Improved $A_1-A_\infty$ and related estimates for commutators of rough singular integrals

TL;DR

This paper improves estimates for commutators of rough singular integrals using sparse domination techniques, providing sharper bounds related to $A_1-A_ abla$ and $A_ abla$ constants, advancing the understanding of these operators.

Contribution

The paper introduces improved $A_1-A_ abla$ and $A_ abla$ estimates for commutators of rough singular integrals, utilizing new sparse domination results.

Findings

Enhanced $A_1-A_ abla$ estimate over previous results

New bounds involving $A_ abla$ constant and exponential $A_q-A_ abla$ constant

Establishment of sparse domination for commutators with rough kernels

Abstract

An $A_1-A_\infty$ estimate improving a previous result in arXiv:1607.06432 is obtained. Also new a result in terms of the ${A_\infty}$ constant and the one supremum $A_q-A_\infty^{\exp}$ constant, is proved, providing a counterpart for the result obained in arXiv:1705.08364. Both of the preceding results rely upon a sparse domination in terms of bilinear forms for $[b,T_\Omega]$ with $\Omega\in L^\infty(\mathbb{S}^{n-1})$ and $b\in BMO$ which is established relying upon techniques from arXiv:1705.07397.

Equations

$$[w{]}_{A_p}=\underset{Q}{\sup }\left(\frac{1}{|Q|}{\int }_{Q}w\right){\left(\frac{1}{|Q|}{\int }_Q{w}^{\frac{1}{1-p}}\right)}^{p-1}<\infty 1<p<\infty$$

$$[w{]}_{A_1}=\underset{x\in {\mathbb{R}}^n}{\mathrm{esssup}}\frac{Mw(x)}{w(x)}<\infty .$$

$$T_{\Omega }f(x)=pv{\int }_{{\mathbb{R}}^n}\frac{\Omega (y^{\prime })}{|y{|}^n}f(x-y)dy$$

$$\Vert T_{\Omega }{\Vert }_{L^2(w)}\le c_n\Vert \Omega {\Vert }_{L^{\infty }({\mathbb{S}}^{n-1})}[w{]}_{A_2}^2.$$

$$\Big{|}\int_{\mathbb{R}^{n}}T_{\Omega}(f)gdx\Big{|}\leq c_{n}C_{T}s^{\prime}\sup_{\mathcal{S}}\sum_{Q\in\mathcal{S}}\Big{(}\int_{Q}|f|\Big{)}\Big{(}\frac{1}{|Q|}\int_{Q}|g|^{s}\Big{)}^{1/s},$$

$$\{\begin{array}{ll}1<s<\infty & \text{if }\Omega \in L^{\infty }({\mathbb{S}}^{n-1})\\ q^{\prime }\le s<\infty & \text{if }\Omega \in L^{q,1}\log L({\mathbb{S}}^{n-1})\end{array}$$

$$C_T=\{\begin{array}{ll}\Vert \Omega {\Vert }_{L^{\infty }({\mathbb{S}}^{n-1}),}& \text{if }\Omega \in L^{\infty }({\mathbb{S}}^{n-1})\\ \Vert \Omega {\Vert }_{L^{q,1}\log L({\mathbb{S}}^{n-1})}& \text{if }\Omega \in L^{q,1}\log L({\mathbb{S}}^{n-1}).\end{array}$$

$$\Vert T_{\Omega }{\Vert }_{L^p(w)}\le c_n\Vert \Omega {\Vert }_{L^{\infty }({\mathbb{S}}^{n-1})}[w{]}_{A_1}^{\frac{1}{p}}[w{]}_{A_{\infty }}^{\frac{1}{p^{\prime }}}.$$

$$\Vert T_{\Omega }{\Vert }_{L^p(w)}\le c_n\Vert \Omega {\Vert }_{L^{\infty }({\mathbb{S}}^{n-1})}[w{]}_{A_1}^{\frac{1}{p}}[w{]}_{A_{\infty }}^{1+\frac{1}{p^{\prime }}}.$$

$$[b,T]f(x)=T(bf)(x)-b(x)Tf(x).$$

$$\Vert [b,T_{\Omega }]{\Vert }_{L^p(w)}\le c_n\Vert \Omega {\Vert }_{L^{\infty }({\mathbb{S}}^{n-1})}[w{]}_{A_1}^{\frac{1}{p}}[w{]}_{A_{\infty }}^{2+\frac{1}{p}}.$$

$$\Vert [b,T_{\Omega }]{\Vert }_{L^p(M_r(w))\to L^p(w)}\le c_n\Vert \Omega {\Vert }_{L^{\infty }({\mathbb{S}}^{n-1})}\Vert b{\Vert }_{\mathrm{BMO}}(p^{\prime }{)}^3{p}^2(r^{\prime }{)}^{1+\frac{1}{p^{\prime }}}$$

$$\Vert [b,T_{\Omega }]{\Vert }_{L^p(M(w))\to L^p(w)}\le c_n\Vert \Omega {\Vert }_{L^{\infty }({\mathbb{S}}^{n-1})}\Vert b{\Vert }_{\mathrm{BMO}}(p^{\prime }{)}^3{p}^2[w{]}_{A_{\infty }}^{1+\frac{1}{p^{\prime }}}$$

$$\Vert [b,T_{\Omega }]{\Vert }_{L^p(w)}\le c_n\Vert \Omega {\Vert }_{L^{\infty }({\mathbb{S}}^{n-1})}\Vert b{\Vert }_{\mathrm{BMO}}(p^{\prime }{)}^3{p}^2[w{]}_{A_1}^{\frac{1}{p}}[w{]}_{A_{\infty }}^{1+\frac{1}{p^{\prime }}}.$$

$$\Vert T{\Vert }_{L^p(w)}\le c_{n,p,q}{c}_T[w{]}_{A_q^{\frac{1}{p}}(A_{\infty }^{\exp }{)}^{\frac{1}{p^{\prime }}}}$$

$$[w{]}_{A_q^{\frac{1}{p}}(A_{\infty }^{\exp }{)}^{\frac{1}{p^{\prime }}}}=\underset{Q}{\sup }⟨w{⟩}_Q⟨w^{\frac{1}{1-q}}{⟩}_Q^{\frac{q-1}{p}}\exp {\left(⟨\log w^{-1}{⟩}_Q\right)}^{\frac{1}{p^{\prime }}}$$

$$c_T=\{\begin{array}{ll}\Vert \Omega {\Vert }_{L^{\infty }({\mathbb{S}}^{n-1})}& \text{if }T=T_{\Omega }\text{ with }\Omega \in L^{\infty }({\mathbb{S}}^{n-1}),\\ c_K+\Vert T{\Vert }_{L^2}+\Vert \omega {\Vert }_{\text{Dini}}& \text{if }T\text{ is an }\omega \text{-Calderoˊn-Zygmund operator}.\end{array}$$

$$[w{]}_{A_q^{\frac{1}{p}}(A_{\infty }^{\exp }{)}^{\frac{1}{p^{\prime }}}}\le c_n[w{]}_{A_q}.$$

$$\Vert [b,T]{\Vert }_{L^p(w)}\le c_{n,p,q}{c}_T[w{]}_{A_{\infty }}[w{]}_{A_q^{\frac{1}{p}}(A_{\infty }^{\exp }{)}^{\frac{1}{p^{\prime }}}}$$

$$\begin{array}{rl}\Vert [b,T]{\Vert }_{L^p(w)}& \le c[w{]}_{A_q}^2,\\ \Vert [b,T_{\Omega }]{\Vert }_{L^p(w)}& \le c[w{]}_{A_q}^3.\end{array}$$

$$\max \{[w{]}_{A_{\infty }},[w{]}_{A_q^{\frac{1}{p}}(A_{\infty }^{\exp }{)}^{\frac{1}{p^{\prime }}}}\}\le c_n[w{]}_{A_q}.$$

$$|⟨[b,T_{\Omega }]f,g⟩|\le C_n{p}^{\prime }\Vert \Omega {\Vert }_{L^{\infty }({\mathbb{S}}^{n-1})}\sum _{j=1}^{\infty }\left({\mathcal{T}}_{{\mathcal{S}}_j,1,p}(b,f,g)+{\mathcal{T}}_{{\mathcal{S}}_j,1,p}^{\ast }(b,f,g)\right)$$

$$\begin{array}{rl}{\mathcal{T}}_{{\mathcal{S}}_j,r,s}(b,f,g)& =\sum _{Q\in {\mathcal{S}}_j}⟨f{⟩}_{r,Q}⟨(b-b_Q)g{⟩}_{s,Q}|Q|\\ {\mathcal{T}}_{{\mathcal{S}}_j,r,s}^{\ast }(b,f,g)& =\sum _{Q\in {\mathcal{S}}_j}⟨(b-b_Q)f{⟩}_{r,Q}⟨g{⟩}_{s,Q}|Q|\end{array}$$

$$\{3Q:Q\in \mathcal{D}\}={\cup }_{j=1}^{3^n}{\mathcal{D}}_j$$

$$[w{]}_{A_{\infty }}=\underset{Q}{\sup }\frac{1}{w(Q)}{\int }_{Q}M(w{\chi }_Q)<\infty ,$$

$${\left(\frac{1}{|Q|}{\int }_Q{w}^{r_w}\right)}^{\frac{1}{r_w}}\le 2\left(\frac{1}{|Q|}{\int }_{Q}w\right)$$

$${\mathcal{B}}_{\mathcal{S}}f(x):=\sum _{Q\in \mathcal{S}}\Vert f{\Vert }_{\Psi (L),Q}{\chi }_Q(x).$$

$$\Vert {\mathcal{B}}_{\mathcal{S}}f{\Vert }_{L^1(w)}\le \frac{4}{\eta }[w{]}_{A_{\infty }}\Vert M_{\Psi (L)}f{\Vert }_{L^1(w)}.$$

$$\Vert f{\Vert }_{\Psi (L)(\mu ),E}=\inf \left\{\lambda >0:\frac{1}{\mu (E)}{\int }_E\Psi \left(\frac{|f|}{\lambda }\right)d\mu \le 1\right\}$$

$$M_{\Psi (L)}f(x)=\underset{x\in Q}{\sup }\Vert f{\Vert }_{\Psi (L)(\mu ),Q}.$$

Full text

This research was supported by the Basque Government through the BERC 2014-2017 program and by the Spanish Ministry of Economy and Competitiveness MINECO through BCAM Severo Ochoa excellence accreditation SEV-2013-0323 and also through the project MTM2014-53850-P

1. Introduction

We recall that a weight $w$, namely a non negative locally integrable function, belongs to $A_{p}$ if

[TABLE]

or in the case $p=1$ if

[TABLE]

Given $\Omega\in L(\mathbb{S}^{n-1})$ with $\int_{\mathbb{S}^{n-1}}\Omega=0$ we define the rough singular integral $T_{\Omega}$ by

[TABLE]

where $y^{\prime}=\frac{y}{|y|}$.

During the last years an increasing interest on the study of the sharp dependence on the $A_{p}$ constants of rough singular integrals has appeared. In particular it was established in [10] that

[TABLE]

Recently the following sparse domination (very recently reproved in [13] for the case $\Omega\in L^{\infty}(\mathbb{S}^{n-1})$) was established in [3].

Theorem**.**

For all $1<p<\infty$, $f\in L^{p}(\mathbb{R}^{n})$ and $g\in L^{p^{\prime}}(\mathbb{R}^{n})$, we have that

[TABLE]

where each $\mathcal{S}$ is a sparse family of a dyadic lattice $\mathcal{D}$,

[TABLE]

and

[TABLE]

The preceding sparse domination was proved to be a very useful result in [20]. Among other estimates, the following $A_{1}-A_{\infty}$ estimate was established in that paper (see Lemma 2.2 in Section 2 for the definition of the $A_{\infty}$ constant)

[TABLE]

The preceding inequality is an improvement of the following estimate established earlier in [22]

[TABLE]

Now we recall that the commutator of an operator $T$ and a symbol $b$ is defined as

[TABLE]

In the case of $T$ being a Calderón-Zygmund operator this operator was introduced by R.R. Coifman, R. Rochberg and G. Weiss in [2]. They established that $b\in\operatorname*{BMO}$ is a sufficient condition for $[b,T]$ to be bounded on $L^{p}$ for every $1<p<\infty$ and also a converse result in terms of the Riesz transforms, namely that the boundedness of $[b,R_{j}]$ on $L^{p}$ for some $1<p<\infty$ and for every Riesz transform implies that $b\in\operatorname*{BMO}$.

In [22] the following estimate for commutators of rough singular integrals and a symbol $b\in\operatorname*{BMO}$ was obtained.

[TABLE]

One of the main goals of this paper is to improve the dependence on the $[w]_{A_{\infty}}$ constant in (1.2). Our result is the following.

Theorem 1.1**.**

Let $T_{\Omega}$ be a rough homogeneous singular integral with $\Omega\in L^{\infty}(\mathbb{S}^{n-1})$ and let $b\in\operatorname*{BMO}$. For every weight $w$ we have that

[TABLE]

where $r>1$. Assuming additionally that $w\in{A_{\infty}}$

[TABLE]

and, furthermore, if $w\in A_{1}$, then

[TABLE]

Very recently a conjecture left open by K. Moen and A. Lerner in [18] was solved by K. Li in [19]. Actually he obtained a more general result.

Theorem**.**

Let $T$ be a Calderón-Zygmund operator or a rough singular integral with $\Omega\in L^{\infty}(\mathbb{S}^{n-1})$. Then for every $1<q<p<\infty$

[TABLE]

where

[TABLE]

and

[TABLE]

This result can be regarded as an improvement of the linear dependence on the $A_{q}$ constant established in [20], and that, as it was stated there, follows from the linear dependence on the $A_{1}$ constant by [5, Corollary 4.3]. Such an improvement stems from the fact that

[TABLE]

In the next Theorem we provide a counterpart of the preceding result for commutators.

Theorem 1.2**.**

Let $T$ be a Calderón-Zygmund operator or a rough singular integral with $\Omega\in L^{\infty}(\mathbb{S}^{n-1})$. Then for every $1<q<p<\infty$

[TABLE]

We would like to recall the following known estimates.

[TABLE]

The first of them can be derived as a consequence of the quadratic dependence on the $A_{1}$ constant of $[b,T]$ obtained in [24] combined with [5, Corollary 4.3], while the second one was established in [22]. In both cases we improve the dependence on the $A_{q}$ constant since we are able to prove a mixed $A_{\infty}-A_{q}^{\frac{1}{p}}(A_{\infty}^{\exp})^{\frac{1}{p^{\prime}}}$ bound and

[TABLE]

In order to establish Theorems 1.2 and 1.1 we will rely upon a suitable sparse domination result for $[b,T_{\Omega}]$. This result will be a natural bilinear counterpart of the result obtained in [17] for $[b,T]$ with $T$ a Calderón-Zygmund operator and also of (1.1). The precise statement is the following.

Theorem 1.3**.**

Let $T_{\Omega}$ be a rough homogeneous singular integral with $\Omega\in L^{\infty}(\mathbb{S}^{n-1})$. Then, for every compactly supported $f,g\in\mathcal{C}^{\infty}({\mathbb{R}}^{n})$ every $b\in\operatorname*{BMO}$ and $1<p<\infty$, there exist $3^{n}$ dyadic lattices $\mathcal{D}_{j}$ and $3^{n}$ sparse families ${\mathcal{S}}_{j}\subset\mathcal{D}_{j}$ such that

[TABLE]

where

[TABLE]

Remark 1.4*.*

In the preceding Theorem and throughout the rest of this work $\langle h\rangle^{w}_{\alpha,Q}=\left(\frac{1}{w(Q)}\int_{Q}|h|^{\alpha}wdx\right)^{\frac{1}{\alpha}}$. We may drop $\alpha$ in the case $\alpha=1$ and $w$ when we consider the Lebesgue measure.

The rest of the paper is organized as follows. We devote Section 2 to gather some results and definitions that will be needed to prove the main theorems. Section 3 is devoted to the proof of Theorem 1.3. In Section 4 we prove Theorem 1.1. We end this work providing a proof of Theorem 1.2 in Section 5.

2. Preliminaries

In this section we gather some definitions and results that will be necessary for the proofs of the main theorems.

We start borrowing some definitions and a basic lemma from [14]. Given a cube $Q_{0}\subset{\mathbb{R}}^{n}$, we denote by ${\mathcal{D}}(Q_{0})$ the family of all dyadic cubes with respect to $Q_{0}$, namely, the cubes obtained subdividing repeatedly $Q_{0}$ and each of its descendants into $2^{n}$ subcubes of the same sidelength.

We say that $\mathcal{D}$ is a dyadic lattice if it is a collection of cubes of ${\mathbb{R}}^{n}$ such that:

1. (1)

If $Q\in\mathcal{D}$, then $\mathcal{D}(Q_{0})\subset\mathcal{D}$. 2. (2)

For every pair of cubes $Q^{\prime},Q^{\prime\prime}\in\mathcal{D}$ there exists a common ancestor, namely, we can find $Q\in\mathcal{D}$ such that $Q^{\prime},Q^{\prime\prime}\in{\mathcal{D}}(Q)$. 3. (3)

For every compact set $K\subset{\mathbb{R}}^{n}$, there exists a cube $Q\in\mathcal{D}$ such that $K\subset Q$.

Lemma 2.1** ($3^{n}$ dyadic lattices lemma).**

Given a dyadic lattice $\mathcal{D}$, there exist $3^{n}$ dyadic lattices $\mathcal{D}_{1},\dots,\mathcal{D}_{3^{n}}$ such that

[TABLE]

and for each cube $Q\in\mathcal{D}$ and $j=1,\dots,3^{n}$, there exists a unique cube $R\in\mathcal{D}_{j}$ with sidelength $l(R)=3l(Q)$ containing $Q$.

Now we gather some results that will be needed to prove Theorem 1.1. The first of them is the so called Reverse Hölder inequality that was proved in [8] (see also [9]).

Lemma 2.2**.**

For every $w\in A_{\infty}$, namely for every weight such that

[TABLE]

the following estimate holds

[TABLE]

where $r_{w}=1+\frac{1}{\tau_{n}[w]_{A_{\infty}}}$ and $\tau_{n}>0$ is a constant independent of $w$.

At this point we would like to recall that if $w\in A_{p}\subseteq A_{\infty}$ then $[w]_{A_{\infty}}\leq c_{n}[w]_{A_{p}}$. This fact makes mixed $A_{\infty}-A_{p}$ bounds interesting, since they provide a sharper dependence than $A_{p}$ bounds. We also need to borrow the following lemma from [22].

Lemma 2.3**.**

Let $w\in A_{\infty}$. Let $\mathcal{D}$ be a dyadic lattice and $\mathcal{S}\subset\mathcal{D}$ be an $\eta$-sparse family. Let $\Psi$ be a Young function. Given a measurable function $f$ on $\mathbb{R}^{n}$ define

[TABLE]

Then we have

[TABLE]

We recall that $\Psi:[0,\infty)\rightarrow[0,\infty)$ is a Young function if it is a convex, increasing function such that $\Psi(0)=0$. We define the local Orlicz norm associated to a Young function $\Psi$ as

[TABLE]

where $E$ is a set of finite measure. We note that in the case $\Psi(t)=t^{r}$ we recover the standard $L^{r}$ local norm. We shall drop $\mu$ from the notation in the case of the Lebesgue measure and write $w$ instead of $wdx$ for measures that are absolutely continuous with respect to the Lebesgue measure.

Using the local the preceding definition of local norm, we can define the maximal function associated to a Young function $\Psi$ in the natural way,

[TABLE]

We end this section recalling two basic estimates that work for doubling measures. The first of them is a particular case of the generalized Hölder inequality and the second can be derived, for example, from [1, Lemma 4.1].

[TABLE]

For a detailed account of local Orlicz norms and maximal functions associated to Young functions we encourage the reader to consult references such as [25], [23], [21] or [4].

3. Proof of Theorem 1.3

The proof of Theorem 1.3 relies upon techniques recently developed by A. K. Lerner in [13]. Given an operator $T$ we define the bilinear operator $\mathcal{M}_{T}$ by

[TABLE]

where the supremum is taken over all cubes $Q\subset{\mathbb{R}}^{n}$ containing $x$. Our first result provides a sparse domination principle based on that bilinear operator.

Theorem 3.1**.**

Let $1\leq q\leq r$ and $s\geq 1$. Assume that $T$ is a sublinear operator of weak type $(q,q)$, and ${\mathcal{M}}_{T}$ maps $L^{r}\times L^{s}$ into $L^{\nu,\infty}$, where $\frac{1}{\nu}=\frac{1}{r}+\frac{1}{s}$. Then, for every compactly supported $f,g\in\mathcal{C}^{\infty}({\mathbb{R}}^{n})$ and every $b\in\operatorname*{BMO}$, there exist $3^{n}$ dyadic lattices $\mathcal{D}_{j}$ and $3^{n}$ sparse families ${\mathcal{S}}_{j}\subset\mathcal{D}_{j}$ such that

[TABLE]

where

[TABLE]

and

[TABLE]

It is possible to relax the condition imposed on $b$ for this result and the subsequent ones, but we restrict ourselves to this choice for the sake of clarity.

Proof of Theorem 3.1.

By Lemma 2.1, there exist $3^{n}$ dyadic lattices ${\mathcal{D}}_{j}$ such that for every $Q\subset{\mathbb{R}}^{n}$, there is a cube $R=R_{Q}\in{\mathcal{D}}_{j}$ for some $j$, for which $3Q\subset R_{Q}$ and $|R_{Q}|\leq 9^{n}|Q|$.

Let us fix a cube $Q_{0}\subset{\mathbb{R}}^{n}$. Now we can define a local analogue of ${\mathcal{M}}_{T}$ by

[TABLE]

We define the sets $E_{i}$ $i=1,\dots 4$ as follows

[TABLE]

We can choose $A_{i}$ in such a way that

[TABLE]

Actually it suffices to take

[TABLE]

with $c_{n},c_{n,r,\nu}$ large enough. For this choice of $E_{i}$ the set $\Omega=\cup_{i}E_{i}$ satisfies $|\Omega|\leq\frac{1}{2^{n+2}}|Q_{0}|$.

Now applying Calderón-Zygmund decomposition to the function $\chi_{\Omega}$ on $Q_{0}$ at height $\lambda=\frac{1}{2^{n+1}}$ we obtain pairwise disjoint cubes $P_{j}\in{\mathcal{D}}(Q_{0})$ such that

[TABLE]

and also $|\Omega\setminus\cup_{j}P_{j}|=0$. From the properties of the cubes it readily follows that $\sum_{j}|P_{j}|\leq\frac{1}{2}|Q_{0}|$ and $P_{j}\cap\Omega^{c}\not=\emptyset$.

Now, since $|\Omega\setminus\cup_{j}P_{j}|=0$, we have that

[TABLE]

Also, since $P_{j}\cap\Omega^{c}\not=\emptyset$, we obtain

[TABLE]

Our next step is to observe that for any arbitrary pairwise disjoint cubes $P_{j}\in{\mathcal{D}}(Q_{0})$,

[TABLE]

For the first two terms, using that $[b,T]f=[b-c,T]f$ for any $c\in{\mathbb{R}}$, we obtain

[TABLE]

Therefore, combining all the preceding estimates with Hölder’s inequality (here we take into account $q\leq r$ and $s\geq 1$) and calling $A=\sum_{i}A_{i}$ we have that

[TABLE]

Since $\sum_{j}|P_{j}|\leq\frac{1}{2}|Q_{0}|$, iterating the above estimate, we obtain that there is a $\frac{1}{2}$-sparse family ${\mathcal{F}}\subset{\mathcal{D}}(Q_{0})$ such that

[TABLE]

To end the proof, take now a partition of ${\mathbb{R}}^{n}$ by cubes $R_{j}$ such that $\text{supp}\,(f)\subset 3R_{j}$ for each $j$. One way to do that is the following. We take a cube $Q_{0}$ such that $\text{supp}\,(f)\subset Q_{0}$ and cover $3Q_{0}\setminus Q_{0}$ by $3^{n}-1$ congruent cubes $R_{j}$. Each of them satisfies $Q_{0}\subset 3R_{j}$. We continue covering in the same way $9Q_{0}\setminus 3Q_{0}$, and so on. The family of the resulting cubes of this process, including $Q_{0}$, satisfies the desired property.

Having such a partition, apply (3.2) to each $R_{j}$. We obtain a $\frac{1}{2}$-sparse family ${\mathcal{F}}_{j}\subset{\mathcal{D}}(R_{j})$ such that

[TABLE]

Therefore, setting $\mathcal{F}=\cup_{j}\mathcal{F}_{j}$

[TABLE]

Now since $3Q\subset R_{Q}$ and $|R_{Q}|\leq 3^{n}|3Q|$, clearly $\langle h\rangle_{\alpha,{3Q}}\leq c_{n}\langle h\rangle_{\alpha,{R_{Q}}}$. Further, setting ${\mathcal{S}}_{j}=\{R_{Q}\in{\mathcal{D}}_{j}:Q\in{\mathcal{F}}\}$, and using that ${\mathcal{F}}$ is $\frac{1}{2}$-sparse, we obtain that each family ${\mathcal{S}}_{j}$ is $\frac{1}{2\cdot 9^{n}}$-sparse. Hence

[TABLE]

and (3.1) holds. ∎

Given $1\leq p\leq\infty$, we define the maximal operator ${\mathcal{M}}_{p,T}$ by

[TABLE]

(in the case $p=\infty$ we call ${\mathcal{M}}_{p,T}f(x)=M_{T}f(x)$).

Our next step is to provide a suitable version of [13, Corollary 3.2] for the commutator. The result is the following.

Corollary 3.2**.**

Let $1\leq q\leq r$ and $s\geq 1$. Assume that $T$ is a sublinear operator of weak type $(q,q)$, and ${\mathcal{M}}_{s^{\prime},T}$ is of weak type $(r,r)$. Then, for every compactly supported $f,g\in\mathcal{C}^{\infty}({\mathbb{R}}^{n})$ and every $b\in\operatorname*{BMO}$, there exist $3^{n}$ dyadic lattices $\mathcal{D}_{j}$ and $3^{n}$ sparse families ${\mathcal{S}}_{j}\subset\mathcal{D}_{j}$ such that

[TABLE]

where

[TABLE]

and

[TABLE]

Proof.

The proof is the same as [13, Corollary 3.2]. It suffices to observe that

[TABLE]

and to apply Theorem 3.1. ∎

Remark 3.3*.*

At this point we would like to note that if $T$ is an $\omega$-Calderón-Zygmund operator, with $\omega$ satisfying a Dini condition, since $M_{T}$ is of weak-type $(1,1)$ with

[TABLE]

(see [12], also for the notation) and we have that

[TABLE]

then from the preceding Corollary we recover a bilinear version of the sparse domination established in [17].

In order to use Corollary 3.2 to obtain Theorem 1.3, we need to borrow some results from [13]. Given an operator $T$, we define the maximal operator $M_{\lambda,T}$ by

[TABLE]

That operator was proved to be of weak type $(1,1)$ in [13] where the following estimate was established.

Theorem 3.4**.**

If $\Omega\in L^{\infty}(\mathbb{S}^{n-1})$, then

[TABLE]

Also in [13] the following result showing the relationship between the $L^{1}\to L^{1,\infty}$ norms of the operators $M_{\lambda,T}$ and ${\mathcal{M}}_{p,T}$ was provided.

Lemma 3.5**.**

Let $0<\gamma\leq 1$ and let $T$ be a sublinear operator. The following statements are equivalent:

1. (1)

there exists $C>0$ such that for all $p\geq 1$,

[TABLE] 2. (2)

there exists $C>0$ such that for all $0<\lambda<1$,

[TABLE]

At this point we are in the position to prove that Theorem 1.3 follows as a corollary from the previous results.

Proof of Theorem 1.3.

Theorem 3.4 combined with Lemma 3.5 with $\gamma=1$ yields

[TABLE]

with $p\geq 1$. Also, by [26], we have that

[TABLE]

Hence, by Corollary 3.2 with $q=r=1$ and $s=p>1$, there exist $3^{n}$ dyadic lattices $\mathcal{D}_{j}$ and $3^{n}$ sparse families ${\mathcal{S}}_{j}\subset{\mathcal{D}}_{j}$ such that

[TABLE]

∎

4. Proof of Theorem 1.1

We start providing a proof for (1.3). We follow some of the key ideas from [15, 16] (see also [22]). By duality, it suffices to prove (1.3) it suffices to show that

[TABLE]

We can calculate the norm by duality. Then,

[TABLE]

Let us define now a Rubio de Francia algorithm suited for this situation (see [6, Chapter IV.5] and [4] for plenty of applications of the Rubio de Francia algorithm). First we consider the operator

[TABLE]

and we observe that $S$ is bounded on $L^{p}(M_{r}w)$ with norm bounded by a dimensional multiple of $p^{\prime}$. Relying upon $S$ we define

[TABLE]

This operator has the following properties:

1. (a)

$0\leq h\leq R(h)$, 2. (b)

$\|Rh\|_{L^{p}(M_{r}w)}\leq 2\|h\|_{L^{p}(M_{r}w)}$, 3. (c)

$R(h)(M_{r}w)^{\frac{1}{p}}\in A_{1}$ with $[R(h)(M_{r}w)^{\frac{1}{p}}]_{A_{1}}\leq cp^{\prime}$. We also note that $[Rh]_{A_{\infty}}\leq[Rh]_{A_{3}}\leq c_{n}p^{\prime}$.

Using Theorem 1.3 and taking into account (a) we have that,

[TABLE]

and it suffices to obtain estimates for

[TABLE]

First we focus on $I$. Now we choose $r,s>1$ such that $sr=1+\frac{1}{\tau_{n}[Rh]_{A_{\infty}}}$. For instance, choosing $r=1+\frac{1}{2\tau_{n}[Rh]_{A_{\infty}}}$ we have that $s=2\frac{1+\tau_{n}[Rh]_{A_{\infty}}}{1+2\tau_{n}[Rh]_{A_{\infty}}}$ and also that $sr^{\prime}=2(1+\tau_{n}[Rh]_{A_{\infty}})\simeq[Rh]_{A_{\infty}}$. Now we recall that for every $0<t<\infty$ it was established in [7, Corollary 3.1.8] that

[TABLE]

For $t>1$ we have that $\left(t\Gamma(t)\right)^{\frac{1}{t}}e^{\frac{1}{t}+1}2^{n}\leq c_{n}t$. Taking into account the preceding estimate, the choices for $r$ and $s$, the reverse Hölder inequality (Lemma 2.2), and the property (c) above, we have that

[TABLE]

An application of Lemma 2.3 with $\Psi(t)=t$ yields

[TABLE]

From here

[TABLE]

Now by [15, Lemma 3.4] (see also [24, Lemma 2.9])

[TABLE]

Gathering all the preceding estimates we have that

[TABLE]

Now we turn our attention to $II$. Recalling that we have chosen $rs=1+\frac{1}{\tau_{n}[Rh]_{A_{\infty}}}$, taking into account the Reverse Hölder inequality and applying also (2.1) we have that

[TABLE]

Then a direct application of Lemma 2.3 with $\Psi(t)=t\log(e+t)$ yields the following estimate

[TABLE]

Arguing as in the estimate of $I$,

[TABLE]

Now [24, Proposition 3.2] gives

[TABLE]

Combining all the estimates we have that

[TABLE]

Finally, collecting the estimates we have obtained for $I$ and $II$, we arrive at the desired bound, namely

[TABLE]

We end the proof observing that the $A_{\infty}$ and the $A_{1}-A_{\infty}$ results are a direct consequence of the estimate we have just established and of the Reverse-Hölder inequality (see [15, 16, 8] for this kind of argument). ∎

5. Proof of Theorem 1.2

Let us consider first the case in which $T$ is a Calderón-Zygmund operator. Calculating the norm by duality we have that

[TABLE]

Now taking into account Remark 3.3 (or [17]) we have that

[TABLE]

so it suffices to provide estimates for

[TABLE]

First we work on $\mathcal{T}_{\mathcal{S}_{j},1,1}(b,f,gw)$. Following ideas in [19] we have that

[TABLE]

where $\overline{A}(t)=t^{\frac{p}{q-1}}$ and $\sigma=w^{1-p^{\prime}}$. Then, choosing $s<p^{\prime}$ and taking into account [11, Lemma 6], (2.1) and (2.2),

[TABLE]

where in the last step we use the Carleson embedding Theorem [8, Theorem 4.5] and the sparsity of $\mathcal{S}$.

Now we turn our attention to $\mathcal{T}^{*}_{\mathcal{S},1,1}(b,f,gw)$. We observe that for any $r>1$

[TABLE]

and from this point it suffices to follow the proof of [19, Theorem 3.1] to obtain the following estimate

[TABLE]

Combining the estimates for $\mathcal{T}_{\mathcal{S},1,1}(b,f,gw)$ and $\mathcal{T}^{*}_{\mathcal{S},1,1}(b,f,gw)$ we obtain (1.4) in the case of $T$ being a Calderón-Zygmund operator.

Let us consider now the remaining case. Assume that $T$ is a rough singular integral with $\Omega\in L^{\infty}(\mathbb{S}^{n-1})$. Calculating the norm by duality and denoting by $[b,T]^{t}$ the adjoint of $[b,T]$ we have that

[TABLE]

Taking into account that $[b,T]^{t}$ is also a commutator we can use the sparse domination obtained in Theorem 1.3 so we have that

[TABLE]

and then the question reduces to control both

[TABLE]

We begin observing that, arguing as before, choosing $1<s<p^{\prime}$

[TABLE]

On the other hand we have that for $s_{1}>1$ to be chosen later

[TABLE]

By Hölder inequality, we have that both $B_{1}$ and $B_{2}$ are controlled by

[TABLE]

We note that we can choose $us_{1}$ as close to $1$ as we want so let us rename $us_{1}=r$. Now denoting $\overline{B}(t)=t^{\frac{p}{r(q-1)}}$ and arguing as in [19, Theorem 3.1] we have that

[TABLE]

where in the last step we have used again the sparsity of $\mathcal{S}$ and the Carleson embedding theorem ([8, Theorem 4.5]). Collecting all the estimates

[TABLE]

This ends the proof of Theorem 1.2.∎

acknoledgements

The author would like to thank Kangwei Li for some insightful discussions during the elaboration of this paper.