Logarithmic Mean Oscillation on the Polydisc, Multi-Parameter Paraproducts and Iterated Commutators

Abstract

We introduce another notion of bounded logarithmic mean oscillation in the $N$-torus and give an equivalent definition in terms of boundedness of multi-parameter paraproducts from the dyadic little $\mathrm {BMO}$, $\mathrm {bmo}^d(\mathbb {T}^N)$ to the dyadic product $\mathrm {BMO}$ space, $\mathrm {BMO}^d(\mathbb {T}^N)$. We also obtain a sufficient condition for the boundedness of the iterated commutators from the subspace of $\mathrm {bmo}(\mathbb {R}^N)$ consisting of functions with support in $[0,1]^N$ to $\mathrm {BMO}(\mathbb {R}^N)$.

1 Introduction

Multi-parameter paraproducts are among the most studied operators in modern harmonic analysis as they appear as the building blocks of several other operators. Their study has been at the origin of several results in the literature and it is still attracting a lot of attention [1, 5, 9–11, 14].

Our first interest in this paper will be essentially for the boundedness of dyadic paraproducts from the dyadic little $\mathrm {BMO}(\mathbb {T}^N)$, $\mathrm {bmo}^d(\mathbb {T}^N)$ of Cotlar and Sadosky [4] to the dyadic product $\mathrm {BMO}(\mathbb {T}^N)$ of Chang and Fefferman [3], $\mathrm {BMO}^d(\mathbb {T}^N)$. We will focus on the so-called main paraproduct denoted below by $\Pi$. We prove a characterization of boundedness of $\Pi$ from $\mathrm {bmo}^d(\mathbb {T}^N)$ to $\mathrm {BMO}^d(\mathbb {T}^N)$ in terms of a new notion of logarithmic mean oscillation in the polydisc. In the two-parameter case, this notion is in fact implicit in [14] and [16].

Some other notions of logarithmic mean oscillation were discussed in [14–16]. The notion of logarithmic mean oscillation that we are considering in this paper is the natural one when the first space is the little space of functions of bounded mean oscillation, the target space being the product $\mathrm {BMO}$.

Our second interest is for the boundedness of the iterated commutators with the Hilbert transforms from $\mathrm {bmo}([0,1]^N)$ to $\mathrm {BMO}(\mathbb {R}^N)$. We prove that a sufficient condition for these commutators to be bounded is given by our notion of logarithmic oscillation adapted to $\mathbb {R}^N$. This last interest is in the scope of the works [5, 6, 14] which can be seen as a motivation for this paper.

Our presentation is close to the one of [14]. In the next section, we provide various definitions and notations, and we give the statement of the result for the main paraproduct. Section 3 is devoted to the proof of the boundedness of the main paraproduct from $\mathrm {bmo}^d(\mathbb {T}^N)$ to $\mathrm {BMO}^d(\mathbb {T}^N)$. In Sect. 4, we deal with the other paraproducts. The study of iterated commutators is in Sect. 5 where for simplicity of the presentation, we will mainly discuss the two-parameter case as the $N$-parameter case follows the same steps.

2 Preliminaries and Main Result

As usual, $\mathcal D$ will be the set of all dyadic intervals of the unit circle $\mathbb {T}$ that we identify with the interval $[0,1)$. The set of all dyadic rectangles $R=R_1\times \cdots \times R_N$, where $R_j\in \mathcal D$, $j=1,\ldots , N$ is denoted $\mathcal R=\mathcal {D}^N$. The Haar wavelet adapted to the dyadic interval $I$ is given by

$$\begin{aligned} h_I=|I|^{-1/2}\left( \chi _{I^+}-\chi _{I^-}\right) , \end{aligned}$$

where $I^+$ and $I^-$ are the right and left halves of $I$, respectively.

The product Haar wavelet $h_R$ adapted to the rectangle $R=R_1\times \cdots \times R_N\in \mathcal R$ is defined by $h_R(t_1,\ldots , t_N)=h_{R_1}(t_1)\ldots h_{R_N}(t_N)$. We denote by $L_0^2(\mathbb {T}^N)$ the subspace of $L^2(\mathbb {T}^N)$ defined as follows

$$\begin{aligned}&L_0^2(\mathbb {T}^N)\\&\quad = \left\{ f \in L^2(\mathbb {T}^N): \int \limits _\mathbb {T}f(\ldots ,t_j,\ldots ) dt_j =0,\quad j=1,\ldots , N \text { for a.e. }t_1,\ldots , t_N \in \mathbb {T}\right\} , \end{aligned}$$

so that

$$\begin{aligned} f=\sum _{R\in \mathcal {R}} \langle f,h_R \rangle h_R \quad \left( f \in L^2_0(\mathbb {T}^N)\right) . \end{aligned}$$

The Haar coefficient $\langle f, h_R \rangle$ will be quite often denoted $f_R$ or $f_{S\times T}$ whenever $R=S\times T$. The mean of $f\in L^2(\mathbb {T}^N)$ over the dyadic rectangle $R$ is denoted $m_Rf$.

The dyadic product Hardy space $H^{1}_d(\mathbb T^N)$ is defined by

$$\begin{aligned} H_d^{1}(\mathbb T^N)=\left\{ f\in L_0^1(\mathbb T^N): \mathcal {S}[f]\in L^1(\mathbb T^N)\right\} , \end{aligned}$$

where $\mathcal S$ is the dyadic square function,

$$\begin{aligned} \mathcal {S}[f] = \left( \sum _{R \in \mathcal R} \frac{\chi _R}{|R|} |f_R|^2 \right) ^{1/2}. \end{aligned}$$

(1)

The dual space of the dyadic product Hardy space $H^{1}_d(\mathbb T^N)$ is the space of functions of dyadic bounded mean oscillations in $\mathbb T^N$, $\mathrm {BMO}^d(\mathbb T^N)$ (see e.g. [1, 2, 17]) and it consists of all functions $f\in L_0^2(\mathbb T^N)$ such that

$$\begin{aligned} ||f||_{\mathrm {BMO}^d(\mathbb {T}^N)}^2:=\sup _{\Omega \subset \mathbb {T}^N}\frac{1}{|\Omega |}\sum _{R\subset \Omega }|f_R|^2=\sup _{\Omega \subset \mathbb {T}^N}\frac{1}{|\Omega |}||P_\Omega f||_{L^2(\mathbb {T}^N)}^2 < \infty , \end{aligned}$$

(2)

where the supremum is taken over all open sets $\Omega \subset \mathbb T^N$ and $P_\Omega$ is the orthogonal projection on the subspace spanned by Haar functions $h_R$, $R\in \mathcal R$ and $R\subset \Omega$.

The little $\mathrm {BMO}$ of Cotlar and Sadosky is defined by

$$\begin{aligned} \mathrm {bmo}(\mathbb {T}^N)= \left\{ f \in L^2(\mathbb {T}^N): ||f||_{*,N} < \infty \right\} , \end{aligned}$$

(3)

where

$$\begin{aligned} ||f||_{*,N}:=\sup _{R \subset \mathbb {T}^N, \text { rectangle }} \frac{1}{|R|}\int \limits _R |f(s,t) - m_R f| ds dt \end{aligned}$$

with $m_Rf=\frac{1}{|R|}\int \limits _R f(t_1,\ldots , t_N) dt_1\ldots dt_N$.

The dyadic $\mathrm {bmo}(\mathbb {T}^N)$ denoted $\mathrm {bmo}^d(\mathbb {T}^N)$ is defined as above by taking the supremum only over dyadic rectangles.

Let us introduce some further notions in the dyadic setting. For $\vec {j} =(j_1,\ldots , j_N) \in \mathbb {N}_0 \times \cdots \times \mathbb {N}_0=\mathbb {N}_0^N$ we define the $j_1$-th generation of dyadic intervals and the $\vec {j}$-th generation of dyadic rectangles as follows.

$$\begin{aligned} \mathcal D_{j_1}&= \{I \in \mathcal D: |I|= 2^{-j_1} \},\\ \mathcal R_{\vec {j}}&= \mathcal D_{j_1} \times \cdots \times \mathcal D_{j_N} = \{ I_1 \times \cdots \times I_N \in \mathcal R: |I_k|= 2^{-j_k}\}. \end{aligned}$$

We will be also using the following notations

$$\begin{aligned} \mathcal D^K=\mathcal D\times \cdots \times \mathcal D\;(\mathrm K-factors ),\quad K\in \mathbb {N}_0, \end{aligned}$$

for $\vec {j}\in \mathbb {N}_0^K$,

$$\begin{aligned} \mathcal D_{\vec {j}}^K=\mathcal D_{j_1} \times \cdots \times \mathcal D_{j_K}. \end{aligned}$$

The product Haar martingale differences are given by

$$\begin{aligned} \Delta _{\vec {j}} f = \sum _{R\in \mathcal R_{\vec {j}}} \langle f, h_R \rangle h_R, \end{aligned}$$

and the expectations are defined by

$$\begin{aligned} E_{\vec {j}} f = \sum _{\vec {k} \in \mathbb {N}_0^N , \vec {k} < \vec {j} } \Delta _{\vec {k}}f, \end{aligned}$$

where we write $\vec {k} < \vec {j}$ for $k_l < j_l$, $l=1,\ldots ,N$ and correspondingly $\vec {k} \le \vec {j}$ for $k_l \le j_l$, for $f \in L^2(\mathbb {T}^N)$.

If we care about the variable on which we are acting, then we need the following operators

$$\begin{aligned} E^{(l)}_{\vec {j}} f = \sum _{\vec {k} \in \mathbb {N}_0 \times \mathbb {N}_0, k_l <j_l} \Delta _{\vec {k}}f, \end{aligned}$$

for $f \in L^2(\mathbb {T}^N)$.

The following operators defined on $L^2(\mathbb {T}^N)$ will be also needed

$$\begin{aligned} Q_{\vec {j}}f =\sum _{\vec {k} \ge \vec {j} } \Delta _{\vec {k}}f. \end{aligned}$$

(4)

The operators $Q^{(l)}_{\vec {j}}$ are defined as for $E^{(l)}_{\vec {j}}$.

For $I$ a dyadic interval and $\varepsilon \in \{0,1\}$, we define $h^{\varepsilon }_I$ by

$$\begin{aligned} h^{\varepsilon }_I =\left\{ \begin{array}{ll} h_I &{}\quad \text {if}\;\varepsilon =0\\ |I|^{-1/2}|h_I| &{}\quad \text {if}\; \varepsilon =1. \end{array} \right. \end{aligned}$$

For $R=R_1\times \cdots \times R_N\in \mathcal R$ and $\vec {\varepsilon }=(\varepsilon _1,\ldots ,\varepsilon _N)$, with $\varepsilon _j\in \{0,1\}$, we write

$$\begin{aligned} h^{\vec {\varepsilon }}_R(t)=h^{\varepsilon _1}_{R_1}(t_1)\ldots h^{\varepsilon _N}_{R_N}(t_N),\quad t=(t_1,\ldots ,t_N). \end{aligned}$$

Several operators appearing in Fourier analysis are related to the following family of operators

$$\begin{aligned} B_{\vec {\varepsilon },\vec {\delta },\vec {\beta }}(\phi ,f):=\sum _{R\in \mathcal R}\langle \phi ,h^{\vec {\varepsilon }}_R\rangle \langle f,h^{\vec {\delta }}_R\rangle h^{\vec {\beta }}_R. \end{aligned}$$

(5)

The paraproducts we are interested in here are of the above form. They correspond to the operators $B_{\vec {\varepsilon },\vec {\delta },\vec {\beta }}(\phi , \cdot )$ with symbol $\phi$ corresponding to triples $(\vec {\varepsilon }, \vec {\delta }, \vec {\beta })$ such that $\vec {\varepsilon }=(0,\ldots ,0)$ and

$$\begin{aligned} \delta _j=\left\{ \begin{array}{ll} 1 &{}\quad \text {if}\;\beta _j=0\\ 0 &{}\quad \text {otherwise}. \end{array}\right. \end{aligned}$$

For simplicity, we denote these paraproducts by $\Pi ^{\vec {\beta }}$. We will be using the notations $\vec {1}=(1,\ldots ,1)$, $\vec {0} = (0,\ldots ,0)$.

Let $\phi \in L^2(\mathbb {T}^N)$. The (main) paraproduct $\Pi _\phi$ is defined by

$$\begin{aligned} \Pi _\phi f = \Pi (\phi ,f):= \sum _{\vec {j} \in \mathbb {N}_0 \times \cdots \times \mathbb {N}_0} \left( \Delta _{\vec {j}} \phi \right) \left( E_{\vec {j}} f\right) = \sum _{R\in \mathcal R} h_R \phi _R m_R f \end{aligned}$$

on functions with finite Haar expansion; it is just the paraproduct $\Pi ^{(0,\ldots ,0)}$ above.

Next, we define the space of functions of dyadic logarithmic mean oscillation on $\mathbb {T}^N$, $\mathrm {LMO}^d(\mathbb {T}^N)$.

Definition 2.1

Let $\phi \in L^2(\mathbb {T}^N)$. We say that $\phi \in \mathrm {LMO}^d(\mathbb {T}^N)$, if there exists $C >0$ with

$$\begin{aligned} \Vert Q_{\vec {j}} \phi \Vert _{\mathrm {BMO}^d(\mathbb {T}^N)} \le C \frac{1}{\left( \sum _{k=1}^Nj_k\right) +N} \end{aligned}$$

for all $\vec {j} = (j_1,\ldots , j_N) \in \mathbb {N}_0^N$. The infinimum of such constants is denoted by $\Vert \phi \Vert _{\mathrm {LMO}^d(\mathbb {T}^N)}$.

We have the following alternative characterization of our space.

Proposition 2.2

Let $\phi \in L^2(\mathbb {T}^N)$. Then $\phi \in \mathrm {LMO}^d(\mathbb {T}^N)$, if and only if there exists a constant $C >0$ such that for each dyadic rectangle $R= I_1 \times \cdots \times I_N$ and each open set $\Omega \subseteq R$,

$$\begin{aligned} \frac{\left( \log \frac{4}{|I_1|}+\cdots +\log \frac{4}{|I_N|}\right) ^2}{|\Omega |} \sum _{Q \in \mathcal R, Q \subseteq \Omega } |\phi _Q|^2 \le C. \end{aligned}$$

(6)

Proof

Let $\phi \in \mathrm {LMO}^d(\mathbb {T}^N)$ in the sense of Definition 2.1, let $R= I_1 \times \cdots \times I_N$ be a dyadic rectangle with $|I_j| = 2^{-k_j}$, $j=1,\ldots ,N$, and let $\Omega \subseteq R$ be open. Let $\vec {k} = (k_1,\ldots , k_N)$. Then

$$\begin{aligned} \sum _{Q \in \mathcal R, Q \subseteq \Omega } |\phi _Q|^2&= \Vert P_\Omega \phi \Vert _{L^2(\mathbb {T}^N)}^2 = \Vert P_\Omega Q_{\vec {k}} \phi \Vert _{L^2(\mathbb {T}^N)}^2 \le |\Omega | \Vert Q_{\vec {k}} \phi \Vert ^2_{\mathrm {BMO}^d(\mathbb {T}^N)} \\&\le |\Omega | \frac{1}{\left( k_1 +\cdots +k_N \right) ^2} \Vert \phi \Vert ^2_{\mathrm {LMO}^d(\mathbb {T}^N)} \\&\lesssim \left( \log \frac{4}{|I_1|}+\cdots +\log \frac{4}{|I_N|}\right) ^{-2} |\Omega | \Vert \phi \Vert ^2_{\mathrm {LMO}^d(\mathbb {T}^N)}. \end{aligned}$$

Conversely, suppose that $\phi \in L^2(\mathbb {T}^N)$ and that (6) holds. Let $\vec {k} = (k_1,\ldots , k_N) \in \mathbb {N}_0^N$, and let $\Omega \subseteq \mathbb {T}^N$ open. Then

$$\begin{aligned} \Vert P_\Omega Q_{\vec {k}} \phi \Vert _{L^2(\mathbb {T}^N)}^2&= \sum _{R \in \mathcal R_{\vec {k}}} \Vert P_{R \cap \Omega } Q_{\vec {k}} \phi \Vert _{L^2(\mathbb {T}^N)}^2 \\&\lesssim C \frac{1}{\left( k_1\cdots \!+\!k_N\!+\!N\right) ^2} \sum _{R \in \mathcal R_{\vec {k}}} |R \cap \Omega | = C \frac{1}{\left( k_1 \!+\!\cdots \!+\!k_N\!+\!N\right) ^2} |\Omega |. \end{aligned}$$

This holds for all $\Omega \subseteq \mathbb {T}^N$ open, hence $\Vert Q_{\vec {k}} \phi \Vert _{\mathrm {BMO}^d(\mathbb {T}^N)} \lesssim \frac{1}{\left( k_1 +\cdots +k_N +N\right) }$. $\square$

We would like to observe an important equivalent definition of the above space, $\mathrm {LMO}^d(\mathbb {T}^N)$ . For this we introduce further definitions.

Definition 2.3

Let $\phi \in L^2(\mathbb {T}^N)$, $j\in \{1,\ldots ,N\}$. We say that $\phi \in \mathrm {LMO}_{j}^d(\mathbb {T}^N)$, if there exists $C >0$ with

$$\begin{aligned} \Vert Q^{(j)}_{i} \phi \Vert _{\mathrm {BMO}^d(\mathbb {T}^N)} \le C \frac{1}{i+1 } \end{aligned}$$

for all $i \in \mathbb {N}_0$.

The infimum of such constants is denoted by $\Vert \phi \Vert _{\mathrm {LMO}_{j}^d(\mathbb {T}^N)}$.

It is not hard to see that

$$\begin{aligned} \mathrm {LMO}^d(\mathbb {T}^N)=\cap _{j=1}^N\mathrm {LMO}_j^d(\mathbb {T}^N). \end{aligned}$$

(7)

One way to see this is by considering the following equivalent definition of $\mathrm {LMO}_j^d(\mathbb {T}^N)$.

Proposition 2.4

Let $\phi \in L^2(\mathbb {T}^N)$ and $j=1,2,\ldots ,N$. Then $\phi \in \mathrm {LMO}_j^d(\mathbb {T}^N)$, if and only if there exists $C >0$ such that for each dyadic rectangle $R= I_1 \times \cdots \times I_N$ and each open set $\Omega \subseteq R$,

$$\begin{aligned} \frac{\left( \log \frac{4}{|I_j|}\right) ^2}{|\Omega |} \sum _{Q \in \mathcal R, Q \subseteq \Omega } |\phi _Q|^2 \le C. \end{aligned}$$

(8)

Here is one of our main results that gives an equivalent definition of $\mathrm {LMO}^d(\mathbb {T}^N)$ in terms of boundedness of the main paraproduct $\Pi$ from $\mathrm {bmo}^d(\mathbb {T}^N)$ to $\mathrm {BMO}^d(\mathbb {T}^N)$.

Theorem 2.5

Let $\phi \in L^2(\mathbb {T}^N)$. Then $\phi \in \mathrm {LMO}^d(\mathbb {T}^N)$, if and only if $\Pi _\phi :\mathrm {bmo}^d(\mathbb {T}^N) \rightarrow \mathrm {BMO}^d(\mathbb {T}^N)$ is bounded. Moreover,

$$\begin{aligned} \Vert \Pi _\phi \Vert _{\mathrm {bmo}^d(\mathbb {T}^N) \rightarrow \mathrm {BMO}^d(\mathbb {T}^N)} \approx \Vert \phi \Vert _{\mathrm {LMO}^d(\mathbb {T}^N)}. \end{aligned}$$

3 The Main Paraproduct

The aim of this section is to prove Theorem 2.5. We start by introducing some further notations.

Given an integrable function $f$ on $\mathbb {T}^N$ and rectangles $Q\subset \mathbb {T}^{N_1}$ and $S\subset \mathbb {T}^{N_2}$, $N=N_1+N_2$, we write $m_Qf=\frac{1}{|Q|}\int _Qf(s,t)ds$, $m_Sf=\frac{1}{|J|}\int _Jf(s,t)dt$, $s\in \mathbb {T}^{N_1},t\in \mathbb {T}^{N_2}$ and, $m_Rf=\frac{1}{|R|}\int _Rf(s,t)dsdt=\frac{1}{|R|}\int _Rf(t_1,\ldots , t_N)dt_1\ldots dt_N$, $t_j\in \mathbb {T}$, $j=1,\ldots ,N$, $R=Q\times S$.

We remark that if $Q$ is a rectangle in the $N_1$ first variables, then $m_Qf$ is in fact a function of the last $N-N_1$ variables. It is not hard to see that the space $\mathrm {bmo}(\mathbb {T}^N)$ has the following property.

Proposition 3.1

Let $f\in \mathrm {bmo}(\mathbb {T}^N)$. Then for any $R\in \mathbb {T}^K$, $N>K\in \mathbb {N}_0$, $m_Rf\in \mathrm {bmo}(\mathbb {T}^{N-K})$ uniformly. Moreover,

$$\begin{aligned} \Vert m_Rf\Vert _{\mathrm {bmo}(\mathbb {T}^{N-K})}\lesssim \Vert f\Vert _{\mathrm {bmo}(\mathbb {T}^{N})}. \end{aligned}$$

Let us also observe the following.

Lemma 3.2

The following assertions hold.

• (1) Given an interval $I$ in $\mathbb {T}$, there is a function in $\mathrm {BMO}(\mathbb {T})$, denoted $\log _I$ such that

  • the restriction of $\log _I$ to $I$ is $\log \frac{4}{|I|}$.

  • $\Vert \log _I\Vert _{\mathrm {BMO}(\mathbb {T})}\le C$ where $C$ is a constant that does not depend on $I$.

• (2) For any $f_j\in \mathrm {BMO}(\mathbb {T})$, $j=1,\ldots ,N$, the function $b(t_1,\ldots ,t_N)=\sum _{j=1}^Nf_j(t_j)$ belongs to $\mathrm {bmo}(\mathbb {T}^N)$. Moreover,

  $$\begin{aligned} \left\| \sum _{j=1}^Nf_j(t_j)\right\| _{\mathrm {bmo}(\mathbb {T}^N)}\le \sum _{j=1}^N\Vert f_j\Vert _{\mathrm {BMO}(\mathbb {T})}. \end{aligned}$$

Proof

We refer to [16, Chapter 3] for the proof of $(1)$. Assertion $(2)$ follows directly from the definition of $\mathrm {bmo}(\mathbb {T}^N)$. $\square$

We will need the following.

Lemma 3.3

Let $b \in \mathrm {bmo}^d(\mathbb {T}^N)$, $k\in \mathbb {N}_0$, $\vec {k} = (k_1,\ldots , k_N) \in \mathbb {N}_0 \times \cdots \times \mathbb {N}_0$. Then

$$\begin{aligned} |m_{I\times R} b| \lesssim (k+1) \Vert b\Vert _{\mathrm {bmo}^d(\mathbb {T}^N)} \quad (I \in \mathcal D_k, R\in \mathcal D^{N-1});\\ \Vert \chi _{I\times R} b \Vert _{L^2(\mathbb {T}^N)}^2 \lesssim (k+1)^2 |I||R| \Vert b\Vert ^2_{\mathrm {bmo}^d(\mathbb {T}^N)} \quad (I \in \mathcal D_k, R\in \mathcal D^{N-1}); \end{aligned}$$

$$\begin{aligned} \Vert \chi _R P_T b \Vert _{L^2(\mathbb {T}^N)}^2 \lesssim |R| |T| \Vert b\Vert ^2_{\mathrm {bmo}^d(\mathbb {T}^N)} \quad (R\times T \in \mathcal R_{\vec {k}}). \end{aligned}$$

(9)

Proof

Let $S = I \times R$, $|I|=2^{-k}$. The one parameter estimate of the mean of a $\mathrm {BMO}^d(\mathbb {T})$-function and the properties of functions in $\mathrm {bmo}^d(\mathbb {T}^N)$ give directly,

$$\begin{aligned} |m_Sb|=|m_I(m_Rb)|\lesssim (k+1)||m_Rb||_{\mathrm {BMO}^d(\mathbb {T})}\lesssim (k+1)||b||_{\mathrm {bmo}^d(\mathbb {T}^N)}, \end{aligned}$$

giving the first inequality.

For the second inequality, we observe that for $L\in \mathcal D^N$, the following identity holds:

$$\begin{aligned} \chi _Lb= P_L b+\sum _{S\;\mathrm a factor of \; L}\varepsilon _S\chi _L(s,t) m_S b(t) \end{aligned}$$

(10)

where $\varepsilon _S=(-1)^K$, $K\in \mathbb {N}_0$, $S\in \mathcal D^K$. Here and below, we say $S$ is a factor of $L=L_1\times \cdots \times L_N$, if $S=L_{j_1}\times \cdots \times L_{j_M}$, $j_l\in \{1,\ldots ,N\}$, $M\le N$. When $M<N$, we say $S$ is a subfactor of $L$. Hence,

$$\begin{aligned} \Vert \chi _Lb\Vert _{L^2(\mathbb {T}^N)}&\lesssim \Vert P_L b(s,t)\Vert _{L^2(\mathbb {T}^N)}+\sum _{S\;\mathrm a subfactor of \; L}\Vert \chi _L(s,t) m_S b(t)\Vert _{L^2(\mathbb {T}^N)}\\&+\Vert \chi _L(s,t) m_L b\Vert _{L^2(\mathbb {T}^N)}. \end{aligned}$$

Thus, we only need to estimate the first and the last terms in the right hand side of the above inequality and the norm $\Vert \chi _{S\times Q}(s,t) m_S b(t)\Vert _{L^2(\mathbb {T}^N)}$, with $Q\times S=L=I\times R$.

Clearly $\Vert P_L b \Vert _{L^2(\mathbb {T}^N)}^2 \le |L| \Vert b\Vert ^2_{\mathrm {BMO}^d(\mathbb {T}^N)}\le |I||R| \Vert b\Vert ^2_{\mathrm {bmo}^d(\mathbb {T}^N)}$ and

$$\begin{aligned} \Vert \chi _L m_Lb \Vert _{L^2(\mathbb {T}^N)}^2 = |m_L b|^2 |L| \lesssim (k+1)^2|I| |R| \Vert b\Vert ^2_{\mathrm {bmo}^d(\mathbb {T}^N)} \end{aligned}$$

by the first inequality in Lemma 3.3. To estimate $\Vert \chi _{S\times Q}(s,t) m_S b(t)\Vert _{L^2(\mathbb {T}^N)}$, we suppose that $S\in \mathcal D^{N-K}$ and $Q\in \mathcal D^K$. It follows that

$$\begin{aligned} \Vert \chi _{S\times Q}(s,t) m_S b(t)\Vert _{L^2(\mathbb {T}^N)}&= |S|^{1/2} \Vert \chi _Q(t) m_S b(t) \Vert _{L^2(\mathbb {T}^K)} \\&\lesssim |S|^{1/2} |Q|^{1/2}\left( \Vert m_S b \Vert _{\mathrm {bmo}^d(\mathbb {T}^K)}+|m_{I\times R}b|\right) \\&\lesssim |I|^{1/2}|R|^{1/2} (k+1) \Vert b \Vert _{\mathrm {bmo}^d(\mathbb {T}^N)} \end{aligned}$$

by the first inequality and the fact that $\Vert m_S b(t) \Vert _{\mathrm {bmo}^d(\mathbb {T}^K)}\lesssim \Vert b\Vert _{\mathrm {bmo}^d(\mathbb {T}^N)}$.

For the last inequality, we use an induction argument on the number of parameters $N$. Starting from $N=2$, we first show that for $I,J\in \mathcal D$, we have

$$\begin{aligned} \Vert \chi _I(s)P_Jb(s,t)\Vert _{L^2(\mathbb {T}^2)}^2\lesssim |I||J|\Vert b\Vert _{\mathrm {bmo}^d(\mathbb {T}^2)}^2. \end{aligned}$$

(11)

Using the decomposition (10) above, we find that

$$\begin{aligned} \Vert \chi _I(s)P_Jb(s,t)\Vert _{L^2(\mathbb {T}^2)}^2\lesssim \Vert P_{IJ}b\Vert _{L^2(\mathbb {T}^2)}^2+\Vert \chi _I(s)P_Jm_Ib(t)\Vert _{L^2(\mathbb {T}^2)}^2. \end{aligned}$$

We have already seen that $\Vert P_{IJ}b\Vert _{L^2(\mathbb {T}^2)}^2\le |I||J|\Vert b\Vert _{\mathrm {bmo}^d(\mathbb {T}^2)}$, so we only have to take care of $\Vert \chi _I(s)P_Jm_Ib(t)\Vert _{L^2(\mathbb {T}^2)}^2$.

We clearly obtain

$$\begin{aligned} \Vert \chi _I(s)P_Jm_Ib(t)\Vert _{L^2(\mathbb {T}^2)}^2&= |I|\Vert P_Jm_Ib(t)\Vert _{L^2(\mathbb {T})}^2\le |I||J|\Vert m_Ib\Vert _{\mathrm {BMO}^d(\mathbb {T})}\\&\le |I||J|\Vert b\Vert _{\mathrm {bmo}^d(\mathbb {T}^2)}. \end{aligned}$$

Let us now suppose that the inequality holds in $(N-1)$-parameter and prove that this also holds in $N$-parameter.

From the identity (10), we get

$$\begin{aligned} \Vert \chi _R(s,t)P_Tb(s,t,u)\Vert _{L^2(\mathbb {T}^N)}&\lesssim \Vert P_{R\times T} b\Vert _{L^2(\mathbb {T}^N)}+\sum _{S\;\mathrm a subfactor of \; R}\\&\Vert \chi _R(s,t) m_S P_Tb(t,u)\Vert _{L^2(\mathbb {T}^N)}\\&\quad +\Vert \chi _R(s,t) m_R P_Tb(u)\Vert _{L^2(\mathbb {T}^N)}. \end{aligned}$$

We first consider the term $\Vert \chi _R(s,t) m_S P_Tb(t,u)\Vert _{L^2(\mathbb {T}^N)}$, we suppose that $R=S\times Q\in \mathcal D^K$, $K=K_1+K_2<N$, and $S\in \mathcal D^{K_1}$. It follows from our hypothesis that

$$\begin{aligned} \Vert \chi _R(s,t) m_S (P_T b)(t,u)\Vert ^2_{L^2(\mathbb {T}^N)}&= |S| \Vert \chi _Q(t) m_S(P_T b)(t,u) \Vert _{L^2(\mathbb {T}^{N-K_1})}^2 \\&\lesssim |S||Q| |T| \Vert m_S b \Vert _{\mathrm {bmo}^d(\mathbb {T}^{N-K_1})}^2 \\&\lesssim |R| |T| \Vert b \Vert _{\mathrm {bmo}^d(\mathbb {T}^N)}^2. \end{aligned}$$

Next we have

$$\begin{aligned} \Vert \chi _R(s,t) m_R P_Tb(u)\Vert _{L^2(\mathbb {T}^N)}^2&= |R|\Vert m_R P_Tb(u)\Vert _{L^2(\mathbb {T}^{N-K})}^2\\&\le |R||T|\Vert m_Rb\Vert _{\mathrm {BMO}^d(\mathbb {T}^{N-K})}^2\\&\le |R||T|\Vert m_Rb\Vert _{\mathrm {bmo}^d(\mathbb {T}^{N-K})}^2\\&\le |R||T|\Vert b\Vert _{\mathrm {bmo}^d(\mathbb {T}^{N})}^2. \end{aligned}$$

The proof is complete. $\square$

Remark 3.4

From the first inequality in the above lemma, we obtain that

$$\begin{aligned} |m_{ R} b| \lesssim \left( k_1+\cdots +k_N+N\right) \Vert b\Vert _{\mathrm {bmo}^d(\mathbb {T}^N)} \quad (R\in \mathcal {R}_{\vec {k}}, \vec {k}\in \mathbb {N}_0^N) \end{aligned}$$

and this is sharp. The sharpness is obtained by testing with the function

$$\begin{aligned} \log _R(t_1,\ldots ,t_N)=\log _{R_1}(t_1)+\cdots +\log _{R_N}(t_N) \end{aligned}$$

where for any interval $I\in \mathcal D$, the function $\log _I$ is given in Lemma 3.2.

Next, for $k,l \in \mathbb {N}_0$ and $b\in L^2(\mathbb {T}^N)$, we consider on $L^2(\mathbb {T}^N)$ the operator $\Pi _bE_{k}^{(l)} = \Pi (b, E_{k}^{(l)}\, \cdot )$ given by

$$\begin{aligned} \Pi _bE_{k}^{(l)}f=\Pi (b,E_{k}^{(l)}f),\quad f\in L^2(\mathbb {T}^N). \end{aligned}$$

The next lemma is proved in [14].

Lemma 3.5

Let $b \in L^2(\mathbb {T}^N)$ and let $k,l \in \mathbb {N}_0$. Then

$$\begin{aligned} \left\| \Pi _b E^{(l)}_{k}\right\| _{L^2(\mathbb {T}^N) \rightarrow L^2(\mathbb {T}^N)} =\left\| \Pi _{\sigma ^{(l)}_{k} b}\right\| _{L^2(\mathbb {T}^N) \rightarrow L^2(\mathbb {T}^N)}, \end{aligned}$$

where for $R=\prod _{j=1}^NR_j=R_l\times S\in \mathcal D^N$, $S\in \mathcal D^{N-1}$,

$$\begin{aligned} \left( \sigma ^{(l)}_{k}b\right) _{R} =\left\{ \begin{array}{ll} b_{R} &{}\quad \text {if}\;R_l| > 2^{-k}\\ \left( \sum _{R_l' \subseteq R_l} |b_{R_l'\times S}|^2\right) ^{1/2} &{}\quad \text {if}\;|R_l| = 2^{-k}\\ 0 &{}\quad \text {otherwise.} \end{array} \right. \end{aligned}$$

The following lemma is the bedrock of our proof.

Lemma 3.6

Let $\phi \in \mathrm {BMO}^d(\mathbb {T}^N)$, $b\in \mathrm {bmo}^d(\mathbb {T}^N)$, and $k,j \in \mathbb {N}_0$. Then

$$\begin{aligned} \left\| \Pi \left( \Pi (\phi ,b), E_{k}^{(j)}\cdot \right) \right\| _{L^2(\mathbb {T}^N) \rightarrow L^2(\mathbb {T}^N)} \lesssim (k+1) \, \Vert \phi \Vert _{\mathrm {BMO}^d(\mathbb {T}^N)} \Vert b \Vert _{\mathrm {bmo}^d(\mathbb {T}^N)}. \end{aligned}$$

Proof

Following Lemma 3.5, we have to estimate the $\mathrm {BMO}^d(\mathbb {T}^N)$ norm of $\sigma _{k}^{(j)} (\Pi _{\phi } b)=\sigma _{k}^{(j)} (\Pi (\phi , b))$. Without loss of generality, we can suppose that $j=1$. For simplicity, we remove the supscript $(1)$ and write $E_k, Q_k$ and $\sigma _k$ for $E_k^{(1)}$, $Q_k^{(1)}$ and $\sigma _k^{(1)}$ respectively. Clearly

$$\begin{aligned} \sigma _{k} ( \Pi _\phi b) = \sigma _{k} (E_{k} \Pi _\phi b) + \sigma _{k}(Q_{k} \Pi _\phi b) = I + II. \end{aligned}$$

Let us start with term I.

For any open set $\Omega \subseteq \mathbb {T}^N$,

$$\begin{aligned} \frac{1}{|\Omega |} \Vert P_{\Omega } E_{k} \Pi _\phi b\Vert _{L^2(\mathbb {T}^N)}^2&= \frac{1}{|\Omega |} \sum _{R = R_1 \times \cdots \times R_N, |R_1| > 2^{-k}, R \subset \Omega } |\phi _R|^2 |m_R b|^2 \\&\lesssim \frac{(k+1)^2}{|\Omega |} \sum _{R_1 \times \cdots \times R_N, |R_1| > 2^{-k}, R \subset \Omega } |\phi _R|^2 \Vert b\Vert ^2_{\mathrm {bmo}^d(\mathbb {T}^N)} \\&\lesssim (k+1)^2 \Vert \phi \Vert ^2_{\mathrm {BMO}^d(\mathbb {T}^N)} \Vert b\Vert ^2_{\mathrm {bmo}^d(\mathbb {T}^N)} \end{aligned}$$

with the help of Lemma 3.3 at the second inequality.

To deal with term II, we observe that $\sigma _{k}( Q_{k} \Pi _\phi b)$ has only nontrivial Haar coefficients for $R=R_1\times \cdots \times R_N \in \mathcal R$, with $|R_1|=2^{-k}$. We first compute $\Vert P_R \sigma _{k}(Q_{k} \Pi _\phi b )\Vert _{L^2(\mathbb {T}^N)}^2$ for $R\in \mathcal R$ a rectangle of this type.

$$\begin{aligned} \int \limits _R |P_R \sigma _{k}(Q_{k} \Pi _\phi b ) |^2 ds dt&\le \sum _{S \subseteq R} |\phi _{S}|^2 |m_{S} b|^2 \\&= \Vert \Pi _\phi \chi _R b\Vert ^2_{L^2(\mathbb {T}^N)} \\&\lesssim \Vert \phi \Vert ^2_{\mathrm {BMO}^d(\mathbb {T}^N)} \Vert \chi _R b\Vert _{L^2(\mathbb {T}^N)}^2 \\&\lesssim (k+1)^2 |R|\Vert \phi \Vert ^2_{\mathrm {BMO}^d(\mathbb {T}^N)} \Vert b\Vert ^2_{\mathrm {bmo}^d(\mathbb {T}^N)} \end{aligned}$$

where we used Lemma 3.3 at the last inequality.

Now for $\Omega \subseteq \mathbb {T}^N$ open, we denote by $\mathcal {M}_k(\Omega )$ the set of all rectangles $R=R_1\times \cdots \times R_N\in \mathcal R,\,\,\, R\subseteq \Omega$ which are maximal in $\Omega$ with respect to $|R_1|=2^{-k}$.

If $\mathcal {M}_k(\Omega )=\emptyset$, then

$$\begin{aligned} \Vert P_\Omega \sigma _k(Q_{k} \Pi _\phi b )\Vert _{L^2(\mathbb {T}^N)}^2=0\le (k+1)^2 |\Omega |\Vert \phi \Vert ^2_{\mathrm {BMO}^d(\mathbb {T}^N)} \Vert b\Vert ^2_{\mathrm {bmo}^d(\mathbb {T}^N)}. \end{aligned}$$

If $\mathcal {M}_k(\Omega )\ne \emptyset$, then using the above estimate of the $L^2$ norm of $P_R\sigma _{k}( Q_{k} \Pi _\phi b)$ for $R=R_1\times \cdots \times R_N \in \mathcal R$, with $|R_1|=2^{-k}$, we obtain

$$\begin{aligned} \Vert P_\Omega \sigma _k(Q_{k} \Pi _\phi b )\Vert _{L^2(\mathbb {T}^N)}^2&= \sum _{R\in \mathcal {M}_k(\Omega )}\sum _{R'\subseteq R, |R'_1|=2^{-k}}|\left( \sigma _{k}( Q_{k} \Pi _\phi b)\right) _{R'}|^2\\&\le \sum _{R\in \mathcal {M}_k(\Omega )}\Vert P_R \sigma _{k}(Q_{k} \Pi _\phi b )\Vert _{L^2(\mathbb {T}^N)}^2\\&\lesssim (k+1)^2\Vert \phi \Vert ^2_{\mathrm {BMO}^d(\mathbb {T}^N)} \Vert b\Vert ^2_{\mathrm {bmo}^d(\mathbb {T}^N)}\sum _{R\in \mathcal {M}_k(\Omega )}|R|\\&\lesssim (k+1)^2 |\Omega |\Vert \phi \Vert ^2_{\mathrm {BMO}^d(\mathbb {T}^N)} \Vert b\Vert ^2_{\mathrm {bmo}^d(\mathbb {T}^N)}. \end{aligned}$$

The proof is complete. $\square$

We deduce the following from Definition 2.1, the equality (7) and, Lemma 3.6.

Lemma 3.7

Let $\phi \in \mathrm {LMO}^d(\mathbb {T}^N)$, $b \in \mathrm {bmo}^d(\mathbb {T}^N)$ and $j,k,l \in \mathbb {N}_0$. Then

$$\begin{aligned} \left\| \Pi \left( \Pi (Q_{j}^{(l)}\phi ,b), E_{k}^{(l)}\cdot \right) \right\| _{L^2(\mathbb {T}^N) \rightarrow L^2(\mathbb {T}^N)} \lesssim \frac{k + 1}{j + 1} \, \Vert \phi \Vert _{\mathrm {LMO}^d(\mathbb {T}^N)} \Vert b \Vert _{\mathrm {bmo}^d(\mathbb {T}^N)}. \end{aligned}$$

[Proof of Theorem 2.5]. We begin by proving necessity. Suppose that $\Pi _\phi : \mathrm {bmo}^d(\mathbb {T}^N) \rightarrow \mathrm {BMO}^d(\mathbb {T}^N)$ is bounded. Let $R = R_1 \times \cdots \times R_N$ be a given dyadic rectangle, and let $\Omega \subseteq R$ be open. We take as test function, $b(t_1,\ldots ,t_N)=\log _R(t_1,\ldots ,t_N)=\sum _{j=1}^N\log _{R_j}(t_j)$, where for an interval $I$, the function $\log _I(x)$ is given in Lemma 3.2. Then

$$\begin{aligned} \frac{\left( \sum _{j=1}^N\log \frac{4}{|R_j|}\right) ^2}{|\Omega |} \sum _{Q \in \mathcal R, Q \subseteq \Omega } |\phi _Q|^2&\approx \frac{1}{|\Omega |} \sum _{Q \in \mathcal R, Q \subseteq \Omega } |\phi _Q|^2 |m_Qb|^2\\&\le \Vert \Pi _\phi b\Vert ^2_{\mathrm {BMO}^d(\mathbb {T}^N)}\\&\le C^2 \Vert \Pi _\phi \Vert ^2_{\mathrm {bmo}^d(\mathbb {T}^N) \rightarrow \mathrm {BMO}^d(\mathbb {T}^N)}. \end{aligned}$$

Thus $\phi \in \mathrm {LMO}^d(\mathbb {T}^N)$ by Proposition 2.2, with the appropriate norm estimate.

To prove sufficiency of the $\mathrm {LMO}^d(\mathbb {T}^N)$ condition for the boundedness of the paraproduct from $\mathrm {bmo}^d(\mathbb {T}^N)$ to $\mathrm {BMO}^d(\mathbb {T}^N)$, we recall that $\phi \in \mathrm {LMO}^d(\mathbb {T}^N)$ implies that $\phi \in \mathrm {LMO}_j^d(\mathbb {T}^N)$, $j=1,\ldots ,N$. Let $\phi \in \mathrm {LMO}^d(\mathbb {T}^N)$ and $b \in \mathrm {bmo}^d(\mathbb {T}^N)$. We recall that the following estimate holds

$$\begin{aligned} \Vert \Pi _\phi b \Vert _{\mathrm {BMO}^d(\mathbb {T}^N)} \approx \Vert \Pi _{\Pi (\phi ,b)}\Vert _{L^2(\mathbb {T}^N) \rightarrow L^2(\mathbb {T}^N)}=\Vert \Pi \left( \Pi (\phi ,b),\cdot \right) \Vert _{L^2(\mathbb {T}^N) \rightarrow L^2(\mathbb {T}^N)}. \end{aligned}$$

Motivated by the equality (7) and Lemma 3.6, we would like to apply Cotlar’s Lemma in one direction (parameter).

For $N \in \mathbb {N}_0$, let

$$\begin{aligned} P_{N}&= \sum _{j=2^N-1}^{2^{N+1}-2} \Delta _{j}^{(1)}, \nonumber \\ P^{N}&= \sum _{j=2^N}^\infty \Delta _{j}^{(1)}, \end{aligned}$$

(12)

and

$$\begin{aligned} T_{N} = \Pi _{\Pi (\phi , b)} P_{N}=\Pi \left( \Pi (\phi ,b), P_{N}\, \cdot \right) . \end{aligned}$$

Our operator can be written now $\Pi (\Pi (\phi , b), \cdot ) = \sum _{N=0}^\infty T_{N}$. Following exactly [14] with the help of Lemma 3.6, as $T_{N} T_{N'}^* =0$ for $N \ne N'$, we obtain that

$$\begin{aligned} \Vert T_{N}^* T_{N'}\Vert \le 2^{-|N-N'|} \Vert \phi \Vert ^2_{\mathrm {LMO}_1^d(\mathbb {T}^N)}\Vert b\Vert ^2_{\mathrm {bmo}^d(\mathbb {T}^N)}. \end{aligned}$$

Thus, by applying Cotlar’s Lemma, we obtain that $T= \Pi (\Pi (\phi , b),\cdot )$ is bounded, and there exists an absolute constant $C>0$ with

$$\begin{aligned} \Vert \Pi \left( \Pi (\phi , b), \cdot \right) \Vert _{L^2(\mathbb {T}^N)\rightarrow L^2(\mathbb {T}^N)}\le C \Vert \phi \Vert _{\mathrm {LMO}^d(\mathbb {T}^N)}\Vert b\Vert _{\mathrm {bmo}^d(\mathbb {T}^N)}. \end{aligned}$$

Consequently,

$$\begin{aligned} \Vert \Pi (\phi , b)\Vert _{\mathrm {BMO}^d(\mathbb {T}^N)} \le C \Vert \phi \Vert _{\mathrm {LMO}^d(\mathbb {T}^N)}\Vert b\Vert _{\mathrm {bmo}^d(\mathbb {T}^N)}. \end{aligned}$$

$\square$

4 The Other Paraproducts

The other paraproducts correspond to $\vec {\beta }=\vec {1}=(1,\ldots ,1)$ and $\vec {\beta }\ne \vec {0}, \vec {1}$. The first one is the adjoint of $\Pi =\Pi ^{\vec {0}}$, it is defined on $L^2(\mathbb {T}^N)$ by

$$\begin{aligned} \Pi ^{\vec {1}}(\phi ,f)=\Delta _\phi f=\Delta (\phi ,f) = \sum _{R \in \mathcal R} \frac{\chi _R}{|R|} \phi _R f_R, \end{aligned}$$

then there are the mixed paraproducts given by the following general form

$$\begin{aligned} \Pi ^{\vec {\beta }}(\phi ,f)=\sum _{R=Q \times S \in \mathcal R}\frac{\chi _Q(s)}{|Q|} h_S(t) \phi _{R} m_S f_Q, \end{aligned}$$

where $R_j=Q_{j}$ if $\beta _j=1$ and $R_j=S_j$ if $\beta _j=0$.

Let us introduce some further definitions and notations.

Definition 4.1

Let $\phi \in L^2(\mathbb {T}^N)$, $\vec {\delta }=(\delta _1,\ldots ,\delta _N)$, $\delta _j\in \{0,1\}$. Then $\phi \in \mathrm {LMO}_{\vec {\delta }}^d(\mathbb {T}^N)$, if and only if there exists $C >0$ such that for each dyadic rectangle $R= R_1 \times R_2 \times \cdots \times R_N \in \mathcal D^N$ and each open set $\Omega \subseteq R$,

$$\begin{aligned} \frac{\left( \log \left( \frac{4}{|R_{\delta _1}|}\right) + \cdots \log \left( \frac{4}{|R_{\delta _N}|}\right) \right) ^2}{|\Omega |} \sum _{Q \in \mathcal R, Q \subseteq \Omega } |\phi _Q|^2 \le C, \end{aligned}$$

where

$$\begin{aligned} R_{\delta _j}=\left\{ \begin{array}{ll} R_j &{}\quad \text {if}\;\delta _j=0\\ \mathbb {T}&{}\quad \text {otherwise.} \end{array}\right. \end{aligned}$$

When $\vec {\delta }=\vec {0}=(0,\ldots ,0)$, $\mathrm {LMO}_{\vec {\delta }}^d(\mathbb {T}^N)=\mathrm {LMO}^d(\mathbb {T}^N)$. One easily sees that for $\vec {\delta }=(1,\ldots ,1)= \vec {1}$, the corresponding space is just the space $\mathrm {BMO}^d(\mathbb {T}^N)$. We observe that

$$\begin{aligned} \mathrm {LMO}_{\vec {\delta }}^d(\mathbb {T}^N)=\bigcap _{j=1, \delta _j=0}^N\mathrm {LMO}_j^d(\mathbb {T}^N). \end{aligned}$$

Let us recall with [1] the following result.

Proposition 4.2

Let $\phi \in L_0^2(\mathbb {T}^N)$. Then $\Pi ^{(1,\dots ,1)}_\phi = \Delta _\phi : \mathrm {BMO}^d(\mathbb {T}^N) \rightarrow \mathrm {BMO}^d(\mathbb {T}^N)$ is bounded, if and only if $\phi \in \mathrm {BMO}^d(\mathbb {T}^N)$. Moreover,

$$\begin{aligned} \Vert \Delta _\phi \Vert _{\mathrm {BMO}^d(\mathbb {T}^N) \rightarrow \mathrm {BMO}^d(\mathbb {T}^N)} \approx \Vert \phi \Vert _{\mathrm {BMO}^d(\mathbb {T}^N)}. \end{aligned}$$

Our main result in this section is the following.

Theorem 4.3

Let $\phi \in L_0^2(\mathbb {T}^N)$, $\vec {\beta }=(\beta _1,\ldots ,\beta _n)$, $\beta _j\in \{0,1\}$. Then for $\vec {\beta }\ne (0,\ldots ,0), (1,\ldots ,1)$,

$\Pi ^{\vec {\beta }}_\phi :\mathrm {bmo}^d(\mathbb {T}^N) \rightarrow \mathrm {BMO}^d(\mathbb {T}^N)$ is bounded if $\phi \in \mathrm {LMO}_{\vec {\beta }}^d(\mathbb {T}^N)$. Moreover,

$$\begin{aligned} \Vert \Pi ^{\vec {\beta }}\phi \Vert _{\mathrm {bmo}^d(\mathbb {T}^N) \rightarrow \mathrm {BMO}^d(\mathbb {T}^N)} \lesssim \Vert \phi \Vert _{\mathrm {LMO}_{\vec {\beta }}^d(\mathbb {T}^N)}. \end{aligned}$$

The proof of the above result requires the following lemma.

Lemma 4.4

Let $\phi \in \mathrm {BMO}^d(\mathbb {T}^N)$, $b\in \mathrm {bmo}^d(\mathbb {T}^N)$, and $j, k \in \mathbb {N}$, $\vec {\beta }\ne \vec {0}, \vec {1}$, with $\beta _j=1$. Then

$$\begin{aligned} \left\| \Pi \left( \Pi ^{(\vec {\beta })}(\phi , b), E^{(j)}_{k}\, \cdot \right) \right\| _{L^2(\mathbb {T}^N) \rightarrow L^2(\mathbb {T}^N)} \lesssim (k+1)\Vert \phi \Vert _{\mathrm {BMO}^d(\mathbb {T}^N)} \Vert b \Vert _{\mathrm {bmo}^d(\mathbb {T}^N)}. \end{aligned}$$

Proof

We can suppose that $j=1$ and write $E_k$ for $E_k^{(1)}$, $\sigma _k$ for $\sigma _k^{(1)}$, and $Q_k$ for $Q_k^{(1)}$.

We have to estimate

$$\begin{aligned} \left\| \sigma _k\left( {\Pi ^{\vec {\beta }}}(\phi , b)\right) \right\| _{\mathrm {BMO}^d(\mathbb {T}^N)}&\le \left\| \sigma _k\left( {\Pi ^{\vec {\beta }}}({E_k\phi }, b)\right) \right\| _{\mathrm {BMO}^d(\mathbb {T}^N)}\\&+ \left\| \sigma _k\left( {\Pi ^{\vec {\beta }}}({Q_k\phi }, b)\right) \right\| _{\mathrm {BMO}^d(\mathbb {T}^N)}. \end{aligned}$$

We start with the second term. Since ${\Pi ^{\vec {\beta }}}(Q_k\phi , b)$ has no nontrivial Haar terms in the first variable for rectangle $R$ with $|R_1| > 2^{-k}$,

$$\begin{aligned} \sigma _k\left( {\Pi ^{\vec {\beta }}}(Q_k\phi , b)\right) \!=\! \sum _{Q } \sum _{S\!=\!S_1\!\times \! T, |S_1|\!= 2^{-k}} h_S(s) \left( \sum _{S_1' \subseteq S_1} \left| \phi _{S_1'TQ}\right| ^2 \left| m_{S_1'T} b_Q\right| ^2\right) ^{1/2} \frac{\chi _Q}{|Q|}(t) \end{aligned}$$

and this has only nontrivial Haar terms in the first variable for rectangles $R=R_1\times \cdots \times R_N$ with $|R_1|=2^{-k}$ (here $R_j$ is understood as an interval in $j$th variable). Hence we first compute $\Vert P_R \sigma _k( {\Pi ^{\vec {\beta }}}(Q_k\phi , b))\Vert _{L^2(\mathbb {T}^N)}$ for $R\in \mathcal R$ of the form $R=R_1\times \cdots \times R_N=Q \times S$, with $|R_1| = 2^{-k}$, $Q\in \mathcal D^K$. We have

$$\begin{aligned} \left\| P_R \sigma _k\left( {\Pi ^{\vec {\beta }}}(Q_k\phi , b)\right) \right\| _{L^2(\mathbb {T}^N)}^2&= \left\| P_R \sigma _k\left( {\Pi ^{\vec {\beta }}}({P_RQ_k\phi }, b)\right) \right\| _{L^2(\mathbb {T}^N)}^2 \\&\le \left\| \sigma _k\left( {\Pi ^{\vec {\beta }}}({P_RQ_k\phi }, b)\right) \right\| _{L^2(\mathbb {T}^N)}^2 \\&\le \left\| {\Pi ^{\vec {\beta }}}\left( {P_RQ_k\phi }, b\right) \right\| _{L^2(\mathbb {T}^N)}^2 \\&= \left\| \sum _{Q' \subseteq Q} \sum _{S' \subseteq S} h_{S'} \frac{\chi _{Q'}}{|Q'|}(t) \phi _{S'Q'} m_{S'} b_{Q'} \right\| _{L^2(\mathbb {T}^N)}^2\\&= \sum _{S' \subseteq S}\left\| \sum _{Q' \subseteq Q}\frac{\chi _{Q'}}{|Q'|}(t) \phi _{S'Q'} m_{S'} b_{Q'}\right\| _{L^2(\mathbb {T}^K)}^2\\&= \sum _{S' \subseteq S}\left\| \Delta _{P_Q\phi _{S'}}m_{S'}b\right\| _{L^2(\mathbb {T}^K)}^2\\&= \sum _{S' \subseteq S}\left\| \Delta _{m_{S'}b}{P_Q\phi _{S'}}\right\| _{L^2(\mathbb {T}^K)}^2\\&\lesssim \sum _{S' \subseteq S}\left\| P_Q\phi _{S'}\right\| _{L^2(\mathbb {T}^K)}^2\left\| m_{S'}b\right\| _{\mathrm {BMO}^d(\mathbb {T}^K)}^2\\&\lesssim \Vert b\Vert _{\mathrm {bmo}^d(\mathbb {T}^N)}^2\sum _{S' \subseteq S}\left\| P_Q\phi _{S'}\right\| _{L^2(\mathbb {T}^K)}^2\\&= \Vert b\Vert _{\mathrm {bmo}^d(\mathbb {T}^N)}^2\left\| P_{Q\times S}\phi \right\| _{L^2(\mathbb {T}^N)}^2\\&\lesssim |R|\Vert \phi \Vert _{\mathrm {BMO}^d(\mathbb {T}^N)}^2 \Vert b\Vert ^2_{\mathrm {bmo}^d(\mathbb {T}^N)}. \end{aligned}$$

Next, for general open set $\Omega \subseteq \mathbb {T}^N$, we still denote by $\mathcal {M}_k(\Omega )$ the set of all rectangles $R=R_1\times \cdots \times R_N\in \mathcal R,\,\,\, R\subseteq \Omega$ which are maximal in $\Omega$ with respect to $|R_1|=2^{-k}$. If $\mathcal {M}_k(\Omega )=\emptyset$, then

$$\begin{aligned} \left\| P_\Omega \sigma _k\left( {\Pi ^{\vec {\beta }}}(Q_k\phi , b)\right) \right\| _{L^2(\mathbb {T}^N)}^2=0\le |\Omega |\Vert \phi \Vert ^2_{\mathrm {BMO}^d(\mathbb {T}^N)} \Vert b\Vert ^2_{\mathrm {bmo}^d(\mathbb {T}^N)}. \end{aligned}$$

If $\mathcal {M}_k(\Omega )\ne \emptyset$, then using the above estimate of the $L^2$ norm of $P_R\sigma _k( {\Pi ^{\vec {\beta }}}(Q_k\phi , b))$ for $R=R_1\times \cdots \times R_N \in \mathcal R$, with $|R_1|=2^{-k}$, we obtain

$$\begin{aligned} \left\| P_\Omega \sigma _k\left( {\Pi ^{\vec {\beta }}}(Q_k\phi , b)\right) \right\| _{L^2(\mathbb {T}^N)}^2&= \sum _{R\in \mathcal {M}_k(\Omega )}\sum _{R'\subseteq R, |R'_1|=2^{-k}}\left| \left( \sigma _k( {\Pi ^{\vec {\beta }}}(Q_k\phi , b))\right) _{R'}\right| ^2\\&\le \sum _{R\in \mathcal {M}_k(\Omega )}\left\| P_R \sigma _k\left( {\Pi ^{\vec {\beta }}}(Q_k\phi , b)\right) \right\| _{L^2(\mathbb {T}^N)}^2\\&\lesssim \Vert \phi \Vert ^2_{\mathrm {BMO}^d(\mathbb {T}^N)} \Vert b\Vert ^2_{\mathrm {bmo}^d(\mathbb {T}^N)}\sum _{R\in \mathcal {M}_k(\Omega )}|R|\\&\lesssim |\Omega |\Vert \phi \Vert ^2_{\mathrm {BMO}^d(\mathbb {T}^N)} \Vert b\Vert ^2_{\mathrm {bmo}^d(\mathbb {T}^N)}. \end{aligned}$$

Let us go back to the first term $\Vert \sigma _k( {\Pi ^{\vec {\beta }}}({E_k\phi }, b))\Vert _{\mathrm {BMO}^d(\mathbb {T}^N)}$. Let $\Omega \subseteq \mathbb {T}^N$ be open, $R=Q\times S$, $Q\in \mathcal D^K$. $\mathcal J_S = \cup _{Q \in \mathcal D^K, Q \times S \subseteq \Omega } Q$ for $S \in \mathcal D^{N-K}$. Then

$$\begin{aligned} \left\| P_\Omega \left( \sigma _k\left( {\Pi ^{\vec {\beta }}}\left( E_k\phi , b\right) \right) \right) \right\| _{L^2(\mathbb {T}^N)}^2&= \left\| P_{\Omega } \sigma _k \left( {\Pi ^{\vec {\beta }}}\left( P_\Omega E_k\phi , b\right) \right) \right\| _{L^2(\mathbb {T}^N)}^2 \\&\le \left\| \sigma _k \left( {\Pi ^{\vec {\beta }}}\left( {P_\Omega E_k\phi }, b\right) \right) \right\| _{L^2(\mathbb {T}^N)}^2\\&= \left\| {\Pi ^{\vec {\beta }}}\left( {P_\Omega E_k\phi }, b\right) \right\| _{L^2(\mathbb {T}^N)}^2 \\&= \left\| \sum _{S \in \mathcal D^{N-K}, |S_1| > 2^{-k}} \sum _{Q \in \mathcal D^K: S \times Q \subseteq \Omega } \right. \\&\left. h_{S}(s) \frac{\chi _{Q}}{|Q|}(t) \phi _{SQ} m_{S} b_{Q} \right\| _{L^2(\mathbb {T}^N)}^2\\&= \sum _{S \in \mathcal D^{N\!-\! K}, |S_1| > 2^{\!-k}} \left\| \sum _{Q \subseteq \mathcal J_S } \frac{\chi _{Q}}{|Q|}(t) \phi _{QS} m_{S}b_{Q} \right\| _{L^2(\mathbb {T}^K)}^2\\&= \sum _{S \in \mathcal D^{N\!-\!K}, |S_1| > 2^{\!-\!k}} \left\| \Delta _{m_S b} P_{\mathcal J_S} \phi _S \right\| _{L^2(\mathbb {T}^K)}^2\\&\lesssim \sum _{S \in \mathcal D^{N\!-\!K}, |S_1| > 2^{\!-\!k}} \Vert m_S b\Vert ^2_{\mathrm {BMO}^d(\mathbb {T}^K)} \Vert P_{\mathcal J_S} \phi _S \Vert _{L^2(\mathbb {T}^K)}^2\\&\lesssim \Vert b\Vert ^2_{\mathrm {bmo}^d(\mathbb {T}^N)} \sum _{S \in \mathcal D^{N-K}} \Vert P_{\mathcal J_S} \phi _S \Vert _{L^2(\mathbb {T}^K)}^2\\&\lesssim \Vert b\Vert ^2_{\mathrm {bmo}^d(\mathbb {T}^N)}\Vert P_\Omega \phi \Vert _{L^2(\mathbb {T}^N)}^2\\&\lesssim \Vert b\Vert ^2_{\mathrm {bmo}^d(\mathbb {T}^N)} \Vert \phi \Vert ^2_{\mathrm {BMO}^d(\mathbb {T}^N)}|\Omega |. \end{aligned}$$

$\square$

As in the last section, we immediately deduce that

$$\begin{aligned} \left\| \Pi \left( \Pi ^{\vec {\beta }}(Q^{(1)}_j \phi , b), E^{(1)}_{k}\, \cdot \right) \right\| _{L^2(\mathbb {T}^N) \rightarrow L^2(\mathbb {T}^N)}&\lesssim \frac{k+1}{j+1} \Vert \phi \Vert _{\mathrm {LMO}_1^d(\mathbb {T}^N)} \Vert b \Vert _{\mathrm {bmo}^d(\mathbb {T}^N)}\\&< \frac{k+1}{j+1} \Vert \phi \Vert _{\mathrm {LMO}_{\vec {\beta }}^d(\mathbb {T}^N)} \Vert b \Vert _{\mathrm {bmo}^d(\mathbb {T}^N)}. \end{aligned}$$

The remainder of the proof of Theorem 4.3 is now exactly analogous to the proof of Therem 2.5, defining $T_N = \Pi ({\Pi ^{\vec {\beta }}}(\phi , \cdot ), P_N\cdot )$, where $P_{N} = \sum _{i=2^N-1}^{2^{N+1}-2} \Delta ^{(1)}_{i}$, and using Cotlar’s Lemma in one parameter once more.

5 Commutators and Dyadic Shifts

In this section, for convenience, we restrict ourselves to the two-parameter case. Our interest here is for the iterated commutators with the Hilbert transforms. We would like to mention some facts that can also be seen as a motivation for this paper. We will be writing $H_1$ and $H_2$ for the Hilbert transform in the first and second variables respectively. The first fact is a result of Ferguson–Lacey–Sadosky.

Theorem 5.1

[2, 5, 6] Let $\phi \in \mathrm {BMO}(\mathbb {R}^2)$. Then

$$\begin{aligned}{}[H_1, [H_2, \phi ]]: L^2(\mathbb {R}^2) \rightarrow L^2(\mathbb {R}^2) \end{aligned}$$

is bounded, and $\Vert [H_1, [H_2, \phi ]]\Vert _{L^2 \rightarrow L^2} \approx \Vert \phi \Vert _{\mathrm {BMO}}$.

From this result arises the question of the boundedness of iterated commutators on the endpoints. Our second fact then is the observation that the commutator $[b,H]$ is not in general bounded on $H^1(\mathbb R)$ for $b\in \mathrm {BMO}(\mathbb {R})$ (see [8]). It comes that if we are really considering the boundedness of these commutators on the endpoints ($H^1$ and $\mathrm {BMO}$), then we are only allowed to deal with functions with compact support. This is the idea in [14] from which comes our third fact that will be given after introducing several definitions and notations.

Let

$$\begin{aligned} \mathrm {BMO}\left( [0,1]^2\right) :=\left\{ f\in \mathrm {BMO}(\mathbb {R}^2): \mathrm {supp}f\subseteq [0,1]^2\right\} . \end{aligned}$$

(13)

and

$$\begin{aligned} \mathrm {BMO}^d\left( [0,1]^2\right) = \left\{ f \in \mathrm {BMO}^d(\mathbb {R}^2): \mathrm {supp}f\subseteq [0,1]^2\right\} . \end{aligned}$$

We say that $f \in \mathrm {LMO}^d([0,1]^2)$, if $f \in \mathrm {BMO}^d([0,1]^2)$ and there exists $C>0$ with

$$\begin{aligned} \Vert Q_{\vec {k}} f\Vert _{\mathrm {BMO}^d([0,1]^2)} \le C\frac{1}{(k_1+k_2+2)} \text { for } \vec {k}=(k_1, k_2) \in \mathbb {N}_0 \times \mathbb {N}_0. \end{aligned}$$

The spaces $\mathrm {LMO}_{\vec {\beta }}^d([0,1]^2)$ are defined correspondingly. We also recall the notion of logarithmic mean oscillation introduced in [14].

We say that $f \in \mathcal {LMO}^d([0,1]^2)$, if $f \in \mathrm {BMO}^d([0,1]^2)$ and there exists $C>0$ with

$$\begin{aligned} \Vert Q_{\vec {k}} f\Vert _{\mathrm {BMO}^d([0,1]^2)} \le C\frac{1}{(k_1+1)(k_2+1)} \text { for } \vec {k}=(k_1, k_2) \in \mathbb {N}_0 \times \mathbb {N}_0. \end{aligned}$$

We next recall the relation between various spaces and their dyadic counterparts. Given $\alpha =(\alpha _j)_{j\in \mathbb {Z}}\in \{0,1\}^{\mathbb {Z}}$ and $r\in [1,2)$, we denote by $\mathcal {D}^{\alpha ,r}=r\mathcal {D}^{\alpha }$ the dilated and translated standard dyadic grid $\mathcal {D}$ of $\mathbb {R}$ in the sense of [7]. For $\vec {\alpha }=(\alpha ^1,\alpha ^2)\in \{0,1\}^{\mathbb {Z}}\times \{0,1\}^{\mathbb {Z}}$ and $\vec {r}=(r_1,r_2)\in [1,2)^2$, we define $\mathcal {D}^{\vec {\alpha },\vec {r}}$ to be the dilated and translated product dyadic grid in $\mathbb {R}^2$. This means that $Q=Q_1\times Q_2\in \mathcal {D}^{\vec {\alpha },\vec {r}}$ if $Q_1\in r_1\mathcal {D}^{\alpha ^1}$ and $Q_2\in r_2\mathcal {D}^{\alpha ^2}$.

Following [13, 17], we have the following relations between the strong notions of bounded mean oscillation and their dyadic versions.

$$\begin{aligned} \mathrm {BMO}(\mathbb {R}^2)&= \bigcap _{\vec {\alpha }\in \{0,1\}^{\mathbb {Z}}\times \{0,1\}^{\mathbb {Z}}, \vec {r}\in [1,2)^2 }\mathrm {BMO}^{d,\vec {\alpha }, \vec {r}}(\mathbb {R}^2) \\&= \bigcap _{\vec {\alpha }\in \{0,1\}^{\mathbb {Z}}\times \{0,1\}^{\mathbb {Z}} }\mathrm {BMO}^{d,\vec {\alpha }, \vec {r_0}}(\mathbb {R}^2) \text { for any } \vec {r_0} \in [0,1)^2 , \end{aligned}$$

$$\begin{aligned} \mathrm {bmo}(\mathbb {R}^2)&= \bigcap _{\vec {\alpha }\in \{0,1\}^{\mathbb {Z}}\times \{0,1\}^{\mathbb {Z}}, \vec {r}\in [1,2)^2 }\mathrm {bmo}^{d,\vec {\alpha }, \vec {r}}(\mathbb {R}^2) \\&= \bigcap _{\vec {\alpha }\in \{0,1\}^{\mathbb {Z}}\times \{0,1\}^{\mathbb {Z}} }\mathrm {bmo}^{d,\vec {\alpha }, \vec {r_0}}(\mathbb {R}^2) \text { for any } \vec {r_0} \in [0,1)^2 , \end{aligned}$$

where $\mathrm {BMO}^{d,\vec {\alpha }, \vec {r}}(\mathbb {R}^2)$ and $\mathrm {bmo}^{d,\vec {\alpha }, \vec {r}}(\mathbb {R}^2)$ are the dyadic (with respect to the product dyadic grid $\mathcal {D}^{\vec {\alpha },\vec {r}}$) $\mathrm {BMO}(\mathbb {R}^2)$ and $\mathrm {bmo}(\mathbb {R}^2)$ respectively. One also obtains that

$$\begin{aligned} \mathrm {BMO}\left( [0,1]^2\right) =\bigcap _{\vec {\alpha }\in \{0,1\}^{\mathbb {Z}}\times \{0,1\}^{\mathbb {Z}}, \vec {r}\in [1,2)^2 }\mathrm {BMO}^{d,\vec {\alpha }, \vec {r}}\left( [0,1]^2\right) \end{aligned}$$

and

$$\begin{aligned} \mathrm {bmo}\left( [0,1]^2\right) =\bigcap _{\vec {\alpha }\in \{0,1\}^{\mathbb {Z}}\times \{0,1\}^{\mathbb {Z}}, \vec {r}\in [1,2)^2 }\mathrm {bmo}^{d,\vec {\alpha }, \vec {r}}\left( [0,1]^2\right) . \end{aligned}$$

For the purpose of our last fact, we introduce the space $\mathcal {LMO}([0,1]^2)$.

Definition 5.2

Let $f\in L^2(\mathbb {R}^2)$. We say that $f\in \mathcal {LMO}([0,1]^2)$ if $\mathrm {supp}f\subseteq [0,1]^2$, and there exists a constant $C>0$ such that for any $\vec {\alpha }\in \{0,1\}^{\mathbb {Z}}\times \{0,1\}^{\mathbb {Z}}$, $\vec {r}\in [1,2)^2$, and $\vec {j} = (j_1, j_2) \in \mathbb {N}_0\times \mathbb {N}_0$,

$$\begin{aligned} \left\| Q^{\vec {\alpha }, \vec {r}}_{\vec {j}}f\right\| _{\mathrm {BMO}^{d,\vec {\alpha }, \vec {r}}([0,1]^2)}\le C\frac{1}{(j_1+1)(j_2+1)}. \end{aligned}$$

Here, $Q^{\vec {\alpha }, \vec {r}}_{\vec {j}}$ denotes the projection as in (4), but relative to the dyadic grid $\mathcal {D}^{\vec {\alpha },\vec {r}}$; that is

$$\begin{aligned} Q^{\vec {\alpha }, \vec {r}}_{\vec {j}}f(s,t)=\sum _{r_1|I|\le 2^{-j_1}, r_2|J|\le 2^{-j_2}}\langle f,h_I^{\alpha _1,r_1}h_J^{\alpha _2,r_2}\rangle h_I^{\alpha _1,r_1}(s)h_J^{\alpha _2,r_2}(t), \end{aligned}$$

where $h_I^{\alpha _l,r_l}$ is the Haar wavelet adapted to $I\in r_l\mathcal {D}^{\alpha _l}$, $l=1,2$.

Our last fact is the following pretty recent result by S. Pott and the author [14].

Theorem 5.3

Let $\phi \in \mathcal {LMO}([0,1]^2)$. Then

$$\begin{aligned}{}[H_1, [H_2, \phi ]]: \mathrm {BMO}([0,1]^2) \rightarrow \mathrm {BMO}(\mathbb {R}^2), \end{aligned}$$

is bounded, and $\Vert [H_1, [H_2, \phi ]]\Vert _{\mathrm {BMO}([0,1]^2) \rightarrow \mathrm {BMO}(\mathbb {R}^2)} \lesssim \Vert \phi \Vert _{\mathcal {LMO}([0,1]^2)}$.

We aim to replace in the last theorem, the space $\mathrm {BMO}([0,1]^2)$ by $\mathrm {bmo}([0,1]^2)$, keeping $\mathrm {BMO}(\mathbb {R}^2)$ as the target space. For this we will need to introduce the right concept of logarithmic mean oscillation here.

Definition 5.4

Let $f\in L^2(\mathbb {R}^2)$. We say that $f\in \mathrm {LMO}([0,1]^2)$ if $\mathrm {supp}f\subseteq [0,1]^2$, and there exists a constant $C>0$ such that for any $\vec {\alpha }\in \{0,1\}^{\mathbb {Z}}\times \{0,1\}^{\mathbb {Z}}$, $\vec {r}\in [1,2)^2$, and $\vec {j} = (j_1, j_2) \in \mathbb {N}_0\times \mathbb {N}_0$,

$$\begin{aligned} \left\| Q^{\vec {\alpha }, \vec {r}}_{\vec {j}}f\right\| _{\mathrm {BMO}^{d,\vec {\alpha }, \vec {r}}([0,1]^2)}\le C\frac{1}{(j_1+j_2+2)}. \end{aligned}$$

We do the following observations. First $\mathrm {LMO}([0,1]^2)$ continuously embeds into $\mathrm {BMO}([0,1]^2)$. Secondly, if we denote by $\mathrm {LMO}^{d,\vec {\alpha }, \vec {r}}([0,1]^2)$ the subset of $\mathrm {BMO}^{d,\vec {\alpha }, \vec {r}}([0,1]^2)$ of functions $f$ such that there exists $C>0$ with

$$\begin{aligned} \left\| Q_{\vec {j}}^{\vec {\alpha }, \vec {r}}f\right\| _{\mathrm {BMO}^{d,\vec {\alpha }, \vec {r}}([0,1]^2)}\le C\frac{1}{(j_1+j_2+2)} \quad \text { for any } \vec {j}\in \mathbb {N}_0\times \mathbb {N}_0, \end{aligned}$$

then

$$\begin{aligned} \mathrm {LMO}\left( [0,1]^2\right) =\bigcap _{\vec {\alpha }\in \{0,1\}^{\mathbb {Z}}\times \{0,1\}^{\mathbb {Z}}, \vec {r}\in [1,2)^2 }\mathrm {LMO}^{d,\vec {\alpha }, \vec {r}}\left( [0,1]^2\right) . \end{aligned}$$

$\mathrm {LMO}_1([0,1]^2)$ and $\mathrm {LMO}_2([0,1]^2)$ along with their dyadic counterparts are defined analogously. We are ready to give our main result of this section.

Theorem 5.5

Let $\phi \in \mathrm {LMO}([0,1]^2)$. Then

$$\begin{aligned}{}[H_1, [H_2, \phi ]]: \mathrm {bmo}\left( [0,1]^2\right) \rightarrow \mathrm {BMO}(\mathbb {R}^2), \end{aligned}$$

is bounded, and $\Vert [H_1, [H_2, \phi ]]\Vert _{\mathrm {bmo}([0,1]^2) \rightarrow \mathrm {BMO}(\mathbb {R}^2)} \lesssim \Vert \phi \Vert _{\mathrm {LMO}([0,1]^2)}$.

Before proving Theorem 5.5, let us start by considering its dyadic counterpart. We introduce the dyadic shift operators $S^{d, {\alpha }, {r}}$, $\alpha \in \{0,1\}^{\mathbb Z}$, $r\in [1,2)$. These are the bounded linear operators $S^{d, {\alpha }, {r}}: L^2( \mathbb {R}) \rightarrow L^2(\mathbb {R})$ defined by $S^{d,{\alpha }, \vec {r}} h_I = h_{I^+} - h_{I^-}$, $I \in \mathcal D^{ {\alpha }, {r}}$. For simplicity, we restrict to the standard dyadic grid and write $S^{(1)} = S^d \otimes \mathbf {1}$, $S^{(2)} = \mathbf {1}\otimes S^d$, as operators on $L^2(\mathbb {R}^2) = L^2(\mathbb {R}) \otimes L^2(\mathbb {R})$. The corresponding dyadic version of the above theorem is the following.

Theorem 5.6

Let $\phi \in \mathrm {LMO}^d([0,1]^2)$. Then

$$\begin{aligned} \left[ S^{(1)}, \left[ S^{(2)},\phi \right] \right] : \mathrm {bmo}^d\left( [0,1]^2\right) \rightarrow \mathrm {BMO}^d(\mathbb {R}^2) \end{aligned}$$

is bounded, and $\Vert [S^{(1)}, [S^{(2)},\phi ]]\Vert _{\mathrm {bmo}^d([0,1]^2) \rightarrow \mathrm {BMO}^d(\mathbb {R}^2)} \lesssim \Vert \phi \Vert _{\mathrm {LMO}^d([0,1]^2)}$.

Proof

Let us follow the ideas of [14, 16]. First, we decompose the multiplication operator by $\phi$ into nine parts: ${\Pi }_\phi$, ${\Delta }_\phi$, ${\Pi ^{(0,1)}}_\phi$, ${\Pi ^{(1,0)}}_\phi$, ${R_\Delta }_\phi$, ${R_\Pi }_\phi$, ${\Delta _R}_\phi$, ${\Pi _R}_\phi$, ${R_R}_\phi$, corresponding to the matrix elements $\langle M_\phi h_I(s) h_J(t), h_{I'}(s) h_{J'}(t) \rangle$ for $I' \subset I$, $I' = I$, $I' \subset I$, $I' \supset I$, $J' \subset J$, $J' = J$, $J' \supset J$.

Let us point out the symmetric pairs $({R_\Delta },{\Delta _R})$, $({R_\Pi },{\Pi _R})$, where

$$\begin{aligned} R_{R_\phi }b(s,t)=\sum _{I,J}b_{IJ}m_{IJ}(\phi )h_I(s)h_J(t),\\ \Pi _{R_\phi }b(s,t)=\sum _{I,J}m_J(\phi _I)m_I(b_J)h_I(s)h_J(t) \end{aligned}$$

and

$$\begin{aligned} \Delta _{R_\phi }b(s,t)=\sum _{I,J}m_J(\phi _I) b_{I,J} h_I(s)h_J^2(t). \end{aligned}$$

The following was already proved in [14].

Lemma 5.7

Let $\phi \in L^2([0,1]^2)$. Then the following estimates hold.

$$\begin{aligned} \left\| \left[ S^{(1)}, \left[ S^{(2)},{R_R}_\phi \right] \right] \right\| _{\mathrm {BMO}^d\left( [0,1]^2\right) \rightarrow \mathrm {BMO}^d(\mathbb {R}^2)} \le 2 \Vert \phi \Vert _{\mathrm {BMO}^d\left( [0,1]^2\right) }, \end{aligned}$$

(14)

$$\begin{aligned} \left\| \left[ S^{(1)}, \left[ S^{(2)},{\Delta _R}_\phi \right] \right] \right\| _{\mathrm {BMO}^d\left( [0,1]^2\right) \rightarrow \mathrm {BMO}^d(\mathbb {R}^2)} \lesssim \Vert \phi \Vert _{\mathrm {BMO}^d\left( [0,1]^2\right) }, \end{aligned}$$

(15)

$$\begin{aligned} \left\| \left[ S^{(1)}, \left[ S^{(2)},{\Pi _R}_\phi \right] \right] \right\| _{\mathrm {BMO}^d\left( [0,1]^2\right) \rightarrow \mathrm {BMO}^d(\mathbb {R}^2)} \lesssim \Vert \phi \Vert _{\mathrm {LMO}_{1}^d\left( [0,1]^2\right) }. \end{aligned}$$

(16)

Swapping variables yields

$$\begin{aligned} \left\| \left[ S^{(1)}, \left[ S^{(2)},{R_\Delta }_\phi \right] \right] \right\| _{\mathrm {BMO}^d\left( [0,1]^2\right) \rightarrow \mathrm {BMO}^d(\mathbb {R}^2)} \lesssim \Vert \phi \Vert _{\mathrm {BMO}^d\left( [0,1]^2\right) } \end{aligned}$$

(17)

and

$$\begin{aligned} \left\| \left[ S^{(1)}, \left[ S^{(2)},R_{\Pi _\phi }\right] \right] \right\| _{\mathrm {BMO}^d\left( [0,1]^2\right) \rightarrow \mathrm {BMO}^d\left( \mathbb {R}^2\right) } \lesssim \Vert \phi \Vert _{\mathrm {LMO}_{2}^d\left( [0,1]^2\right) }. \end{aligned}$$

(18)

It comes that as $S^{(1)}$ and $S^{(2)}$ are bounded on $\mathrm {BMO}^d(\mathbb {R}^2)$, it only remains to prove that for $\phi \in \mathrm {LMO}^d([0,1]^2)$, the operators ${\Pi }_\phi$, ${\Delta }_\phi$, ${\Pi ^{(0,1)}}_\phi$, ${\Pi ^{(1,0)}}_\phi$ are bounded from $\mathrm {bmo}^d([0,1]^2)$ to $\mathrm {BMO}^d(\mathbb {R}^2)$.

Again, we already have from [14] that

Lemma 5.8

Let $\phi \in L^2([0,1]^2)$. Then

• (1) $\Pi ^{(1,1)}_\phi = \Delta _\phi : \mathrm {BMO}^d([0,1]^2) \rightarrow \mathrm {BMO}^d(\mathbb {R}^2)$ is bounded, if and only if $\phi \in \mathrm {BMO}^d([0,1]^2)$. Moreover,

  $$\begin{aligned} \Vert \Delta _\phi \Vert _{\mathrm {BMO}^d([0,1]^2) \rightarrow \mathrm {BMO}^d(\mathbb {R}^2)} \approx \Vert \phi \Vert _{\mathrm {BMO}^d([0,1]^2)}. \end{aligned}$$

• (2) $\Pi ^{(1,0)}_\phi :\mathrm {BMO}^d([0,1]^2) \rightarrow \mathrm {BMO}^d(\mathbb {R}^2)$ is bounded if $\phi \in \mathrm {LMO}_{1}^d([0,1]^2)$. Moreover,

  $$\begin{aligned} \left\| \Pi ^{(1,0)}_\phi \right\| _{\mathrm {BMO}^d([0,1]^2) \rightarrow \mathrm {BMO}^d(\mathbb {R}^2)} \lesssim \Vert \phi \Vert _{\mathrm {LMO}_{1}^d([0,1]^2)}. \end{aligned}$$

• (3) $\Pi ^{(0,1)}_\phi :\mathrm {BMO}^d([0,1]^2) \rightarrow \mathrm {BMO}^d(\mathbb {R}^2)$ is bounded if $\phi \in \mathrm {LMO}_{2}^d([0,1]^2)$. Moreover,

  $$\begin{aligned} \left\| \Pi ^{(0,1)}_\phi \right\| _{\mathrm {BMO}^d([0,1]^2) \rightarrow \mathrm {BMO}^d(\mathbb {R}^2)} \lesssim \Vert \phi \Vert _{\mathrm {LMO}_{2}^d([0,1]^2)}. \end{aligned}$$

Hence to finish the proof, we prove the following.

Theorem 5.9

Let $\phi \in L^2([0,1]^2)$. Then $\Pi ^{(0, 0)}_\phi = \Pi _\phi :\mathrm {bmo}^d([0,1]^2) \rightarrow \mathrm {BMO}^d(\mathbb {R}^2)$ is bounded if $\phi \in \mathrm {LMO}^d([0,1]^2)$. Moreover,

$$\begin{aligned} \Vert \Pi _\phi \Vert _{\mathrm {bmo}^d([0,1]^2) \rightarrow \mathrm {BMO}^d(\mathbb {R}^2)} \lesssim \Vert \phi \Vert _{\mathrm {LMO}^d([0,1]^2)}. \end{aligned}$$

Proof

An important ingredient of the proof is a local version of the first assertion in Lemma 3.3. For a bounded (not necessarily dyadic) interval $I \subset \mathbb {R}$, we define $s(I)$ as follows

$$\begin{aligned} s(I) = \left\{ \begin{array}{ll} {\log |I|^{-1} +1} &{}\quad \text {for}\; |I| \le 1 \\ 1 &{}\quad \text {for}\; |I| >1. \end{array} \right. \end{aligned}$$

For any $b\in \mathrm {bmo}([0,1]^2)$ and each rectangle $R = I \times J \subset \mathbb {R}^2$, as $m_Jb$ is uniformly in $\mathrm {BMO}([0,1])$, we get from the one parameter estimate of the mean of a function of bounded mean oscillation (see [14]) the following estimate.

$$\begin{aligned} |m_R b|=|m_I(m_Jb)| \lesssim s(I) \Vert m_Jb\Vert _{\mathrm {BMO}([0,1])}\le s(I) \Vert b\Vert _{\mathrm {bmo}([0,1]^2)}. \end{aligned}$$

(19)

Recall that we are looking to prove that given $\phi \in \mathrm {LMO}^d([0,1]^2)$, $b\in \mathrm {BMO}^d([0,1]^2)$ and $f\in L^2(\mathbb R^2)$, the function $\Pi \left( \Pi (\phi ,b),f\right)$ belongs to $L^2(\mathbb R^2)$.

We recall that $\mathcal D(\mathbb {R})$ is the standard system of dyadic intervals in $\mathbb {R}$. The Haar basis of $L^2(\mathbb {R}^2)$ is $(h_I \otimes h_J)_{I, J \in \mathcal D(\mathbb {R})} = (h_R)_{R \in \mathcal D(\mathbb {R}) \times \mathcal D(\mathbb {R})}$. We have the following decomposition (see also [14])

$$\begin{aligned} f = \sum _{j_1 = - \infty }^\infty \sum _{j_2 = - \infty }^\infty \Delta _{\vec {j}} f , \end{aligned}$$

with

$$\begin{aligned} \Delta _{\vec {j}} f&= \sum _{|I |= 2^{-j_1}, |J|= 2^{- j_2} } h_I(s) h_J(t) \langle f, h_I \otimes h_J \rangle \\&= \sum _{ R \in \mathcal D_{j_1}(\mathbb {R}) \times \mathcal D_{j_2}(\mathbb {R})} h_R \langle f, h_R \rangle \end{aligned}$$

$j_1, j_2 \in \mathbb Z.$

Let $T:=\Pi \left( \Pi (\phi ,b), \cdot \right)$. Then $T$ decomposes as follows

$$\begin{aligned} T=P_{(0,1)^2}T +P_{(0,1)\times (0,1)^c}T + P_{(0,1)^c\times (0,1)} T +P_{(0,1)^c\times (0,1)^c}T , \end{aligned}$$

(20)

where

$$\begin{aligned} P_{(0,1)\times (0,1)}&= \sum _{j_1 = 0}^{\infty }\sum _{j_2=0}^{\infty } \Delta _{\vec {j}}, \\ P_{(0,1)\times (0,1)^c}&= \sum _{j_1 =0}^{\infty }\sum _{j_2=-\infty }^{-1} \Delta _{\vec {j}}, \\ P_{(0,1)^c\times (0,1)}&= \sum _{j_1 = - \infty }^{-1}\sum _{j_2=0}^{\infty } \Delta _{\vec {j}}, \\ P_{(0,1)^c\times (0,1)^c}&= \sum _{j_1 = - \infty }^{-1}\sum _{j_2=-\infty }^{-1} \Delta _{\vec {j}} \end{aligned}$$

(see [14]).

Let us prove that each of the terms in the right hand side of the identity (20) is bounded on $L^2(\mathbb {R}^2)$.

We start with the last term. We observe that as

$$\begin{aligned} P_{(0,1)^c\times (0,1)^c}\Pi \left( \Pi (\phi ,b), \cdot \right) =\Pi \left( \Pi (P_{(0,1)^c\times (0,1)^c}\phi ,b), \cdot \right) , \end{aligned}$$

we only have to prove that given $\phi \in \mathrm {LMO}^d([0,1]^2)$ and $b\in \mathrm {bmo}^d([0,1]^2)$, $P_{(0,1)^c\times (0,1)^c}\Pi (\phi ,b)$ belongs to $\mathrm {BMO}^d(\mathbb R^2)$. Observing with (19) that for $R=I\times J\in \mathcal R$ with $|I|,|J|\ge 1$, $|m_Rb|\lesssim \Vert b\Vert _{\mathrm {bmo}^d([0,1]^2)}$, one obtains directly that for any open set $\Omega \subset \mathbb R^2$,

$$\begin{aligned} \left\| P_\Omega \left( P_{(0,1)^c\times (0,1)^c}\Pi (\phi ,b)\right) \right\| _{L^2(\mathbb {R}^2)}^2\le \left\| P_\Omega \phi \right\| _{L^2(\mathbb {R}^2)}^2\Vert b\Vert _{\mathrm {bmo}^d([0,1]^2)}^2, \end{aligned}$$

which proves that this term is bounded on $L^2(\mathbb R^2)$.

For $f\in L^2(\mathbb {R}^2)$, that

$$\begin{aligned} P_{(0,1)^2}\Pi \left( \Pi (\phi ,b),f\right) = \Pi \left( \Pi (P_{(0,1)^2} \phi ,b),f\right) \end{aligned}$$

is in $L^2(\mathbb R^2)$ is obtained exactly as in the proof of Theorem 2.5, with the help of the growth estimate (19).

The second and third terms are symmetric, hence we only prove the boundedness of the second one. For this, we observe that as

$$\begin{aligned} P_{(0,1)\times (0,1)^c}\Pi \left( \Pi (\phi ,b), \cdot \right) =\Pi \left( \Pi ( P_{(0,1)\times (0,1)^c} \phi ,b), \cdot \right) , \end{aligned}$$

it is enough to prove that $\Pi ( P_{(0,1)\times (0,1)^c} \phi ,b)\in \mathrm{BMO}^\mathrm{d}(\mathbb {R}^2)$. The estimate (19) tells us that for $R=I\times J\in \mathcal R$ with $|J|>1$, $|m_Rb|\lesssim \Vert b\Vert _{\mathrm {bmo}^d([0,1]^2)}$. Hence, for any open set $\Omega \in \mathbb {R}^2$,

$$\begin{aligned} \left\| P_\Omega \left( P_{(0,1)\times (0,1)^c}\Pi (\phi ,b)\right) \right\| _{L^2(\mathbb {R}^2)}^2\le \Vert P_\Omega \phi \Vert _{L^2(\mathbb {R}^2)}^2\Vert b\Vert _{\mathrm {bmo}^d([0,1]^2)}^2, \end{aligned}$$

which proves that this term is also bounded on $L^2(\mathbb R^2)$. The proof is complete. $\square$

We finish this section with the proof of Theorem 5.5.

[Proof of Theorem 5.5] The proof follows exactly as in [14] for the case of $[H_1,[H_2,b]]:\mathrm {BMO}([0,1]^2)\rightarrow \mathrm {BMO}(\mathbb {R}^2)$; we give it here for completeness. We use the fact that the Hilbert transform can be represented as averages of dyadic shifts (see [7, 12]). This allows us to write for $b \in \mathrm {bmo}([0,1]^2)$ and $\phi \in \mathrm {LMO}([0,1]^2)$,

$$\begin{aligned}{}[H_1, [H_2, \phi ]] b=\frac{64}{\pi ^2}\int \limits _1^2\int \limits _1^2\int \limits _{\{0, 1\}^{\mathbb {Z}}}\int \limits _{\{0, 1\}^{\mathbb {Z}}}\left[ S^{\alpha ^1, r_1},\left[ S^{\alpha ^2, r_2}, \phi \right] \right] \, b \,d\mu (\alpha ^1)\frac{dr_1}{r_1}\,d\mu (\alpha ^2)\frac{dr_2}{r_2} \end{aligned}$$

(see [7]). Next, we recall that for $b \in \mathrm {bmo}([0,1]^2)$ and $\phi \in \mathrm {LMO}([0,1]^2)$, we have $b \in \mathrm {bmo}^{d,\vec {\alpha }, \vec {r}}([0,1]^2)$, and $\phi \in \mathrm {LMO}^{d,\vec {\alpha }, \vec {r}}([0,1]^2)$ for each $\vec {\alpha }=(\alpha ^1, \alpha ^2)\in \{0, 1\}^{\mathbb {Z}}\times \{0, 1\}^{\mathbb {Z}}$ and $\vec {r}=(r_1, r_2)\in [1,2)^2$ with uniformly bounded norms. It follows from Theorem 5.6 that there exists a constant $C>0$ such that

$$\begin{aligned} \left\| \left[ S^{\alpha ^1, r_1},\left[ S^{\alpha ^2, r_2}, \phi \right] \right] b\right\| _{\mathrm {BMO}^{d,\vec {\alpha }, \vec {r}}(\mathbb {R}^2)}&\le C \Vert b\Vert _{\mathrm {bmo}([0,1]^2)} \Vert \phi \Vert _{\mathrm {LMO}([0,1]^2)}\\&\text { for all } (\alpha ^1, \alpha ^2, r_1, r_2). \end{aligned}$$

Hence, using [17, Remark 0.5], we obtain that

$$\begin{aligned} \frac{64}{\pi ^2}\int \limits _1^2\int \limits _1^2\int \limits _{\{0, 1\}^{\mathbb {Z}}}\int \limits _{\{0, 1\}^{\mathbb {Z}}}\left[ S^{\alpha ^1, r_1},\left[ S^{\alpha ^2, r_2}, \phi \right] \right] \, b \,d\mu (\alpha ^1)\frac{dr_1}{r_1}\,d\mu (\alpha ^2)\frac{dr_2}{r_2} \in \mathrm {BMO}(\mathbb {R}^2) \end{aligned}$$

with norm controlled by $\Vert b\Vert _{\mathrm {bmo}([0,1]^2)} \Vert \phi \Vert _{\mathrm {LMO}([0,1]^2)}$. The proof is complete. $\square$