A note on weak factorization of Meyer-type Hardy space via Cauchy integral operator

TL;DR

This paper establishes a weak factorization for Meyer-type Hardy spaces and characterizes their duals and preduals via boundedness and compactness of commutators with the Cauchy integral operator associated to a Lipschitz curve.

Contribution

It introduces a weak factorization for $H^1_b( eal)$ and characterizes ${ m BMO}_b( eal)$ and ${ m VMO}_b( eal)$ via commutator boundedness and compactness.

Findings

Weak factorization for Meyer-type Hardy space $H^1_b( eal)$.

Characterizations of ${ m BMO}_b( eal)$ and ${ m VMO}_b( eal)$ via commutator properties.

Connection between the Cauchy integral operator and function space dualities.

Abstract

This paper provides a weak factorization for the Meyer-type Hardy space $H^1_b(\mathbb{R})$, and characterizations of its dual ${\rm BMO}_b(\mathbb{R})$ and its predual ${\rm VMO}_b(\mathbb{R})$ via boundedness and compactness of a suitable commutator with the Cauchy integral $\mathscr{C}_{\Gamma}$, respectively. Here $b(x)=1+iA'(x)$ where $A'\in L^{\infty}(\mathbb{R})$, and the Cauchy integral $\mathscr{C}_{\Gamma}$ is associated to the Lipschitz curve $\Gamma=\{x+iA(x)\, : \, x\in \mathbb{R}\}$.

Equations

$${\mathcal{C}}_{\Gamma }(f)(x):=p.v.\frac{1}{\pi i}{\int }_{\mathbb{R}}\frac{(1+i{A}^{\prime }(y))f(y)}{y-x+i(A(y)-A(x))}dy,$$

$${\tilde{\mathcal{C}}}_{\Gamma }(f)(x):=p.v.\frac{1}{\pi i}{\int }_{\mathbb{R}}\frac{f(y)}{y-x+i(A(y)-A(x))}dy,$$

$$\Pi_{b}(g,h)(x):=\frac{1}{b(x)}\big{(}g(x)\cdot\mathscr{C}_{\Gamma}(h)(x)-h(x)\cdot\mathscr{C}_{\Gamma}^{*}(g)(x)\big{)},$$

$$f=\sum _{k=1}^{\infty }\sum _{j=1}^{\infty }{\lambda }_j^k{\Pi }_b(g_j^k,h_j^k).$$

$${\Vert f\Vert }_{H_b^1(\mathbb{R})}\approx \inf \left\{\sum _{k=1}^{\infty }\sum _{j=1}^{\infty }\left|{\lambda }_j^k\right|{\Vert g_j^k\Vert }_{L^2(\mathbb{R})}{\Vert h_j^k\Vert }_{L^2(\mathbb{R})}:f=\sum _{k=1}^{\infty }\sum _{j=1}^{\infty }{\lambda }_j^k{\Pi }_b(g_j,h_j)\right\}.$$

$${\Vert \left[\mathfrak{A}/b,{\mathcal{C}}_{\Gamma }\right]\Vert }_{L^p(\mathbb{R})\to L^p(\mathbb{R})}\lesssim \Vert \mathfrak{A}{\Vert }_{{\mathrm{BMO}}_b(\mathbb{R})}.$$

$$\Vert \mathfrak{A}{\Vert }_{{\mathrm{BMO}}_b(\mathbb{R})}\lesssim {\Vert \left[\mathfrak{A}/b,{\mathcal{C}}_{\Gamma }\right]\Vert }_{L^p(\mathbb{R})\to L^p(\mathbb{R})}.$$

$$\Vert f{\Vert }_{\mathrm{BMO}}:=\underset{I}{\sup }\frac{1}{|I|}{\int }_I|f(x)-f_I|dx<\infty .$$

$$\|f\|_{H^{1}(\mathbb{R})}:=\inf\Big{\{}\sum_{j=1}^{\infty}|\lambda_{j}|\,:\,f=\sum_{j=1}^{\infty}\lambda_{j}a_{j},\;\mbox{$a_{j}$ are $L^{2}$-atoms}\Big{\}}.$$

$$bf\in H^1(\mathbb{R}),\text{mbox}moreover\Vert f{\Vert }_{H_b^1(\mathbb{R})}:=\Vert bf{\Vert }_{H^1(\mathbb{R})}.$$

$$\mathfrak{A}/b\in \mathrm{BMO}(\mathbb{R}),$$

$$\mathfrak{A}/b\in \mathrm{VMO}(\mathbb{R}).$$

$${\mathcal{C}}_{\Gamma }(f)(z):=\frac{1}{2\pi i}{\int }_{\Gamma }\frac{f(\zeta )}{z-\zeta }d\zeta .$$

$$|A(x_1)-A(x_2)|\le L|x_1-x_2|\text{for all }{x}_1,x_2\in \mathbb{R}.$$

$${\mathcal{C}}_{\Gamma }(f)(x):=p.v.\frac{1}{\pi i}{\int }_{\mathbb{R}}\frac{(1+i{A}^{\prime }(y))f(y)}{y-x+i(A(y)-A(x))}dy,$$

$$C_{\Gamma }(x,y)=\frac{1}{\pi i}\frac{1+i{A}^{\prime }(y)}{y-x+i(A(y)-A(x))}.$$

$${\tilde{\mathcal{C}}}_{\Gamma }(f)(x):=p.v.\frac{1}{\pi i}{\int }_{\mathbb{R}}\frac{f(y)}{y-x+i(A(y)-A(x))}dy.$$

$${\tilde{C}}_{\Gamma }(x,y)=\frac{1}{\pi i}\frac{1}{y-x+i(A(y)-A(x))}.$$

$$⟨{\mathcal{C}}_{\Gamma }(f),g{⟩}_{L^2(\mathbb{R})}$$

$$={\int }_{\mathbb{R}}p.v.\frac{1}{\pi i}{\int }_{\mathbb{R}}\frac{1}{y-x+i(A(y)-A(x))}g(x)dx(1+i{A}^{\prime }(y))f(y)dy$$

$$={\int }_{\mathbb{R}}b(y)({\tilde{\mathcal{C}}}_{\Gamma }{)}^{\ast }(g)(y)f(y)dy=⟨f,{\mathcal{C}}_{\Gamma }^{\ast }(g){⟩}_{L^2(\mathbb{R})}.$$

$${\mathcal{C}}_{\Gamma }^{\ast }(g)(x)=b(x)\cdot ({\tilde{\mathcal{C}}}_{\Gamma }{)}^{\ast }(g)(x).$$

$$\Pi (G,H)(x)=G(x)\cdot {\tilde{\mathcal{C}}}_{\Gamma }(H)(x)-H(x)\cdot ({\tilde{\mathcal{C}}}_{\Gamma }{)}^{\ast }(G)(x).$$

$$\|b\,f\|_{H^{1}(\mathbb{R})}\approx\inf\Big{\{}\sum_{k=1}^{\infty}\sum_{s=1}^{\infty}|\lambda_{s,k}|\|G^{k}_{s}\|_{L^{2}(\mathbb{R})}\|H^{k}_{s}\|_{L^{2}(\mathbb{R})}:b\,f=\sum_{k=1}^{\infty}\sum_{s=1}^{\infty}\lambda_{s,k}\,\Pi(G^{k}_{s},H^{k}_{s})\Big{\}}.$$

$$f(x)=\sum_{k=1}^{\infty}\sum_{s=1}^{\infty}\lambda_{s,k}\,\frac{1}{b}\,\Pi(G^{k}_{s},H^{k}_{s})(x)=\sum_{k=1}^{\infty}\sum_{s=1}^{\infty}\lambda_{s,k}\,\Pi_{b}\big{(}G^{k}_{s},\frac{H^{k}_{s}}{b}\big{)}(x).$$

$$\frac{1}{b}\Pi (G,H)(x)$$

$$f(x)=\sum_{k=1}^{\infty}\sum_{s=1}^{\infty}\lambda_{s,k}\,\Pi_{b}\big{(}g^{k}_{s},h^{k}_{s}\big{)}(x).$$

$$\|f\|_{H^{1}_{b}(\mathbb{R})}\approx\inf\Big{\{}\sum_{k=1}^{\infty}\sum_{s=1}^{\infty}|\lambda_{s,k}|\|g^{k}_{s}\|_{L^{2}(\mathbb{R})}\|h^{k}_{s}\|_{L^{2}(\mathbb{R})}\,:\,f=\sum_{k=1}^{\infty}\sum_{s=1}^{\infty}\lambda_{s,k}\,\Pi_{b}(g^{k}_{s},h^{k}_{s})\Big{\}}.$$

$$\Vert a{\Vert }_{\mathrm{BMO}}\approx \Vert [a,{\tilde{\mathcal{C}}}_{\Gamma }]{\Vert }_{L^p(\mathbb{R})\to L^p(\mathbb{R})}.$$

$$\left[\mathfrak{A}/b,{\mathcal{C}}_{\Gamma }\right](f)(x)$$

$${\Vert \left[\mathfrak{A}/b,{\mathcal{C}}_{\Gamma }\right](f)\Vert }_{L^p(\mathbb{R})}$$

Full text

2010* Mathematics Subject Classification:* Primary: 42B35, 42B25.

Key words: Cauchy integral, commutator, weak factorization, Hardy space $H^{1}$, ${\rm BMO}$, ${\rm VMO}$.

1. Introduction and Statement of Main Results

Given a bounded function $b:\mathbb{R}^{n}\to\mathbb{C}$ such that ${\rm Re}\,b(x)\geq 1$ for all $x\in\mathbb{R}^{n}$, the Meyer-type Hardy space $H^{1}_{b}(\mathbb{R}^{n})$ consists of those functions $f:\mathbb{R}^{n}\to\mathbb{C}$ such that the product $bf$ belongs to the Hardy space $H^{1}(\mathbb{R}^{n})$. The Meyer-type space of bounded mean oscillation, space ${\rm BMO}_{b}(\mathbb{R}^{n})$ consists of all functions $\mathfrak{A}:\mathbb{R}^{n}\to\mathbb{C}$ such that $\mathfrak{A}/b$ belongs to ${\rm BMO}(\mathbb{R}^{n})$ and is the dual of $H^{1}_{b}(\mathbb{R}^{n})$. These spaces were introduced by Yves Meyer [Me, Chapter XI, Section 10, p. 358], in dimension one in connection to the study of the Cauchy integral associated with a Lipschitz curve and the $T(b)$ theorem.

In this note we study the Meyer-type Hardy space $H^{1}_{b}(\mathbb{R})$ and its dual ${\rm BMO}_{b}(\mathbb{R})$ for $b(x)=1+iA^{\prime}(x)$ where $A^{\prime}\in L^{\infty}(\mathbb{R})$, via the Cauchy integral $\mathscr{C}_{\Gamma}$ associated to the Lipschitz curve $\Gamma=\{x+iA(x)\,:\,x\in\mathbb{R}\}$. We present a weak factorization of $H^{1}_{b}(\mathbb{R})$ in terms of the Cauchy integral $\mathscr{C}_{\Gamma}$. We also obtain a characterization of ${\rm BMO}_{b}(\mathbb{R})$ and of ${\rm VMO}_{b}(\mathbb{R})$, the Meyer-type space of vanishing mean oscillation, via boundedness and compactness of a suitable commutator with the Cauchy integral respectively.

The Cauchy integral associated with the Lipschitz curve $\Gamma$ is the integral operator $\mathscr{C}_{\Gamma}$ given by

[TABLE]

where $f\in C_{c}^{\infty}(\mathbb{R})$. Note that the Cauchy integral associated with the Lipschitz curve $\Gamma$ is not a standard Calderón-Zygmund operator because it lacks smoothness. The $L^{p}$-boundedness of $\mathscr{C}_{\Gamma}$ is equivalent to that of the related operator $\widetilde{\mathscr{C}}_{\Gamma}$ defined by

[TABLE]

Moreover, the kernel of $\widetilde{\mathscr{C}}_{\Gamma}$ satisfies standard size and smoothness estimates [LNWW] and is therefore bounded on $L^{p}(\mathbb{R})$ for $p\in(1,\infty)$. Note that in [LNWW] the operator $\mathscr{C}_{\Gamma}$ was denoted $\widetilde{\mathscr{C}}_{\Gamma}$ and viceversa. Hence while $\mathscr{C}_{\Gamma}(f)$ is initially defined for $f\in C_{c}^{\infty}(\mathbb{R})$, the operator $\mathscr{C}_{\Gamma}$ can be extended to all $f\in L^{p}(\mathbb{R})$, for each $p\in(1,\infty)$.

The related operator $\widetilde{\mathscr{C}}_{\Gamma}$ and its commutator with functions in ${\rm BMO}(\mathbb{R})$ were studied by Li, Nguyen, Ward, and Wick in [LNWW]. In this setting, one could appeal to a weak factorization for $H^{1}(\mathbb{R}^{n})$ in terms of multilinear Calderón-Zygmund operators, due to Li and Wick [LW, Theorem 1.3], to obtain the desired characterization of ${\rm BMO}(\mathbb{R})$ via boundedness of the commutator, and of ${\rm VMO}(\mathbb{R})$ via compactness of the commutator.

We want to study the Meyer-type Hardy space, bounded mean oscillation space, and vanishing mean oscillation space: $H^{1}_{b}(\mathbb{R})$, ${\rm BMO}_{b}(\mathbb{R})$, and ${\rm VMO}_{b}(\mathbb{R})$, via the rougher operator $\mathscr{C}_{\Gamma}$. As it turns out, we can derive these results from the results for the related Cauchy integral operator [LNWW]. Nevertheless we also present a direct constructive proof of the weak factorization valid for $H^{1}_{b}(\mathbb{R})$ that maybe of independent interest.

We now state our main results. For $b(x)=1+iA^{\prime}(x)$, we introduce the bilinear form associated with $b$ as follows:

[TABLE]

where $\mathscr{C}_{\Gamma}^{*}$ is the adjoint operator to $\mathscr{C}_{\Gamma}$.

Theorem 1.1**.**

For any $f\in H^{1}_{b}(\mathbb{R})$ there exist sequence $\{\lambda_{j}^{k}\}_{j,k\in\mathbb{Z}}\in\ell^{1}$ and functions $g_{j}^{k},h_{j}^{k}\in L^{\infty}(\mathbb{R})$ with compact supports such that

[TABLE]

Moreover, we have that:

[TABLE]

The commutator $[g,T]$ of a function $g$ and an operator $T$ denotes the new operator acting on suitable functions $f$ defined by $[g,T](f):=g\,T(f)-T(gf)$. It is well known that a function $a\in\rm BMO(\mathbb{R})$ (respectively $a\in{\rm VMO}(\mathbb{R})$) if and only if $[a,H]$, the commutator of $a$ with $H$ the Hilbert transform, is a bounded operator on $L^{p}(\mathbb{R})$ [CRW] (respectively is a compact operator [U1]). In [LNWW], functions $a$ in $\rm BMO_{b}(\mathbb{R})$ (respectively in ${\rm VMO}_{b}(\mathbb{R})$) were characterized via boundedness (respectively compactness) of $[a,\widetilde{\mathscr{C}}_{\Gamma}]$, the commutator with the related Cauchy operator. To characterize $\rm BMO_{b}(\mathbb{R})$ and ${\rm VMO}_{b}(\mathbb{R})$ we will consider the commutator of the Cauchy integral not with functions in $\rm BMO_{b}(\mathbb{R})$ or ${\rm VMO}_{b}(\mathbb{R})$ but with those functions divided by the accretive function $b$. In other words we will consider for the next theorems the commutator $[\mathfrak{A}/b,\mathscr{C}_{\Gamma}]$ where $\mathfrak{A}$ is in ${\rm BMO}_{b}(\mathbb{R})$ or in ${\rm VMO}_{b}(\mathbb{R})$.

Theorem 1.2**.**

Let $b(x)=1+iA^{\prime}(x)$, and $p\in(1,\infty)$. If $\mathfrak{A}\in{\rm BMO}_{b}(\mathbb{R})$, then we have

[TABLE]

Conversely, for any complex function $\mathfrak{A}$ such that ${\mathfrak{A}/b}$ is a real-valued function and ${\mathfrak{A}/b}\in L^{1}_{loc}(\mathbb{R})$, if $\left\|\left[{\mathfrak{A}/b},\mathscr{C}_{\Gamma}\right]\right\|_{L^{p}(\mathbb{R})\to L^{p}(\mathbb{R})}<\infty$, then $\mathfrak{A}\in{\rm BMO}_{b}(\mathbb{R})$ with

[TABLE]

Theorem 1.3**.**

Let $b(x)=1+iA^{\prime}(x)$ and $p\in(1,\infty)$. If $\mathfrak{A}\in{\rm VMO}_{b}(\mathbb{R})$, then we have $\left[{\mathfrak{A}/b},C_{\Gamma}\right]$ is compact on $L^{p}(\mathbb{R})$. Conversely, for any complex function $\mathfrak{A}\in{\rm BMO}_{b}(\mathbb{R})$ such that ${\mathfrak{A}/b}$ is a real-valued function and ${\mathfrak{A}/b}\in L^{1}_{loc}(\mathbb{R})$, if $\left[{\mathfrak{A}/b},\mathscr{C}_{\Gamma}\right]$ is compact on $L^{p}(\mathbb{R})$, then $\mathfrak{A}\in{\rm VMO}_{b}(\mathbb{R})$.

Note that it is possible to deduce these three theorems as corollaries from the results in [LNWW] and [LW] directly, as we will show in Section 3.

The paper is organized as follows. In Section 2 we collect the necessary preliminaries needed to explain the result. In Section 3 we provide a connection between the classical Hardy and ${\rm BMO}$ spaces and the spaces introduced by Meyer. In Section 4 we provide a second proof of Theorem 1.1 using a clever construction due to Uchiyama [U].

We use the standard notation that $A\lesssim B$ to mean that there exists an absolute constant $C$ such that $A\leq CB$; $A\gtrsim B$ has the analogous definition. Finally, $A\approx B$ if $A\lesssim B$ and $B\lesssim A$. We use $\langle f,g\rangle_{L^{2}(\mathbb{R})}$ to denote the $L^{2}$-pairing $\int_{\mathbb{R}}f(x)\,g(x)\,dx$. We denote $C_{c}^{\infty}(\mathbb{R})$ the space of compactly supported infinitely differentiable functions on $\mathbb{R}$. We denote by $\chi_{I}$ the characteristic function of the set $I\subset\mathbb{R}$, defined by $\chi_{I}(x)=1$ if $x\in I$ and $\chi_{I}(x)=0$ otherwise.

2. Preliminaries

In this section we introduce basic notions of accretive functions; the classical spaces: Hardy space $H^{1}(\mathbb{R})$, the space of bounded mean oscillation functions ${\rm BMO}(\mathbb{R})$, and the space of vanishing mean oscillation ${\rm VMO}(\mathbb{R})$; and their counterparts, the Meyer-type Hardy spaces: $H^{1}_{b}(\mathbb{R})$, ${\rm BMO}_{b}(\mathbb{R})$, and ${\rm VMO}_{b}(\mathbb{R})$, for $b$ an accretive function. We also introduce the Cauchy integral operator $\mathscr{C}_{\Gamma}$ associated to a Lipschitz curve $\Gamma$ and the related Cauchy integral operator $\widetilde{\mathscr{C}}_{\Gamma}$.

A function $b:\mathbb{R}\to\mathbb{C}$ is accretive if $b\in L^{\infty}(\mathbb{R})$ and ${\rm Re}\,b(x)\geq\delta>0$ for all $x\in\mathbb{R}$.

A locally integrable real-valued function $f:\mathbb{R}\rightarrow\mathbb{R}$ is said to be of bounded mean oscillation, written $f\in\rm BMO$ or $f\in\rm BMO(\mathbb{R})$, if

[TABLE]

Here the supremum is taken over all intervals $I$ in $\mathbb{R}$ and $f_{I}:=\frac{1}{|I|}\int_{I}f(y)\,dy$ is the average of the function $f$ over the interval $I$.

A ${\rm BMO}$ function $f:\mathbb{R}\rightarrow\mathbb{R}$ is said to be of vanishing mean oscillation, written $f\in{\rm VMO}$ or $f\in{\rm VMO}(\mathbb{R})$, if the following three behaviors occur for small, large and far from the origin intervals respectively,

• (i)

$\quad\quad\;\displaystyle{\lim_{\delta\to 0}\sup_{I:|I|<\delta}\frac{1}{|I|}\int_{I}|f(x)-f_{I}|\,dx=0,}$

• (ii)

$\quad\quad\;\displaystyle{\lim_{R\to\infty}\sup_{I:|I|>R}\frac{1}{|I|}\int_{I}|f(x)-f_{I}|\,dx=0,\quad}$ and

• (iii)

$\quad\quad\;\displaystyle{\lim_{R\to\infty}\sup_{I:I\cap I(0,R)=\emptyset}\frac{1}{|I|}\int_{I}|f(x)-f_{I}|\,dx=0.}$

The Hardy space $H^{1}(\mathbb{R})$ consists of those integrable functions $f\,:\,\mathbb{R}\to\mathbb{\mathbb{R}}$ that admit an atomic decomposition $f(x)=\sum_{j=1}^{\infty}\lambda_{j}a_{j}(x)$ where $\sum_{j=1}^{\infty}|\lambda_{j}|<\infty$ and the functions $a_{j}(x)$ are $L^{\infty}$-atoms (respectively $L^{2}$-atoms) in the sense that: each atom $a_{j}$ is supported on an interval $I_{j}$ and the following $L^{\infty}$-size condition $\|a_{j}\|_{L^{\infty}(\mathbb{R})}\leq 1/|I_{j}|$ (respectively, $L^{2}$-size condition $\|a_{j}\|_{L^{2}(\mathbb{R})}\leq C|I_{j}|^{-1/2}$) and cancellation condition $\int_{\mathbb{R}}a_{j}(x)\,b(x)\,dx=0$ hold. The $H^{1}$-norm can be defined using either type of atoms, for example

[TABLE]

If instead we use $L^{\infty}$-atoms we will get an equivalent norm [Gra, Section 6.6.4]. It is well known that ${\rm BMO}$ is the dual of $H^{1}(\mathbb{R})$ [FS].

A function $f:\mathbb{R}\to\mathbb{C}$ is said to be in $H^{1}_{b}(\mathbb{R})$, the Meyer-type Hardy space associated with the accretive function $b$ if and only if

[TABLE]

In other words, the function $f\in H^{1}_{b}(\mathbb{R})$ admits an atomic decomposition $f(x)=\sum_{j=1}^{\infty}\lambda_{j}a_{j}(x)$ where $\sum_{j=1}^{\infty}|\lambda_{j}|<\infty$ and the functions $a_{j}(x)$ are $L^{\infty}$-atoms (respectively $L^{2}$-atoms) in the sense that: each $H^{1}_{b}$ atom $a_{j}$ is supported on an interval $I_{j}$ and the following $L^{\infty}$-size condition $\|a_{j}\|_{L^{\infty}(\mathbb{R})}\leq 1/|I_{j}|$ (respectively, $L^{2}$-size condition $\|a_{j}\|_{L^{2}(\mathbb{R})}\approx\|ba_{j}\|_{L^{2}(\mathbb{R})}\leq C|I_{j}|^{-1/2}$), and cancellation condition $\int_{\mathbb{R}}a_{j}(x)\,b(x)\,dx=0$ hold.

A locally integrable function $\mathfrak{A}:\mathbb{R}\rightarrow\mathbb{C}$ is said to be in $\rm BMO_{b}(\mathbb{R})$, the Meyer-type ${\rm BMO}$ space associated with the accretive function $b$ , if

[TABLE]

and we define its norm naturally to be $\|\mathfrak{A}\|_{\rm BMO_{b}(\mathbb{R})}:=\left\|{\mathfrak{A}/b}\right\|_{\rm BMO(\mathbb{R})}.$ As a consequence of the $H^{1}$-${\rm BMO}$ duality ${\rm BMO}_{b}(\mathbb{R})$ is the dual of $H^{1}_{b}(\mathbb{R})$ [Me].

A locally integrable function $\mathfrak{A}:\mathbb{R}\rightarrow\mathbb{C}$ is said to be in ${\rm VMO}_{b}(\mathbb{R})$, the Meyer-type ${\rm VMO}$ space associated with the accretive function $b$ , if

[TABLE]

Suppose $\Gamma$ is a curve in the complex plane $\mathbb{C}$ and $f$ is a function defined on the curve $\Gamma$. The Cauchy integral of $f$ is the operator $\mathcal{C}_{\Gamma}$ defined on the complex plane for $z\notin\Gamma$ by

[TABLE]

A curve $\Gamma$ is said to be a Lipschitz curve if it can be written in the form $\Gamma=\{x+iA(x):x\in\mathbb{R}\}$ where $A:\mathbb{R}\rightarrow\mathbb{R}$ satisfies a Lipschitz condition

[TABLE]

The best constant $L$ in (2.2) is referred to as the Lipschitz constant of $\Gamma$ or of $A(x)$. One can show that $A$ satisfies a Lipschitz condition if and only if $A$ is differentiable almost everywhere on $\mathbb{R}$ and $A^{\prime}\in L^{\infty}(\mathbb{R})$. The Lipschitz constant is $L=\|A^{\prime}\|_{\infty}$.

The Cauchy integral associated with the Lipschitz curve $\Gamma$ is the singular integral operator $\mathscr{C}_{\Gamma}$ defined for $x\in\mathbb{R}$ and acting on functions $f\in C_{c}^{\infty}(\mathbb{R})$ by

[TABLE]

where $f\in C_{c}^{\infty}(\mathbb{R})$. The kernel of $\mathscr{C}_{\Gamma}$ is given by

[TABLE]

Note that this is not a standard Calderón–Zygmund kernel because the function $1+iA^{\prime}$ does not necessarily possess any smoothness. As noted in [Gra, p.289], the $L^{p}$-boundedness of $\mathscr{C}_{\Gamma}$ is equivalent to that of the related operator $\widetilde{\mathscr{C}}_{\Gamma}$ defined for $x\in\mathbb{R}$ by

[TABLE]

Moreover, the kernel of $\widetilde{\mathscr{C}}_{\Gamma}$ is given by

[TABLE]

The kernel $C_{\Gamma}(x,y)$ of $\widetilde{\mathscr{C}}_{\Gamma}$ satisfies standard size and smoothness111Namely: (size) $|C_{\Gamma}(x,y)|\lesssim 1/|x-y|$ for all $x,y\in\mathbb{R}$ and (smoothness) $|C_{\Gamma}(x,y)-C_{\Gamma}(x_{0},y)|+|C_{\Gamma}(y,x)-C_{\Gamma}(y,x_{0})|\lesssim|x-x_{0}|/|x-y|^{2}$ for all $x,x_{0},y\in\mathbb{R}$ such that $|x-x_{0}|\leq|y-x|/2$. estimates [LNWW, Lemma 3.3] and is therefore bounded on $L^{p}(\mathbb{R})$ for $p\in(1,\infty)$. Therefore, while the operator $\mathscr{C}_{\Gamma}(f)$ is initially defined for $f\in C_{c}^{\infty}(\mathbb{R})$, it can be extended to all $f\in L^{p}(\mathbb{R})$, for each $p\in(1,\infty)$.

An operator $T$ defined on $L^{p}(\mathbb{R})$ is compact on $L^{p}(\mathbb{R})$ if $T$ maps bounded subsets of $L^{p}(\mathbb{R})$ into precompact sets. In other words, for all bounded sets $E\subset L^{p}(\mathbb{R})$, $T(E)$ is precompact. A set S is precompact if its closure is compact.

3. From Classical Spaces to Meyer Hardy spaces

In this section we take advantage of the known weak factorization result for $H^{1}(\mathbb{R})$ in terms of the Calderón-Zygmund singular integral operator $\widetilde{\mathscr{C}}_{\Gamma}$ as well as the characterization of ${\rm BMO}(\mathbb{R})$ via the boundedness of the commutator with $\widetilde{\mathscr{C}}_{\Gamma}$ and of ${\rm VMO}(\mathbb{R})$ via the compactness of the same commutator [LNWW].

We first consider the adjoint operator $\mathscr{C}_{\Gamma}^{*}(g)$. By a direct calculation, we can verify that for $f,g\in L^{2}(\mathbb{R})$,

[TABLE]

We therefore conclude that

[TABLE]

Note that $(\widetilde{\mathscr{C}}_{\Gamma})^{*}=-\widetilde{\mathscr{C}}_{\Gamma}$.

We now use the weak factorization for $H^{1}(\mathbb{R})$ –valid for $m$-linear Calderón-Zygmund operators [LW, Theorem 1.3]– for the particular Calderón-Zygmund operator $\widetilde{\mathscr{C}}_{\Gamma}$ [LNWW], to obtain the desired weak factorization for the Meyer-type Hardy space $H_{b}^{1}(\mathbb{R})$.

First Proof of Theorem 1.1.

The function $f\in H^{1}_{b}(\mathbb{R})$ if and only if $bf\in H^{1}(\mathbb{R})$ but by weak factorization of $H^{1}(\mathbb{R})$ there are a sequence $\{\lambda_{s,k}\}_{s,k\in\mathbb{Z}}$ and compactly supported bounded functions $G^{k}_{s}$ and $H^{k}_{s}$ such that $b\,f=\sum_{k=1}^{\infty}\sum_{s=1}^{\infty}\lambda_{s,k}\,\Pi(G^{k}_{s},H^{k}_{s})$. Where the bilinear form $\Pi(G,H)$ is defined by

[TABLE]

Moreover,

[TABLE]

Therefore

[TABLE]

The last identity since by definition (1.1) of the bilinear form $\Pi_{b}(g,h)$, the fact that $\mathscr{C}_{\Gamma}(f)=\widetilde{\mathscr{C}}_{\Gamma}(bf)$, and identity (3.1), we have that

[TABLE]

Let $g^{k}_{s}:=G^{k}_{s}$ and $h^{k}_{s}:=H^{k}_{s}/b$, both are compactly supported bounded functions and

[TABLE]

Moreover $\|f\|_{H^{1}_{b}(\mathbb{R})}=\|b\,f\|_{H^{1}(\mathbb{R})}$, therefore

[TABLE]

The last identity because $g^{k}_{s}=G^{k}_{s}$ and $\|h^{k}_{s}\|_{L^{2}(\mathbb{R})}\approx\|b\,h^{k}_{s}\|_{L^{2}(\mathbb{R})}=\|H^{k}_{s}\|_{L^{2}(\mathbb{R})}$ since $b$ is an accretive function. This proves Theorem 1.1. ∎

If we know how to construct the functions $G^{k}_{s}$ and $H^{k}_{s}$ then we know how to construct the functions $g^{k}_{s}$ and $h^{k}_{s}$, and viceversa. In the next section we provide an explicit construction of the functions $g^{k}_{s}$ and $h^{k}_{s}$, following Uchiyama’s blueprint directly in our setting.

Before proceeding, we provide proofs of Theorem 1.2 and of Theorem 1.3 relying on the corresponding results for the related Cauchy integral operator $\widetilde{\mathscr{C}}_{\Gamma}$. Namely, $a\in{\rm BMO}$ (respectively in ${\rm VMO}$) if and only if $[a,\widetilde{\mathscr{C}}_{\Gamma}]$ is bounded on $L^{p}(\mathbb{R})$ (respectively, is compact in $L^{p}(\mathbb{R})$) for $p\in(1,\infty)$. Furthermore, the following norm comparability holds

[TABLE]

Proof of Theorem 1.2.

For $b(x)=1+iA^{\prime}(x)$, suppose $\mathfrak{A}$ is in $\rm BMO_{b}(\mathbb{R})$, that is $\mathfrak{A}/b\in\rm BMO(\mathbb{R})$; a direct calculation, using that $\mathscr{C}_{\Gamma}(g)=\widetilde{\mathscr{C}}_{\Gamma}(bg)$, shows that

[TABLE]

Thus, since by [LNWW, Theorem 1.1] the commutator $[\mathfrak{A}/b,\widetilde{\mathscr{C}}_{\Gamma}]$ is bounded on $L^{p}(\mathbb{R})$, we get

[TABLE]

Conversely, for any given complex function $\mathfrak{A}$ such that ${\mathfrak{A}/b}$ is a real-valued function, ${\mathfrak{A}/b}\in L^{1}_{loc}(\mathbb{R})$ and $\left\|\left[{\mathfrak{A}/b},\mathscr{C}_{\Gamma}\right]\right\|_{L^{p}(\mathbb{R})\to L^{p}(\mathbb{R})}<\infty$, we see that

[TABLE]

Hence, the commutator \big{[}{\mathfrak{A}/b},\widetilde{\mathscr{C}}_{\Gamma}\big{]} is bounded on $L^{p}(\mathbb{R})$ and by [LNWW, Theorem 1.1] we conclude that ${\mathfrak{A}/b}$ is in $\rm BMO(\mathbb{R})$ and

[TABLE]

Hence, we conclude that ${\mathfrak{A}}$ is in $\rm BMO_{b}(\mathbb{R})$ and

[TABLE]

This finishes the proof of Theorem 1.2. ∎

Similar considerations yield the proof of Theorem 1.3 from the knowledge that $a\in{\rm VMO}$ if and only if $[a,\widetilde{\mathscr{C}}_{\Gamma}]$ is a compact operator on $L^{p}(\mathbb{R})\to L^{p}(\mathbb{R})$ [LNWW, Theorem 1.2].

Proof Theorem 1.3.

For $b(x)=1+iA^{\prime}(x)\in L^{\infty}(\mathbb{R})$, suppose $\mathfrak{A}$ is in ${\rm VMO}_{b}(\mathbb{R})$, that is $\mathfrak{A}/b\in{\rm VMO}(\mathbb{R})$. Therefore by [LNWW, Theorem 1.2] the commutator $[\mathfrak{A}/b,\widetilde{\mathscr{C}}_{\Gamma}]$ is compact. Let $E$ be a bounded subset of $L^{p}(\mathbb{R})$, then $bE$ is a bounded subset of $L^{p}(\mathbb{R})$ since

[TABLE]

Therefore $[\mathfrak{A}/b,\widetilde{\mathscr{C}}_{\Gamma}](bE)$ is a precompact set. Recall that $\left[{\mathfrak{A}/b},\widetilde{\mathscr{C}}_{\Gamma}\right](bf)(x)=\left[{\mathfrak{A}/b},\mathscr{C}_{\Gamma}\right](f)(x)$ for all $f\in L^{p}(\mathbb{R})$. Thus

[TABLE]

Hence $[\mathfrak{A}/b,\mathscr{C}_{\Gamma}](E)$ is a precompact set for all given bounded subsets $E$ of $L^{p}(\mathbb{R})$. By definition $[\mathfrak{A}/b,\mathscr{C}_{\Gamma}]$ is compact.

Conversely, suppose $[\mathfrak{A}/b,\mathscr{C}_{\Gamma}]$ is compact. Then given a bounded subset $F$ of $L^{p}(\mathbb{R})$, $F/b$ is also a bounded subset of $L^{p}(\mathbb{R})$ since $\|b\|_{L^{\infty}}\geq 1$. Therefore $[\mathfrak{A}/b,\mathscr{C}_{\Gamma}](F/b)$ is precompact, but as before,

[TABLE]

Thus $[\mathfrak{A}/b,\widetilde{\mathscr{C}}_{\Gamma}](F)$ is a precompact set for all bounded subsets $F$ of $L^{p}(\mathbb{R})$. By definition $[\mathfrak{A}/b,\widetilde{\mathscr{C}}_{\Gamma}]$ is a compact operator in $L^{p}(\mathbb{R})$ and by [LNWW, Theorem 1.2] we conclude that $\mathfrak{A}/b\in{\rm VMO}(\mathbb{R})$, and therefore $\mathfrak{A}\in{\rm VMO}_{b}(\mathbb{R})$. This finishes the proof of Theorem 1.3. ∎

4. Weak Factorization of the Meyer Hardy space - Uchiyama’s construction

In this section we present a constructive proof of functions $g^{k}_{s}$ and $h^{k}_{s}$ for $k,s\geq 1$, appearing in the weak factorization of $H^{1}_{b}(\mathbb{R})$. This argument follows Uchiyama’s procedure closely [U].

4.1. The upper bound in Theorem 1.1

Given a function $f\in H^{1}_{b}(\mathbb{R})$, suppose we have a factorization of the form f=\sum_{k=1}^{\infty}\sum_{s=1}^{\infty}\lambda_{s,k}\,\Pi_{b}\big{(}g^{k}_{s},h^{k}_{s}\big{)} with $\{\lambda_{s,k}\}\in\ell^{1}$ and $g^{k}_{s}$ and $h^{k}_{s}$ compactly supported and bounded functions, as claimed in Theorem 1.1. Then it suffices to verify the following Lemma 4.1 to conclude that

[TABLE]

Lemma 4.1**.**

Let $g,h\in L^{\infty}(\mathbb{R})$ with compact supports. Then $\Pi_{b}(g,h)$ is in $H^{1}_{b}(\mathbb{R})$ with

[TABLE]

Proof.

We first point out that for any $g,h\in L^{\infty}(\mathbb{R})$ with compact supports, $\Pi_{b}(g,h)$ is compactly supported in ${\rm supp}(g)\cup{\rm supp}(h)$. Next, it is easy to see that $\Pi_{b}(g,h)\in L^{2}(\mathbb{R})$, using that $\mathscr{C}_{\Gamma}$ is a bounded operator in $L^{2}(\mathbb{R})$, indeed,

[TABLE]

Moreover, since by definition of adjoint $\langle h,\mathscr{C}_{\Gamma}^{*}(g)\rangle_{L^{2}(\mathbb{R})}=\langle\mathscr{C}_{\Gamma}(h),g\rangle_{L^{2}(\mathbb{R})}$, the following cancellation holds,

[TABLE]

Hence, it is clear that up to a multiplication by certain constant, the bilinear form $\Pi_{b}(g,h)(x)$ is a $L^{2}$-atom of $H^{1}_{b}(\mathbb{R})$, that is, $\Pi_{b}(g,h)\in H^{1}_{b}(\mathbb{R})$.

Now it suffices to verify the $H^{1}_{b}(\mathbb{R})$ norm of $\Pi_{b}(g,h)$ is controlled by an absolute multiple of $\left\|g\right\|_{L^{2}(\mathbb{R})}\left\|h\right\|_{L^{2}(\mathbb{R})}$. A simple duality computation shows for $\mathfrak{A}\in{\rm BMO}_{b}(\mathbb{R})$ and for any $g,h\in L^{\infty}(\mathbb{R}^{n})$ with compact supports:

[TABLE]

Remember that $\langle f,g\rangle_{L^{2}(\mathbb{R})}$ denotes the $L^{2}$ pairing $\int_{\mathbb{R}}f(x)\,g(x)\,dx$, not the $L^{2}$ inner product. Thus, from the upper bound as in Theorem 1.2, we obtain that

[TABLE]

This, together with the duality result of [Me], $H^{1}_{b}(\mathbb{R})^{*}={\rm BMO}_{b}(\mathbb{R})$, shows that

[TABLE]

∎

4.2. The factorization and the lower bound in Theorem 1.1

The proof of the factorization and of the lower bound in Theorem 1.1 is more algorithmic in nature and follows a proof strategy pioneered by Uchiyama in [U]. We begin with a fact that will play a prominent role in the algorithm below. It is a modification of a related fact for the standard Hardy space $H^{1}(\mathbb{R})$.

Lemma 4.2**.**

Let $b(x)=1+iA^{\prime}(x)$ with $A^{\prime}\in L^{\infty}(\mathbb{R})$. Suppose $f$ is a function satisfying: $\int_{\mathbb{R}}f(x)\,b(x)\,dx=0$, and $|f(x)|\leq\chi_{I(x_{0},1)}(x)+\chi_{I(y_{0},1)}(x)$, where $|x_{0}-y_{0}|:=M>100$ and $I(z_{0},L):=\{z\in\mathbb{R}\,:\,|z-z_{0}|<L\}$. Then $f\in H^{1}_{b}(\mathbb{R})$ and we have

[TABLE]

The lemma when $b\equiv 1$ is stated in [LW, Lemma 2.2] without proof, the authors refer the reader to [DLWY, Lemma 3.1] and [LW2, Lemma 4.3] where the corresponding lemma, in the Bessel and Neumann Laplacian settings respectively, is stated and proved. We can not apply directly [LW, Lemma 2.2] because although $F=bf$ will satisfy $\int_{\mathbb{R}}F(x)\,dx=0$ by hypothesis, it will not satisfy that $|F(x)|\leq\chi_{I(x_{0},1)}(x)+\chi_{I(y_{0},1)}(x)$, instead it will satisfy |F(x)|\leq|b(x)|\big{(}\chi_{I(x_{0},1)}(x)+\chi_{I(y_{0},1)}(x)\big{)}. But we could apply it to $F_{0}=bf/\|b\|_{L^{\infty}(\mathbb{R})}$, since $|F_{0}(x)|\leq\chi_{I(x_{0},1)}(x)+\chi_{I(y_{0},1)}(x)$, to conclude that $F_{0}\in H^{1}(\mathbb{R})$ and $\|F_{0}\|_{H^{1}(\mathbb{R})}\lesssim\log M$. Finally since $\|F_{0}\|_{H^{1}(\mathbb{R})}=\|bf\|_{H^{1}(\mathbb{R})}/\|b\|_{L^{\infty}(\mathbb{R})}$ we conclude that $f\in H^{1}_{b}(\mathbb{R})$ and

[TABLE]

Nevertheless, for completeness, we present here a direct construction of an atomic decomposition in $H^{1}_{b}(\mathbb{R})$ for $f$ that yields the estimate claimed in Lemma 4.2 that could have an interest in itself, it also provides a proof for [LW, Lemma 2.2] by setting $b\equiv 1$. This construction yields an atomic decomposition for $f=\sum_{j\in\mathbb{Z}}\lambda_{j}a_{j}$. However the $H^{1}_{b}$ $L^{\infty}$-atoms $a_{j}$ built in the proof of Lemma 4.2 for the specific given $f$ are not the $H^{1}$ $L^{\infty}$-atoms one would get by multiplying by $\|b\|_{\infty}/b$ the $H^{1}$ $L^{\infty}$-atoms $A_{j}$ obtained by the same procedure applied to $F_{0}$ when $b\equiv 1$.

Proof of Lemma 4.2 .

Suppose $f$ satisfies the conditions as stated in the lemma above. We will show by construction that $f$ has an atomic decomposition with respect to the $H_{b}^{1}(\mathbb{R})$ $L^{\infty}$-atoms, using an idea from Coifman [CW]. To see this, we first define two functions $f_{1}(x)$ and $f_{2}(x)$ by

[TABLE]

Then we have $f=f_{1}+f_{2}$ and by hypothesis and definition

[TABLE]

Define

[TABLE]

Then we claim that $a_{1}^{1}:=(\alpha_{1}^{1})^{-1}f_{1}^{1}$ is an $H^{1}_{b}(\mathbb{R})$ $L^{\infty}$-atom. First, by definition $a_{1}^{1}$ is supported on $I(x_{0},2)$. Moreover, we have that

[TABLE]

and that

[TABLE]

Thus, $a_{1}^{1}$ is an $H^{1}_{b}(\mathbb{R})$ $L^{\infty}$-atom. We also have the following estimate for the coefficient $\alpha_{1}^{1}$.

[TABLE]

Here we used the facts that $\|b\|_{L^{\infty}(\mathbb{\mathbb{R}})}<\infty$, $f_{1}\leq\chi_{I(x_{0},1)}$, $|I(x_{0},L)|=2L$, and

[TABLE]

Moreover, we see that

[TABLE]

For $g_{1}^{1}(x)$, we further write it as

[TABLE]

with

[TABLE]

Again, we define

[TABLE]

and following similar estimates as for $a_{1}^{1}$, we see that $a_{1}^{2}$ satisfies the compact support condition and the size condition $\|a_{1}^{2}\|_{\infty}\leq\frac{1}{|I(x_{0},4)|}$. Hence, it suffice to see that it also satisfies the cancellation condition with respect to $b$. In fact,

[TABLE]

As a consequence, we see that $a_{1}^{2}$ is an $H^{1}_{b}(\mathbb{R})$ $L^{\infty}$-atom. Moreover, we have the following estimate for the coefficient $a_{1}^{2}$.

[TABLE]

Here again we use the fact that for every $L>0$,

[TABLE]

Then we have

[TABLE]

Continuing in this fashion we see that for $i\in\{1,2,...,i_{0}\}$,

[TABLE]

where for $i\in\{2,...,i_{0}\}$,

[TABLE]

Here we choose $i_{0}$ to be the smallest positive integer such that $I(y_{0},1)\subset I(x_{0},2^{i_{0}})$. Then from the condition that $|x_{0}-y_{0}|=M$, we obtain that

[TABLE]

Moreover, for $i\in\{1,2,...,i_{0}\}$, we have the estimate of the coefficients as follows.

[TABLE]

Following the same steps, we also obtain that for $i\in\{1,2,...,i_{0}\}$,

[TABLE]

where for $i\in\{2,...,i_{0}\}$,

[TABLE]

Similarly, for $i\in\{1,2,...,i_{0}\}$, we can verify that each $a_{2}^{i}$ is an $H^{1}_{b}(\mathbb{R})$ $L^{\infty}$-atom and the coefficient satisfies

[TABLE]

Combining the decompositions above, we obtain that

[TABLE]

We now consider the tail $g_{1}^{i_{0}}(x)+g_{2}^{i_{0}}(x)$. To handle that, consider the interval $\overline{I}$ centered at the point ${x_{0}+y_{0}\over 2}$ with sidelength $2^{i_{0}+1}$. Then, it is clear that $I(x_{0},1)\cup I(y_{0},1)\subset\overline{I}$, and that $I(x_{0},2^{i_{0}}),I(y_{0},2^{i_{0}})\subset\overline{I}$. Thus, since by hypothesis $\int_{\mathbb{R}}f(y)\,b(y)\,dy=0$, we get that

[TABLE]

Hence, we write

[TABLE]

For $j=1,2$, we now define

[TABLE]

Again we can verify that for $j=1,2$, $a_{j}^{i_{0}+1}$ is an $H^{1}_{b}(\mathbb{R})$ $L^{\infty}$-atom supported in $I$ with the appropriate size and cancellation conditions

[TABLE]

Moreover, we also have

[TABLE]

Thus, we obtain an atomic decomposition for $f$

[TABLE]

which implies that $f\in H^{1}_{b}(\mathbb{R})$ and

[TABLE]

This finishes the proof of Lemma 4.2. ∎

Repeating the proof we get that if $r>0$, $\int_{\mathbb{R}}f(x)\,b(x)\,dx=0$ and $|f(x)|\leq\chi_{I(x_{0},r)}(x)+\chi_{I(y_{0},r)}(x)$ where $|x_{0}-y_{0}|\geq rM$, then $f\in H^{1}_{b}(\mathbb{R})$ and

[TABLE]

The additional $r$ comes from the estimates of the coefficients $|\alpha^{i}_{j}|\lesssim r$ for $i=1,\dots,i_{0}+1$ and $j=1,2$ where $i_{0}\sim\log M$.

Ideally, given an $H^{1}_{b}(\mathbb{R})$-atom $a$, we would like to find $g,h\in L^{2}(\mathbb{R})$ such that $\Pi_{b}(g,h)=a$ pointwise. While this can not be accomplished in general, the theorem below shows that it is “almost” true.

Theorem 4.3**.**

For every $H^{1}_{b}(\mathbb{R})$ $L^{\infty}$-atom $a(x)$ and for all $\varepsilon>0$ there exist a large positive number $M$ and $g,h\in L^{\infty}(\mathbb{R})$ with compact supports such that:

[TABLE]

and $\left\|g\right\|_{L^{2}(\mathbb{R})}\left\|h\right\|_{L^{2}(\mathbb{R})}\lesssim M$.

Proof.

Let $a(x)$ be an $H^{1}_{b}(\mathbb{R})$ $L^{\infty}$-atom, supported in $I(x_{0},r)$, the interval centred at $x_{0}$ with radius $r$. We first consider the construction of the explicit bilinear form $\Pi_{b}(h,g)$ and the approximation to $a(x)$. To begin with, fix $\varepsilon>0$. Choose $M\in[100,\infty)$ sufficiently large so that

[TABLE]

Now select $y_{0}\in\mathbb{R}$ such that $y_{0}-x_{0}={Mr}$. For this $y_{0}$ and for any $y\in I(y_{0},r)$ and any $x\in I(x_{0},r)$, we have $|x-y|>{Mr/2}$. We set

[TABLE]

We first note that

[TABLE]

In fact, from the expression of $(\widetilde{\mathscr{C}}_{\Gamma})^{*}(g)(x_{0})=-\widetilde{\mathscr{C}}_{\Gamma}(g)(x_{0})$ we have that

[TABLE]

As a consequence, we get that the claim (4.4) holds.

From the definitions of the functions $g$ and $h$, we obtain that $\operatorname{supp}(g)=I(y_{0},r)$ and $\operatorname{supp}(h)=I(x_{0},r)$. Moreover, from (4.4) and the size estimate for the atom, we obtain that

[TABLE]

And we also get that

[TABLE]

Hence $\|g\|_{L^{2}(\mathbb{R})}\|h\|_{L^{2}(\mathbb{R})}\lesssim M$. Now write

[TABLE]

We first turn to $W_{1}(x)$. By definition and using equation (3.1), we have that

[TABLE]

Thus, since $(\widetilde{\mathscr{C}}_{\Gamma})^{*}=-\widetilde{\mathscr{C}}_{\Gamma}$, we get that for every $x\in I(x_{0},r)$,

[TABLE]

Here we used the standard smoothness estimate for the Calderón-Zygmund kernel $\widetilde{C}_{\Gamma}(x,y)$ of $\widetilde{\mathscr{C}}_{\Gamma}$, see [LNWW, Lemma 3.3.] or [Gra, Example 4.1.6]. Since it is clear that $W_{1}(x)$ is supported in $I(x_{0},r)$, we obtain that

[TABLE]

We next estimate $W_{2}(x)$. By definition, it is clear that $W_{2}(x)$ is supported in $I(y_{0},r)$, and we have

[TABLE]

Here the last equality follows from the cancellation condition of the $H^{1}_{b}(\mathbb{R})$ $L^{\infty}$-atom $a(x)$. Hence, we have for $x\in I(y_{0},r)$ (otherwise $W_{2}(x)=0$ and any estimate will hold)

[TABLE]

Once again using the smoothness of the kernel $\widetilde{C}_{\Gamma}(x,y)$ of $\widetilde{\mathscr{C}}_{\Gamma}$.

Combining the estimates of $W_{1}$ and $W_{2}$, we obtain that

[TABLE]

Next we point out that

[TABLE]

since $a(x)$ has cancellation with respect to $b(x)$ and the same holds for $\Pi_{b}(g,h)(x)$.

Then the size estimate (4.5) and the cancellation (4.6), together with the result in Lemma 4.2, more specifically estimate (4.2), imply that $a-\Pi_{b}(g,h)\in H^{1}_{b}(\mathbb{R})$ and

[TABLE]

This proves the result. ∎

We deduce from the theorem the following corollary concerning $H^{1}(\mathbb{R})$ $L^{\infty}$-atoms.

Corollary 4.4**.**

For every $H^{1}(\mathbb{R})$ $L^{\infty}$-atom $A(x)$ and for all $\varepsilon>0$ there exist $M>0$ and compactly supported $L^{\infty}$ functions $G$ and $H$ such that $\|A-\Pi(H,G)\|_{H^{1}(\mathbb{R})}<\varepsilon$ and $\|G\|_{L^{2}(\mathbb{R})}\|H\|_{L^{2}(\mathbb{R})}\lesssim M$.

Proof.

Note that if $A$ is an $H^{1}(\mathbb{R})$ $L^{\infty}$-atom then $A/b$ is an $H^{1}_{b}(\mathbb{R})$ $L^{\infty}$-atom, hence by Theorem 4.3 for all $\varepsilon>0$ there are $M>0$ and compactly supported $L^{\infty}$ functions $g,h$ such that $\|A/b-\Pi_{b}(g,h)\|_{H^{1}_{b}(\mathbb{R})}\lesssim\varepsilon$. By (3.2) $\Pi_{b}(g,h)=b\,\Pi(g,bh)$, this implies $\|A-b\,\Pi_{b}(g,h)\|_{H^{1}(\mathbb{R})}=\|A-\Pi(g,bh)\|\lesssim\varepsilon$. Let $G=g$ and $H=bh$, these are compactly supported $L^{\infty}$ functions, furthermore $\|G\|_{L^{2}(\mathbb{R})}\|H\|_{L^{2}(\mathbb{R})}\approx\|g\|_{L^{2}(\mathbb{R})}\|h\|_{L^{2}(\mathbb{R})}\lesssim M$. ∎

With this approximation result, we can now prove the main result.

Constructive Proof of Theorem 1.1.

By Theorem 4.1 we have that $\left\|\Pi_{b}(g,h)\right\|_{H^{1}_{b}(\mathbb{R})}\lesssim\left\|g\right\|_{L^{2}(\mathbb{R})}\left\|h\right\|_{L^{2}(\mathbb{R})}$. It follows that if $f\in H^{1}_{b}(\mathbb{R})$ then for any representation of the form

[TABLE]

we have that

[TABLE]

Consequently,

[TABLE]

We turn to show that the other inequality holds and that it is possible to obtain such a decomposition for any $f\in H^{1}_{b}(\mathbb{R}^{n})$. By the definition of $H^{1}_{b}(\mathbb{R})$, for any $f\in H^{1}_{b}(\mathbb{R})$ we can find a sequence $\{\lambda_{j}^{1}\}\in\ell^{1}$ and sequence of $H^{1}_{b}(\mathbb{R})$ $L^{\infty}$-atoms $a_{j}^{1}$ so that $f=\sum_{j=1}^{\infty}\lambda_{j}^{1}a_{j}^{1}$ and $\sum_{j=1}^{\infty}\left|\lambda_{j}^{1}\right|\leq C_{0}\left\|f\right\|_{H^{1}_{b}(\mathbb{R})}$.

We explicitly track the implied absolute constant $C_{0}$ appearing from the atomic decomposition since it will play a role in the convergence of the approach. Fix $\varepsilon>0$ so that $\varepsilon C_{0}<1$. Then we also have a large positive number $M$ with ${M^{-1}\log M}<\epsilon$. We apply Theorem 4.3 to each atom $a_{j}^{1}$. So there exists $g_{j}^{1},h_{j}^{1}\in L^{\infty}(\mathbb{R}^{n})$ with compact supports and satisfying $\left\|g_{j}^{1}\right\|_{L^{2}(\mathbb{R})}\left\|h_{j}^{1}\right\|_{L^{2}(\mathbb{R})}\lesssim M$ and

[TABLE]

Now note that we have

[TABLE]

Observe that we have

[TABLE]

We now iterate the construction on the function $E_{1}$. Since $E_{1}\in H^{1}_{b}(\mathbb{R})$, we can apply the atomic decomposition in $H^{1}_{b}(\mathbb{R})$ to find a sequence $\{\lambda_{j}^{2}\}\in\ell^{1}$ and a sequence of $H^{1}_{b}(\mathbb{R})$ $L^{\infty}$-atoms $\{a_{j}^{2}\}$ so that $E_{1}=\sum_{j=1}^{\infty}\lambda_{j}^{2}a_{j}^{2}$ and

[TABLE]

Again, we will apply Theorem 4.3 to each $L^{\infty}$-atom $a_{j}^{2}$. So there exist $g_{j}^{2},h_{j}^{2}\in L^{\infty}(\mathbb{R})$ with compact supports and satisfying $\left\|g_{j}^{2}\right\|_{L^{2}(\mathbb{R})}\left\|h_{j}^{2}\right\|_{L^{2}(\mathbb{R})}\lesssim M$ and

[TABLE]

We then have that:

[TABLE]

But, as before observe that

[TABLE]

And, this implies for $f$ that we have:

[TABLE]

Repeating this construction for each $1\leq k\leq K$ produces functions $g_{j}^{k},h_{j}^{k}\in L^{\infty}(\mathbb{R})$ with compact supports and satisfying $\left\|g_{j}^{k}\right\|_{L^{2}(\mathbb{R})}\left\|h_{j}^{k}\right\|_{L^{2}(\mathbb{R})}\lesssim M$ for all $j>0$, sequences $\{\lambda_{j}^{k}\}_{j>0}\in\ell^{1}$ with $\left\|\{\lambda_{j}^{k}\}_{j>0}\right\|_{\ell^{1}}\leq\varepsilon^{k-1}C_{0}^{k}\left\|f\right\|_{H^{1}_{b}(\mathbb{R})}$, and a function $E_{K}\in H^{1}_{b}(\mathbb{R})$ with $\left\|E_{K}\right\|_{H^{1}_{b}(\mathbb{R})}\leq\left(\varepsilon C_{0}\right)^{K}\left\|f\right\|_{H^{1}_{b}(\mathbb{R})}$ so that

[TABLE]

Passing $K\to\infty$ gives the desired decomposition of

[TABLE]

We also have that:

[TABLE]

Therefore $\{\lambda_{j}^{k}\}_{j,k\in\mathbb{Z}}$ is in $\ell^{1}$ as claimed. This finishes the proof of Theorem 1.1. ∎

The weak-factorization given by Theorem 1.1 can be used to prove the lower bound of Theorem 1.2, the same way it is done in for example [LW]. However we used the upper bound of Theorem 1.2 to prove Lemma 4.1 responsible for the upper bound in Theorem 1.1.

Acknowledgments: J. Li is supported by ARC DP 160100153 and Macquarie University New Staff Grant. B. D. Wick’s research supported in part by National Science Foundation DMS grants #1560955 and #1800057.