Hausdorff operators on holomorphic Hardy spaces and applications

TL;DR

This paper characterizes functions that ensure the Hausdorff operator is bounded on Hardy spaces of the upper half-plane, providing operator norms and exploring applications in complex analysis.

Contribution

It offers a complete characterization of nonnegative functions for boundedness of Hausdorff operators on Hardy spaces, including explicit operator norms and applications.

Findings

Characterization of functions for bounded Hausdorff operators on Hardy spaces

Explicit formulas for operator norms

Applications to complex analysis and operator theory

Abstract

The aim of this paper is to characterize the nonnegative functions $\varphi$ defined on $(0,\infty)$ for which the Hausdorff operator $$\mathscr H_\varphi f(z)= \int_0^\infty f\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}dt$$ is bounded on the Hardy spaces of the upper half-plane $\mathcal H_a^p(\mathbb C_+)$, $p\in[1,\infty]$. The corresponding operator norms and their applications are also given.

Equations

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

$$H_{\phi }f(x)={\int }_0^{\infty }f\left(\frac{x}{t}\right)\frac{\phi (t)}{t}dt,x\in \mathbb{R}.$$

$$\Vert f{\Vert }_{{\mathcal{H}}_a^p({\mathbb{C}}_{+})}:=\underset{y>0}{\sup }{\left({\int }_{-\infty }^{\infty }|f(x+iy){|}^{p}dx\right)}^{1/p}<\infty$$

$$\Vert f{\Vert }_{{\mathcal{H}}_a^{\infty }({\mathbb{C}}_{+})}:=\underset{z\in {\mathbb{C}}_{+}}{\sup }|f(z)|<\infty .$$

$$f^{\ast }(x)=\underset{y\to 0}{\lim }f(x+iy),\text{mbox}a.e.x\in \mathbb{R}.$$

$${\int }_0^{\infty }{t}^{1/p-1}\phi (t)dt<\infty .$$

$${\mathcal{H}}_{\phi }f(z)={\int }_0^{\infty }f\left(\frac{z}{t}\right)\frac{\phi (t)}{t}dt,z\in {\mathbb{C}}_{+}.$$

$$\Vert {\mathcal{H}}_{\phi }{\Vert }_{{\mathcal{H}}_a^p({\mathbb{C}}_{+})\to {\mathcal{H}}_a^p({\mathbb{C}}_{+})}={\int }_0^{\infty }{t}^{1/p-1}\phi (t)dt$$

$$({\mathcal{H}}_{\phi }f{)}^{\ast }=H_{\phi }(f^{\ast }).$$

$${\int }_0^{\infty }\phi (t)dt\le \Vert H_{\phi }{\Vert }_{{\mathcal{H}}^1(\mathbb{R})\to {\mathcal{H}}^1(\mathbb{R})}<\infty .$$

$$\Vert H_{\phi }{\Vert }_{L^p(\mathbb{R})\to L^p(\mathbb{R})}={\int }_0^{\infty }{t}^{1/p-1}\phi (t)dt$$

$$|f(x+iy)|\le {\left(\frac{2}{\pi y}\right)}^{1/p}\Vert f{\Vert }_{{\mathcal{H}}_a^p({\mathbb{C}}_{+})}$$

$$|f^{\prime }(x+iy)|\le \frac{2}{y}{\left(\frac{4}{\pi y}\right)}^{1/p}\Vert f{\Vert }_{{\mathcal{H}}_a^p({\mathbb{C}}_{+})}$$

$$\mathcal{M}(f)(x)=\underset{|t-x|<y}{\sup }|f(t+iy)|,x\in \mathbb{R}.$$

$$\Vert f^{\ast }{\Vert }_{L^p(\mathbb{R})}=\Vert f{\Vert }_{{\mathcal{H}}_a^p({\mathbb{C}}_{+})}\text{mbox}and\underset{y\to 0}{\lim }\Vert f(\cdot +iy)-f^{\ast }(\cdot ){\Vert }_{L^p(\mathbb{R})}=0.$$

$$\Vert f{\Vert }_{{\mathcal{H}}_a^p({\mathbb{C}}_{+})}\sim \Vert \mathcal{M}(f){\Vert }_{L^p(\mathbb{R})}.$$

$$\Vert {\mathcal{H}}_{\phi }f{\Vert }_{{\mathcal{H}}_a^{\infty }({\mathbb{C}}_{+})}=\underset{z\in {\mathbb{C}}_{+}}{\sup }\left|{\int }_0^{\infty }f\left(\frac{z}{t}\right)\frac{\phi (t)}{t}dt\right|\le {\int }_0^{\infty }{t}^{-1}\phi (t)dt\Vert f{\Vert }_{{\mathcal{H}}_a^{\infty }({\mathbb{C}}_{+})}.$$

$$\Vert {\mathcal{H}}_{\phi }{\Vert }_{{\mathcal{H}}_a^{\infty }({\mathbb{C}}_{+})\to {\mathcal{H}}_a^{\infty }({\mathbb{C}}_{+})}\le {\int }_0^{\infty }{t}^{-1}\phi (t)dt.$$

$$\Vert {\mathcal{H}}_{\phi }{\Vert }_{{\mathcal{H}}_a^{\infty }({\mathbb{C}}_{+})\to {\mathcal{H}}_a^{\infty }({\mathbb{C}}_{+})}\ge \frac{\Vert {\mathcal{H}}_{\phi }(1){\Vert }_{{\mathcal{H}}_a^{\infty }({\mathbb{C}}_{+})}}{\Vert 1{\Vert }_{{\mathcal{H}}_a^{\infty }({\mathbb{C}}_{+})}}={\int }_0^{\infty }{t}^{-1}\phi (t)dt.$$

$$\Vert {\mathcal{H}}_{\phi }{\Vert }_{{\mathcal{H}}_a^{\infty }({\mathbb{C}}_{+})\to {\mathcal{H}}_a^{\infty }({\mathbb{C}}_{+})}={\int }_0^{\infty }{t}^{-1}\phi (t)dt.$$

$$({\mathcal{H}}_{\phi }f{)}^{\ast }(x)=\underset{y\to 0}{\lim }{\int }_0^{\infty }f\left(\frac{x}{t}+\frac{y}{t}i\right)\frac{\phi (t)}{t}dt={\int }_0^1{f}^{\ast }\left(\frac{x}{t}\right)\frac{\phi (t)}{t}dt=H_{\phi }(f^{\ast })(x).$$

$$\Vert {\mathcal{H}}_{\phi }{\Vert }_{{\mathcal{H}}_a^p({\mathbb{C}}_{+})\to {\mathcal{H}}_a^p({\mathbb{C}}_{+})}\le {\int }_0^{\infty }{t}^{1/p-1}\phi (t)dt.$$

$$\Vert {\mathcal{H}}_{\phi }{\Vert }_{{\mathcal{H}}_a^p({\mathbb{C}}_{+})\to {\mathcal{H}}_a^p({\mathbb{C}}_{+})}={\int }_0^1{t}^{1/p-1}\phi (t)dt.$$

$$({\mathcal{H}}_{\phi }f{)}^{\ast }=H_{\phi }(f^{\ast }).$$

$$\mathcal{M}({\mathcal{H}}_{\phi }f)(x)$$

$$\Vert {\mathcal{H}}_{\phi }f{\Vert }_{{\mathcal{H}}_a^p({\mathbb{C}}_{+})}\lesssim \Vert \mathcal{M}({\mathcal{H}}_{\phi }f){\Vert }_{L^p(\mathbb{R})}$$

$$\Vert {\mathcal{H}}_{\phi }{\Vert }_{{\mathcal{H}}_a^p({\mathbb{C}}_{+})\to {\mathcal{H}}_a^p({\mathbb{C}}_{+})}\lesssim {\int }_0^{\infty }{t}^{1/p-1}\phi (t)dt.$$

$$\Vert {\mathcal{H}}_{\phi }{\Vert }_{{\mathcal{H}}_a^p({\mathbb{C}}_{+})\to {\mathcal{H}}_a^p({\mathbb{C}}_{+})}\le {\int }_0^{\infty }{t}^{1/p-1}\phi (t)dt,$$

$${\Vert {\mathcal{H}}_{\phi }f\Vert }_{{\mathcal{H}}_a^p({\mathbb{C}}_{+})}={\Vert ({\mathcal{H}}_{\phi }f{)}^{\ast }\Vert }_{L^p(\mathbb{R})}$$

$$\Vert {\mathcal{H}}_{{\phi }_{\delta }}{\Vert }_{{\mathcal{H}}_a^p({\mathbb{C}}_{+})\to {\mathcal{H}}_a^p({\mathbb{C}}_{+})}\le {\int }_0^{\infty }{t}^{1/p-1}{\phi }_{\delta }(t)dt={\int }_{\delta }^1{t}^{1/p-1}\phi (t)dt<\infty$$

Full text

[TABLE]

is bounded on the Hardy spaces of the upper half-plane $\mathcal{H}_{a}^{p}(\mathbb{C}_{+})$, $p\in[1,\infty]$. The corresponding operator norms and their applications are also given.

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2016.22.

1. Introduction and the main result

Let $\varphi$ be a locally integrable function on $(0,\infty)$. The Hausdorff operator $H_{\varphi}$ is then defined for suitable functions $f$ on ${\mathbb{R}}$ by

[TABLE]

The Hausdorff operator 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 [15] and the references therein. In the recent years, there is an increasing interest in the study of boundedness of the Hausdorff operator and its commuting with the Hilbert transform on the real Hardy spaces and on the Lebesgue spaces, see for example [1, 2, 3, 4, 10, 13, 15, 16, 17, 20].

Let $\mathbb{C}_{+}$ be the upper half-plane in the complex plane. For $0<p\leq\infty$, the Hardy space $\mathcal{H}_{a}^{p}(\mathbb{C}_{+})$ is defined as the set of all holomorphic functions $f$ on $\mathbb{C}_{+}$ such that

[TABLE]

if $0<p<\infty$, and if $p=\infty$, then

[TABLE]

It is classical (see [6, 9]) that if $f\in{\mathcal{H}}_{a}^{p}(\mathbb{C}_{+})$, then $f$ has a boundary value function $f^{*}\in L^{p}({\mathbb{R}})$ defined by

[TABLE]

Let $p\in[1,\infty]$ and let $\varphi$ be a nonnegative function in $L^{1}_{\rm loc}(0,\infty)$ for which

[TABLE]

Then it is well-known (see [1]) that $H_{\varphi}$ is bounded on $L^{p}({\mathbb{R}})$, and thus $H_{\varphi}(f^{*})\in L^{p}({\mathbb{R}})$ for any boundary value function $f^{*}$ of a function $f$ in ${\mathcal{H}}_{a}^{p}(\mathbb{C}_{+})$. A natural question arises is that whether the transformed function $H_{\varphi}(f^{*})$ is also the boundary value function of a function in ${\mathcal{H}}_{a}^{p}(\mathbb{C}_{+})$? In some special cases of $\varphi$ and $1<p<\infty$, using the spectral mapping theorem and the Hille-Yosida-Phillips theorem, Arvanitidis-Siskakis [2] and Ballamoole-Bonyo-Miller-Millerstudied [3] studied and gave affirmative answers to this question.

In the present paper, we give an affirmative answer to the above question by studying a complex version of $H_{\varphi}$ defined by

[TABLE]

Our main result reads as follows.

Theorem 1.1**.**

Let $p\in[1,\infty]$ and let $\varphi$ be a nonnegative function in $L^{1}_{\rm loc}(0,\infty)$. Then ${\mathscr{H}}_{\varphi}$ is bounded on ${\mathcal{H}}_{a}^{p}(\mathbb{C}_{+})$ if and only if (1.2) holds. Moreover, in that case, we obtain

[TABLE]

and, for any $f\in{\mathcal{H}}_{a}^{p}(\mathbb{C}_{+})$,

[TABLE]

It should be pointed out that some main results in [2, 3] (see [2, Theorems 3.1, 3.3 and 4.1] and [3, Theorem 3.4]) can be viewed as special cases of Theorem 1.1 by choosing suitable kernel functions $\varphi$. In the setting of Hardy spaces $\mathcal{H}^{p}(\mathbb{D})$ on the unit disk $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$, Galanopoulos and Papadimitrakis ([8, Theorems 2.3 and 2.4]) studied and obtained some similar results to Theorem 1.1 for $1<p<\infty$ while it is slightly different at the endpoints $p=1$ and $p=\infty$ (see also the survey article [15]).

Furthermore, if we denote by ${\mathcal{H}}^{1}({\mathbb{R}})$ the real Hardy space in the sense of Fefferman-Stein (see the last section), then by using Theorem 1.1, we obtain the following result.

Corollary 1.1** (see Theorem 3.4).**

Let $\varphi$ be a nonnegative function in $L^{1}_{\rm loc}(0,\infty)$ such that $H_{\varphi}$ is bounded on ${\mathcal{H}}^{1}({\mathbb{R}})$. Then,

[TABLE]

The above corollary is not only give an answer to a question posted by Liflyand [13, Problem 4], but also give a lower bound for the norm of $H_{\varphi}$ on ${\mathcal{H}}^{1}({\mathbb{R}})$. Another corollary of Theorem 1.1 is:

Corollary 1.2** (see Theorem 3.1).**

Let $p\in(1,\infty)$ and let $\varphi$ be as in Theorem 1.1. Then $H_{\varphi}$ is bounded on $L^{p}({\mathbb{R}})$ if and only if (1.2) holds. Moreover, in that case,

[TABLE]

and $H_{\varphi}$ commutes with the Hilbert transform $H$ on $L^{p}({\mathbb{R}})$.

Throughout the whole article, we use the symbol $A\lesssim B$ (or $B\gtrsim A$) means that $A\leq CB$ where $C$ is a positive constant which is independent of the main parameters, but it may vary from line to line. If $A\lesssim B$ and $B\lesssim A$, then we write $A\sim B$. For any $E\subset{\mathbb{R}}$, we denote by $\chi_{E}$ its characteristic function.

2. Proof of Theorem 1.1

In the sequel, we always assume that $\varphi$ is a nonnegative function in $L^{1}_{\rm loc}(0,\infty)$. Also we remark that, for any $f\in\mathcal{H}_{a}^{p}(\mathbb{C}_{+})$, the function ${\mathscr{H}_{\varphi}}f$ is well-defined and holomorphic on $\mathbb{C}_{+}$ provided (1.2) holds, since

[TABLE]

and

[TABLE]

for all $z=x+iy\in\mathbb{C}_{+}$. See Garnett’s book [9, p. 57].

Given an holomorphic function $f$ on $\mathbb{C}_{+}$, we define the nontangential maximal function of $f$ by

[TABLE]

The following lemma is classical and can be found in [6, 9].

Lemma 2.1**.**

Let $0<p<\infty$. Then:

1. (i)

For any $f\in\mathcal{H}_{a}^{p}(\mathbb{C}_{+})$, we have

[TABLE] 2. (ii)

$f\in\mathcal{H}_{a}^{p}(\mathbb{C}_{+})$* if and only if $\mathcal{M}(f)\in L^{p}({\mathbb{R}})$. Moreover,*

[TABLE]

Lemma 2.2**.**

Theorem 1.1 is true for $p=\infty$.

Proof.

Suppose that $\int_{0}^{\infty}t^{-1}\varphi(t)dt$ is finite. Then, for any $f\in{\mathcal{H}}^{\infty}_{a}(\mathbb{C}_{+})$,

[TABLE]

Therefore, ${\mathscr{H}}_{\varphi}$ is bounded on $\mathcal{H}^{\infty}_{a}(\mathbb{C}_{+})$, moreover,

[TABLE]

On the other hand, we have

[TABLE]

This, together with (2.3), implies that

[TABLE]

Moreover, by the dominated convergence theorem, for any $x\neq 0$,

[TABLE]

Conversely, suppose that ${\mathscr{H}_{\varphi}}$ is bounded on ${\mathcal{H}}^{\infty}_{a}(\mathbb{C}_{+})$. As the function $f(z)\equiv 1$ is in ${\mathcal{H}}^{\infty}_{a}(\mathbb{C}_{+})$, we obtain that ${\mathscr{H}_{\varphi}}f=\int_{0}^{1}t^{-1}\varphi(t)dt<\infty$. ∎

Lemma 2.3**.**

Let $p\in[1,\infty)$ and let $\varphi$ be such that (1.2) holds. Then

1. (i)

${\mathscr{H}_{\varphi}}$* is bounded on ${{\mathcal{H}}^{p}_{a}(\mathbb{C}_{+})}$, moreover,*

[TABLE] 2. (ii)

If supp $\varphi\subset[0,1]$, then

[TABLE] 3. (iii)

For any $f\in{{\mathcal{H}}^{p}_{a}(\mathbb{C}_{+})}$, we have

[TABLE]

Proof.

(i) For any $f\in\mathcal{H}^{p}_{a}(\mathbb{C}_{+})$, we have

[TABLE]

for all $x\in\mathbb{R}$. Therefore, by the Minkowski inequality and Lemma 2.1(ii),

[TABLE]

This proves that ${\mathscr{H}_{\varphi}}$ is bounded on $\mathcal{H}^{p}_{a}(\mathbb{C}_{+})$, moreover,

[TABLE]

In order to show

[TABLE]

let us first assume that (iii) is proved. Then, by Lemma 2.1(i) and the Minkowski inequality, we get

[TABLE]

This proves that (2.5) holds.

(ii) Let $\delta\in(0,1)$ be arbitrary and let $\varphi_{\delta}(t)=\varphi(t)\chi_{[\delta,\infty)}(t)$ for all $t\in(0,\infty)$. Since (2.5) holds, we see that

[TABLE]

and

[TABLE]

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

[TABLE]

where, and in what follows, $\zeta^{1/p+\varepsilon}=|\zeta|^{1/p+\varepsilon}e^{i(1/p+\varepsilon)\arg\zeta}$ for all $\zeta\in\mathbb{C}$. Then

[TABLE]

For all $z=x+iy\in\mathbb{C}_{+}$, we have

[TABLE]

where $\phi_{\varepsilon,z}(t):=\frac{t^{\varepsilon}}{(z+ti)^{1/p+\varepsilon}}$. For any $t\in[\delta,1]$, a simple calculus gives

[TABLE]

This, together with (2.7), yields

[TABLE]

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

[TABLE]

This, combined with (2.6), allows us to conclude that

[TABLE]

as $\delta\to 0$ since $\int_{0}^{1}t^{1/p-1}\varphi(t)dt<\infty$. Hence, by (2.5),

[TABLE]

(iii) For any $\sigma>0$, it follows from (2.1) that the function

[TABLE]

is in $\mathcal{H}_{a}^{p}(\mathbb{C}_{+})\cap\mathcal{H}^{\infty}_{a}(\mathbb{C}_{+})$. Let $\delta$ and $\varphi_{\delta}$ be as in (ii). Noting that

[TABLE]

Lemma 2.2(ii) gives $(\mathscr{H}_{\varphi_{\delta}}(f_{\sigma}))^{*}=H_{\varphi_{\delta}}(f_{\sigma}^{*})$. Therefore, by Lemma 2.1(i), [1, Theorem 1] and (2.4), we obtain that

[TABLE]

as $\sigma\to 0$ and $\delta\to 0$. This completes the proof of Lemma 2.3.

∎

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

Proof of Theorem 1.1.

By Lemmas 2.2 and 2.3, it suffices to prove that

[TABLE]

whenever ${\mathscr{H}_{\varphi}}$ is bounded on ${{\mathcal{H}}^{p}_{a}(\mathbb{C}_{+})}$ for $1\leq p<\infty$.

Indeed, we first claim that

[TABLE]

Assume (2.10) holds for a moment.

For any $m>0$, we define $\varphi_{m}(t)=\varphi(mt)\chi_{(0,1]}(t)$ for all $t\in(0,\infty)$. Then, by Lemma 2.3(i), we see that

[TABLE]

Noting that

[TABLE]

for all $f\in{{\mathcal{H}}^{p}_{a}(\mathbb{C}_{+})}$, Lemma 2.3(ii) gives

[TABLE]

Combining this with (2.11) allows us to conclude that

[TABLE]

as $m\to\infty$ since $\int_{0}^{\infty}t^{1/p-1}\varphi(t)dt<\infty$. This proves (2.9).

Now we return to prove (2.10). Indeed, we consider the following two cases.

Case 1: $p=1$. Take $f(z)=\frac{1}{(z+i)^{2}}$ for all $z\in\mathbb{C}_{+}$. Then

[TABLE]

Therefore, by the Fatou lemma, we get

[TABLE]

This proves (2.10).

Case 2: $1<p<\infty$. For any $0<\varepsilon<1-1/p$, take

[TABLE]

for all $z\in\mathbb{C}_{+}$. Then

[TABLE]

and

[TABLE]

where we used the Fatou lemma and the fact that

[TABLE]

for all $x,y,t>0$ since $0<1/p+\varepsilon<1$. This, together with (2.12), gives

[TABLE]

Hence,

[TABLE]

Letting $\varepsilon\to 0$, we obtain

[TABLE]

This proves (2.10), and thus ends the proof of Theorem 1.1.

∎

3. Some applications

Let $1\leq p<\infty$, we define (see [19]) the Hilbert transform of $f\in L^{p}({\mathbb{R}})$ by

[TABLE]

Theorem 3.1**.**

Let $p\in(1,\infty)$ and let $\varphi$ be as in Theorem 1.1. Then $H_{\varphi}$ is bounded on $L^{p}({\mathbb{R}})$ if and only if (1.2) holds. Moreover, in that case,

[TABLE]

and $H_{\varphi}$ commutes with the Hilbert transform $H$ on $L^{p}({\mathbb{R}})$.

In order to prove Theorem 3.1, we need the following lemmas.

Lemma 3.1** (see [6, 9]).**

Let $1<p<\infty$. Then:

1. (i)

If $g\in L^{p}({\mathbb{R}})$, then $f^{*}:=g+iH(g)$ is the boundary value function of some function $f\in\mathcal{H}_{a}^{p}(\mathbb{C}_{+})$. 2. (ii)

Conversely, if $f^{*}$ is a boundary value function of $f\in\mathcal{H}_{a}^{p}(\mathbb{C}_{+})$, then there exists a real-valued function $g\in L^{p}({\mathbb{R}})$ such that $f^{*}=g+iH(g)$.

Moreover, in those cases,

[TABLE]

Lemma 3.2** (see [1, 20]).**

Let $p\in(1,\infty)$ and let $\varphi$ be such that (1.2) holds. Then:

1. (i)

$H_{\varphi}$* is bounded on $L^{p}({\mathbb{R}})$, moreover,*

[TABLE] 2. (ii)

If supp $\varphi\subset[0,1]$, then

[TABLE]

Proof of Theorem 3.1.

Suppose that (1.2) holds. By Lemma 3.2(i),

[TABLE]

Conversely, suppose that $H_{\varphi}$ is bounded on $L^{p}({\mathbb{R}})$. We first claim that

[TABLE]

Assume (3.2) holds for a moment.

For any $m>0$, take $\varphi_{m}$ is as in the proof of Theorem 1.1. Then, by a similar argument to the proof of Theorem 1.1, we get

[TABLE]

as $m\to\infty$. This, together with (3.1), yields

[TABLE]

Now let us return to prove (3.2). Indeed, for any $\epsilon\in(0,1)$, take

[TABLE]

and

[TABLE]

for all $x\in{\mathbb{R}}$. Then some simple computations give

[TABLE]

and

[TABLE]

Letting $\epsilon\to 0$, we get

[TABLE]

and

[TABLE]

This proves (3.2).

Finally, we need to show that $H_{\varphi}$ commutes with the Hilbert transform $H$ on $L^{p}({\mathbb{R}})$. To this ends, it suffices to show

[TABLE]

for all real-valued functions $f$ in $L^{p}({\mathbb{R}})$. Indeed, by Theorem 1.1 and Lemma 3.1, there exists a real-valued function $g$ in $L^{p}({\mathbb{R}})$ such that

[TABLE]

This proves (3.3), and thus completes the proof of Theorem 3.1. ∎

Let $1<p<\infty$, we denote by $H^{p}_{+}({\mathbb{R}})$ and $H^{p}_{-}({\mathbb{R}})$ the subspaces of $L^{p}({\mathbb{R}})$ consisting of those functions whose Poisson extensions to the upper half-plane $\mathbb{C}_{+}$ are holomorphic and anti-holomorphic, respectively.

It is well-known (see [6, 9, 19]) that

[TABLE]

and

[TABLE]

Moreover, $L^{p}({\mathbb{R}})=H^{p}_{+}({\mathbb{R}})\oplus H^{p}_{-}({\mathbb{R}})$.

Theorem 3.2**.**

Let $p\in(1,\infty)$ and let $\varphi$ be such that (1.2) holds. Then $H_{\varphi}$ is bounded on the space $H^{p}_{+}({\mathbb{R}})$, moreover,

[TABLE]

and $H_{\varphi}$ commutes with the Hilbert transform $H$ on $H^{p}_{+}({\mathbb{R}})$.

Proof.

It follows from Theorem 3.1 that $H_{\varphi}(f)$ belongs to $H^{p}_{+}({\mathbb{R}})$ for all $f\in H^{p}_{+}({\mathbb{R}})$, and thus

[TABLE]

For any $\varepsilon>0$, by Theorem 1.1, there exists $f_{\varepsilon}\in{{\mathcal{H}}^{p}_{a}(\mathbb{C}_{+})}$ for which

[TABLE]

This, together with (3.6), allows us to conclude that

[TABLE]

Finally, $H_{\varphi}$ commutes with the Hilbert transform $H$ on $H^{p}_{+}({\mathbb{R}})$ is followed from Theorem 3.1 and (3.4). ∎

Theorem 3.3**.**

Let $p\in(1,\infty)$ and let $\varphi$ be such that (1.2) holds. Then $H_{\varphi}$ is bounded on the space $H^{p}_{-}({\mathbb{R}})$, moreover,

[TABLE]

and $H_{\varphi}$ commutes with the Hilbert transform $H$ on $H^{p}_{-}({\mathbb{R}})$.

Proof.

It follows from Theorem 3.2 and the fact that $f\in H^{p}_{+}({\mathbb{R}})$ if and only if $\bar{f}\in H^{p}_{-}({\mathbb{R}})$. ∎

Let $\Phi$ be in the Schwartz space $\mathcal{S}({\mathbb{R}})$ satisfying $\int_{{\mathbb{R}}}\Phi(x)dx\neq 0$. For any $t>0$, set $\Phi_{t}(x):=t^{-1}\Phi(x/t)$. Following Fefferman and Stein [7, 19], we define the real Hardy space ${\mathcal{H}}^{1}({\mathbb{R}})$ as the set of all functions $f\in L^{1}({\mathbb{R}})$ such that

[TABLE]

where $M_{\Phi}(f)$ is the smooth maximal function of $f$ defined by

[TABLE]

Remark that the norm $\|\cdot\|_{{\mathcal{H}}^{1}({\mathbb{R}})}$ depends on the choice of $\Phi$, but the space ${\mathcal{H}}^{1}({\mathbb{R}})$ does not depend on this choice (see Proposition 3.1 below).

The following lemma is well-known.

Lemma 3.3** (see [6, 9, 18]).**

1. (i)

If $g\in{\mathcal{H}}^{1}({\mathbb{R}})$, then $f^{*}:=g+iH(g)$ is the boundary value function of some function $f\in\mathcal{H}_{a}^{1}(\mathbb{C}_{+})$. 2. (ii)

Conversely, if $f^{*}$ is a boundary value function of $f\in\mathcal{H}_{a}^{1}(\mathbb{C}_{+})$, then there exists a real-valued function $g\in{\mathcal{H}}^{1}({\mathbb{R}})$ such that $f^{*}=g+iH(g)$.

Moreover, in those cases,

[TABLE]

Let $P_{t}(x)=\frac{1}{\pi}\frac{t}{x^{2}+t^{2}}$ be the Poisson kernel on ${\mathbb{R}}$. For any $f\in L^{1}({\mathbb{R}})$, we denote $u(y,t)=f*P_{t}(y)$. Then, set

[TABLE]

A function $a$ is called an ${\mathcal{H}}^{1}$-atom related to the interval $B$ if

1. $\bullet$

supp $a\subset B$; 2. $\bullet$

$\|a\|_{L^{\infty}({\mathbb{R}})}\leq|B|^{-1}$; 3. $\bullet$

$\int_{{\mathbb{R}}}a(x)dx=0$.

We define the Hardy space ${\mathcal{H}}^{1}_{\rm at}({\mathbb{R}})$ as the space of functions $f\in L^{1}({\mathbb{R}})$ which can be written as $f=\sum_{j=1}^{\infty}\lambda_{j}a_{j}$ with $a_{j}$’s are ${\mathcal{H}}^{1}$-atoms and $\lambda_{j}$’s are complex numbers satisfying $\sum_{j=1}^{\infty}|\lambda_{j}|<\infty$. The norm on ${\mathcal{H}}^{1}_{\rm at}({\mathbb{R}})$ is then defined by

[TABLE]

The following proposition is classical and can be found in Stein’s book [19].

Proposition 3.1**.**

Let $f\in L^{1}({\mathbb{R}})$. Then the following conditions are equivalent:

1. (i)

$f\in{\mathcal{H}}^{1}({\mathbb{R}})$. 2. (ii)

$\mathcal{M}_{\Phi}(f)(\cdot)=\sup_{|y-\cdot|<t}|f*\Phi_{t}(y)|\in L^{1}({\mathbb{R}})$. 3. (iii)

$\mathcal{M}_{P}(f)\in L^{1}({\mathbb{R}})$. 4. (iv)

$S(f)\in L^{1}({\mathbb{R}})$. 5. (v)

$f\in{\mathcal{H}}^{1}_{\rm at}({\mathbb{R}})$. 6. (vi)

$H(f)\in L^{1}({\mathbb{R}})$.

Moreover, in those cases,

[TABLE]

Of course, the above constants are depending on $\Phi$.

The following gives a lower bound for the norm of $H_{\varphi}$ on ${\mathcal{H}}^{1}({\mathbb{R}})$.

Theorem 3.4**.**

Let $\|\cdot\|_{*}$ be one of the six norms in Proposition 3.1. Assume that $H_{\varphi}$ is bounded on $({\mathcal{H}}^{1}({\mathbb{R}}),\|\cdot\|_{*})$. Then,

[TABLE]

and $H_{\varphi}$ commutes with the Hilbert transform $H$ on ${\mathcal{H}}^{1}({\mathbb{R}})$.

It should be pointed out that, when supp $\varphi\subset[1,\infty)$ and $\|\cdot\|_{*}=\|\cdot\|_{{\mathcal{H}}^{1}_{\rm at}({\mathbb{R}})}$, the above theorem is due to Xiao [20, p. 666] (see also [12, 14]).

In order to prove Theorem 3.4, we need the following lemma.

Lemma 3.4**.**

Let $\varphi$ be such that $\int_{0}^{\infty}\varphi(t)dt<\infty$ and supp $\varphi\subset[0,1]$. Then,

[TABLE]

Proof.

It is well-known (see [1, 10, 16]) that if $\int_{0}^{\infty}\varphi(t)dt<\infty$, then $H_{\varphi}$ is bounded on ${\mathcal{H}}^{1}({\mathbb{R}})$, moreover,

[TABLE]

We now show that

[TABLE]

Indeed, let $\delta\in(0,1)$ and $\varphi_{\delta}$ be as in the proof of Lemma 2.3(ii). For any $\varepsilon>0$, define the function $f_{\varepsilon}:\mathbb{C}_{+}\to\mathbb{C}$ by

[TABLE]

Then, by Lemma 2.3(iii), Lemma 3.3, Proposition 3.1 and (2),

[TABLE]

as $\varepsilon\to 0$. This implies that

[TABLE]

and thus

[TABLE]

since

[TABLE]

as $\delta\to 0$. This ends the proof of Lemma 3.4. ∎

Proof of Theorem 3.4.

It follows from [10, Theorem 3.3] that

[TABLE]

For any $m>0$, let $\varphi_{m}$ be as in the proof of Theorem 1.1. Then, by (3.7),

[TABLE]

where the constant is independent of $m$.

Noting that

[TABLE]

for all $f\in{\mathcal{H}}^{1}({\mathbb{R}})$, Lemma 3.4 gives

[TABLE]

This, together with (3) and $\lim\limits_{m\to\infty}\int_{m}^{\infty}\varphi(t)dt=0$, allows us to conclude that

[TABLE]

Using the Fourier transform, Liflyand and Móricz proved in [17] that $H_{\varphi}$ commutes with the Hilbert transform $H$ on ${\mathcal{H}}^{1}({\mathbb{R}})$. However, we also would like to give a new proof of this fact here. It suffices to prove

[TABLE]

for all real-valued functions $f$ in ${\mathcal{H}}^{1}({\mathbb{R}})$. Indeed, by Theorem 1.1 and Lemma 3.3, there exists a real-valued function $g$ in ${\mathcal{H}}^{1}({\mathbb{R}})$ such that

[TABLE]

This proves (3.9), and thus completes the proof of Theorem 3.4.

∎

Let $a:(0,\infty)\to[0,\infty)$ be a measurable function. Following Carro and Ortiz-Caraballo [5], we define

[TABLE]

for all holomorphic functions $F$ on $\mathbb{C}_{+}$; and define

[TABLE]

for all measurable functions $f$ on ${\mathbb{R}}$.

It is easy to see that

[TABLE]

where $\varphi(t)=t^{-1}a(t^{-1})$ for all $t\in(0,\infty)$. Hence, it follows from Theorems 1.1, 3.1 and 3.4 that:

Theorem 3.5**.**

Let $p\in[1,\infty]$ and let $a:(0,\infty)\to[0,\infty)$ be a measurable function. Then $\mathscr{S}_{a}$ is bounded on ${\mathcal{H}}_{a}^{p}(\mathbb{C}_{+})$ if and only if

[TABLE]

Moreover, when (3.10) holds, we obtain

[TABLE]

and, for any $f\in{\mathcal{H}}_{a}^{p}(\mathbb{C}_{+})$,

[TABLE]

Theorem 3.6**.**

Let $p\in(1,\infty)$ and let $a$ be as in Theorem 3.5. Then $S_{a}$ is bounded on $L^{p}({\mathbb{R}})$ if and only if (3.10) holds. Moreover, in that case,

[TABLE]

and $S_{a}$ commutes with the Hilbert transform $H$ on $L^{p}({\mathbb{R}})$.

Theorem 3.7**.**

Let $a$ be as in Theorem 3.5. Then $S_{a}$ is bounded on ${\mathcal{H}}^{1}({\mathbb{R}})$ if and only if $\int_{0}^{\infty}t^{-1}a(t)dt<\infty.$ Moreover, in that case,

[TABLE]

and $S_{a}$ commutes with the Hilbert transform $H$ on ${\mathcal{H}}^{1}({\mathbb{R}})$.

Also it is easy to see that if (1.2) holds for $1<p<\infty$, then

[TABLE]

whenever $f\in L^{p}({\mathbb{R}})$ and $g\in L^{q}({\mathbb{R}})$, $q=p/(p-1)$. Namely, $S_{\varphi}$ can be viewed as the Banach space adjoint of $H_{\varphi}$ and vice versa. Therefore, by Theorem 3.4, Theorem 3.7 and [12, Theorem 1], a duality argument gives:

Theorem 3.8**.**

1. (i)

$S_{\varphi}$* is bounded on $BMO({\mathbb{R}})$ if and only if $\int_{0}^{\infty}\varphi(t)dt<\infty$. Moreover, in that case,*

[TABLE] 2. (ii)

$H_{\varphi}$* is bounded on $BMO({\mathbb{R}})$ if and only if $\int_{0}^{\infty}t^{-1}\varphi(t)dt<\infty$. Moreover, in that case,*

[TABLE]

Here the space $BMO({\mathbb{R}})$ (see [7, 11]) is the dual space of ${\mathcal{H}}^{1}({\mathbb{R}})$ defined as the space of all functions $f\in L^{1}_{\rm loc}({\mathbb{R}})$ such that

[TABLE]

where the supremum is taken over all intervals $B\subset{\mathbb{R}}$.

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