

∎

Now we are ready to give the proof of Theorem 1.1.

Proof of Theorem 1.1.

By Lemma 3.2(i), it suffices to prove that

[TABLE]

provided ${\mathcal{H}}_{\varphi}$ is bounded on $H^{1}({\mathbb{R}}\times\cdots\times{\mathbb{R}})$. Indeed, by Lemma 3.1, we have

[TABLE]

For any $m>0$, set $\varphi_{m}({\bf t}):=\varphi(m{\bf t})\chi_{(0,1)^{n}}({\bf t})$. Then, by Lemma 3.2(i), we see that

[TABLE]

Noting that

[TABLE]

for all $f\in H^{1}({\mathbb{R}}\times\cdots\times{\mathbb{R}})$, Lemma 3.2(ii) gives

[TABLE]

Combining this with (3) allow us to conclude that

[TABLE]

since $\lim_{m\to\infty}\int_{(0,\infty)^{n}\setminus(0,m)^{n}}\varphi({\bf t})d{\bf t}=0$. This proves (3.5), and thus ends the proof of Theorem 1.1. ∎

From Theorem 3.1, one can define $H^{1}({\mathbb{R}}\times\cdots\times{\mathbb{R}})$ as the space of functions $f\in L^{1}({\mathbb{R}}^{n})$ such that

[TABLE]

Our last result is the following:

Theorem 3.3**.**

${\mathcal{H}}_{\varphi}$* is bounded on $(H^{1}({\mathbb{R}}\times\cdots\times{\mathbb{R}}),\|\cdot\|_{*})$ if and only if (1.1) holds. Moreover, in that case,*

[TABLE]

and, for any ${\bf e}\in\mathbb{E}$, ${\mathcal{H}}_{\varphi}$ commutes with ${\bf H_{e}}$ on $H^{1}({\mathbb{R}}\times\cdots\times{\mathbb{R}})$.

In order to prove Theorem 3.3, we need the following two lemmas.

Lemma 3.3**.**

Let $\varphi$ be such that (1.1) holds. Then, for any ${\bf e}\in\mathbb{E}$, ${\mathcal{H}}_{\varphi}$ commutes with the Hilbert transform ${\bf H_{e}}$ on $H^{1}({\mathbb{R}}\times\cdots\times{\mathbb{R}})$.

Lemma 3.4**.**

Let $\varphi$ be such that (1.1) holds. Then:

1. (i)

${\mathcal{H}}_{\varphi}$* is bounded on $(H^{1}({\mathbb{R}}\times\cdots\times{\mathbb{R}}),\|\cdot\|_{*})$, moreover,*

[TABLE] 2. (ii)

If supp $\varphi\subset[0,1]^{n}$, then

[TABLE]

Proof of Lemma 3.3.

Since Theorem 1.1 and the fact that $H_{j}$’s are bounded on $H^{1}({\mathbb{R}}\times\cdots\times{\mathbb{R}})$, it suffices to prove

[TABLE]

for all $j\in\{1,\ldots,n\}$ and all $f\in H^{1}({\mathbb{R}}\times\cdots\times{\mathbb{R}})$. Indeed, thanks to the ideas from [1, 15, 16] and Lemma 2.1(i), for almost every ${\bf y}=(y_{1},\ldots,y_{n})\in{\mathbb{R}}^{n}$,

[TABLE]

This proves (3.8), and thus ends proof of Lemma 3.3, since the uniqueness of the Fourier transform. ∎

Proof of Lemma 3.4.

(i) For all $f\in H^{1}({\mathbb{R}}\times\cdots\times{\mathbb{R}})$ and all ${\bf e}\in\mathbb{E}$, by Lemma 3.3 and Theorem 1.2, we get

[TABLE]

This proves that

[TABLE]

(ii) The proof is similar to that of Lemma 3.2(ii) and will be omitted. The key point is the estimate (3) and the fact that $\|\cdot\|_{*}\sim\|\cdot\|_{H^{1}({\mathbb{R}}\times\cdots\times{\mathbb{R}})}$.

∎

Proof of Theorem 3.3.

The proof is similar to that of Theorem 1.1 by Lemma 3.4. We leave the details to the interested readers. ∎

Acknowledgements. The authors would like to thank the referees for their carefully reading and helpful suggestions.