A weighted endpoint weak-type estimate for multilinear Calder\'on-Zygmund operators

TL;DR

This paper presents two different proofs for a weighted weak-type estimate of multilinear Calderón-Zygmund operators, extending classical results to a multilinear and weighted setting.

Contribution

It provides novel proofs for a weighted weak-type estimate of multilinear Calderón-Zygmund operators, inspired by classical and modern techniques.

Findings

Two distinct proofs of the weighted weak-type estimate are provided.

The proofs adapt classical Calderón-Zygmund decomposition and modern harmonic analysis ideas.

The results extend classical weak-type estimates to multilinear and weighted contexts.

Abstract

Two proofs of a weighted weak-type $\left(1,\ldots,1;\frac{1}{m}\right)$ estimate for multilinear Calder\'on-Zygmund operators are given. The ideas are motivated by different proofs of the classical weak-type $(1,1)$ estimate for Calder\'on-Zygmund operators. One proof uses the Calder\'on-Zygmund decomposition, and the other proof is motivated by ideas of Nazarov, Treil, and Volberg.

Equations

$$\Vert Tf{\Vert }_{L^{1,\infty }({\mathbb{R}}^n)}:=\underset{t>0}{\sup }t|\{|Tf|>t\}|\lesssim \Vert f{\Vert }_{L^1({\mathbb{R}}^n)}.$$

$$\mu (B(x,2r))\lesssim \mu (B(x,r))$$

$$\mu (B(x,r))\lesssim r^n$$

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

$$\Vert T(fw)w^{-1}{\Vert }_{L^{1,\infty }(w)}\lesssim [w{]}_{A_p}\max \{p,\log (e+[w{]}_{A_p})\}\Vert f{\Vert }_{L^1(w)}.$$

$$\Vert T(f_1,\dots ,f_m){\Vert }_{L^{\frac{1}{m},\infty }({\mathbb{R}}^n)}\lesssim \prod _{i=1}^m\Vert f_i{\Vert }_{L^1({\mathbb{R}}^n)}.$$

$$[\vec{w}{]}_{A_{\vec{P}}}:=\underset{Q}{\sup }{\left(\frac{1}{|Q|}{\int }_Q{v}_{\vec{w}}\right)}^{\frac{1}{p}}\prod _{i=1}^m{\left(\frac{1}{|Q|}{\int }_Q{w}_i^{1-p_i^{\prime }}\right)}^{\frac{1}{p_i^{\prime }}}<\infty ;$$

$${\Vert T\left(f_1{w}_1{v}_{\vec{w}}^{\frac{1-m}{m}},\dots ,f_m{w}_m{v}_{\vec{w}}^{\frac{1-m}{m}}\right)v_{\vec{w}}^{-1}\Vert }_{L^{\frac{1}{m},\infty }(v_{\vec{w}})}\lesssim [v_{\vec{w}}{]}_{A_1}^{2m^2+2m-2}\prod _{i=1}^m\Vert f_i{\Vert }_{L^1(w_i)}.$$

$$|K(x,y_1,\dots ,y_m)|\lesssim \frac{1}{{\left({\sum }_{i=1}^m|x-y_i|\right)}^{nm}}$$

$$|K(x,y_1,\dots ,y_m)-K(x^{\prime },y_1,\dots ,y_m)|\lesssim \frac{|x-x^{\prime }{|}^{\delta }}{({\sum }_{i=1}^m|x-y_i|{)}^{nm+\delta }}$$

$$|K(x,y_1,\dots ,y_j,\dots ,y_m)-K(x,y_1,\dots ,y_j^{\prime },\dots ,y_m)|\lesssim \frac{|y_j-y_j^{\prime }{|}^{\delta }}{({\sum }_{i=1}^m|x-y_i|{)}^{nm+\delta }}$$

$$T(f_1,\dots ,f_m)(x)={\int }_{({\mathbb{R}}^n{)}^m}K(x,y_1,\dots ,y_m)f_1(y_1)\cdots f_m(y_m)d{y}_1\cdots d{y}_m$$

$$\Vert T(f_1,\dots ,f_m){\Vert }_{L^p({\mathbb{R}}^n)}\lesssim \prod _{i=1}^m\Vert f_i{\Vert }_{L^{p_i}({\mathbb{R}}^n)}.$$

$$[v_{\vec{w}}{]}_{A_{mp}}\le [\vec{w}{]}_{A_{\vec{P}}}^p,$$

$${\left[w_i^{1-p_i^{\prime }}\right]}_{A_{m{p}_i^{\prime }}}\le [\vec{w}{]}_{A_{\vec{P}}}^{p_i^{\prime }}\text{when }{p}_i>1,$$

$${\left[w_i^{\frac{1}{m}}\right]}_{A_1}\le [\vec{w}{]}_{A_{\vec{P}}}^{\frac{1}{m}}\text{when }{p}_i=1,\text{and}$$

$$[\vec{w}{]}_{A_{\vec{P}}}\le [v_{\vec{w}}{]}_{A_{mp}}^{\frac{1}{p}}\prod _{i=1}^m{\left[w_i^{1-p_i^{\prime }}\right]}_{A_{m{p}_i^{\prime }}}^{\frac{1}{p_i^{\prime }}},$$

$$\frac{1}{|Q|}{\int }_{Q}w(y{)}^{\gamma }dy\le {\left(\frac{1}{|Q|}{\int }_{Q}w(y)dy\right)}^{\gamma }{\left(\frac{1}{|Q|}{\int }_Q{1}^{{\left(\frac{1}{\gamma }\right)}^{\prime }}dy\right)}^{\frac{1}{{\left(\frac{1}{\gamma }\right)}^{\prime }}}\le [w{]}_{A_1}^{\gamma }\left(\underset{Q}{\inf }{w}^{\gamma }\right).$$

$$M_{w}f(x):=\underset{Q\ni x}{\sup }\frac{1}{w(Q)}{\int }_Q|f(y)|w(y)dy.$$

$$\Vert M_{w}f{\Vert }_{L^{1,\infty }(w)}\lesssim \Vert f{\Vert }_{L^1(w)}.$$

$$(|f|\ast K)(x)\le \Vert K{\Vert }_{L^1({\mathbb{R}}^n)}M(f)(x),$$

$$\sum _{j_1,\dots ,j_l=1}^{\infty }\prod _{i=1}^l{w}^{\frac{1}{m}}(Q_{i,j_i}){\int }_{{\mathbb{R}}^{n(m-l)}}\prod _{i=l+1}^{m}w(y_i{)}^{\frac{1}{m}}$$

$$\times \underset{\begin{array}{c}(y_1,\dots ,y_l)\\ \in {\prod }_{i=1}^l{Q}_{i,j_i}\end{array}}{\sup }{\int }_{{\mathbb{R}}^n\setminus \left({\bigcup }_{i=1}^l{\Omega }_i^{\ast }\right)}|K(x,\text{vv}{y}_{1,m})-K(x,\text{vv}{c}_{(1,j_1),(l,j_l)},\text{vv}{y}_{l+1,m})|dxd\text{vv}{y}_{l+1,m}$$

$$\lesssim [w{]}_{A_1}^{\frac{2m-2}{m}}\sum _{i=1}^{l}w\left({\Omega }_i\right)$$

$$\underset{\begin{array}{c}(y_1,\dots ,y_l)\\ \in {\prod }_{i=1}^l{Q}_{i,j_i}\end{array}}{\sup }$$

$$\lesssim \underset{\begin{array}{c}(y_1,\dots ,y_l)\\ \in {\prod }_{i=1}^l{Q}_{i,j_i}\end{array}}{\sup }{\int }_{{\mathbb{R}}^n\setminus \left({\bigcup }_{i=1}^l{\Omega }_i^{\ast }\right)}\frac{{\sum }_{i=1}^l|y_i-c_{i,j_i}{|}^{\delta }}{({\sum }_{i=1}^m|x-y_i|{)}^{nm+\delta }}dx$$

$$\lesssim \underset{\begin{array}{c}(y_1,\dots ,y_l)\\ \in {\prod }_{i=1}^l{Q}_{i,j_i}\end{array}}{\sup }{\int }_{{\mathbb{R}}^n\setminus \left({\bigcup }_{i=1}^l{\Omega }_i^{\ast }\right)}\frac{{\sum }_{i=1}^l{r}_{i,j_i}^{\delta }}{({\sum }_{i=1}^m|x-y_i|{)}^{nm+\delta }}dx.$$

$$\underset{\begin{array}{c}(y_1,\dots ,y_l)\\ \in {\prod }_{i=1}^l{Q}_{i,j_i}\end{array}}{\sup }$$

$$={\int }_{{\mathbb{R}}^n\setminus \left({\bigcup }_{i=1}^l{\Omega }_i^{\ast }\right)}\frac{{\sum }_{i=1}^l{r}_{i,j_i}^{\delta }}{({\sum }_{i=1}^l|x-y_i^{\ast }|+{\sum }_{i=l+1}^m|x-y_i|{)}^{nm+\delta }}dx,$$

$$\underset{\begin{array}{c}(y_1,\dots ,y_l)\\ \in {\prod }_{i=1}^l{Q}_{i,j_i}\end{array}}{\inf }$$

$$={\int }_{{\mathbb{R}}^n\setminus \left({\bigcup }_{i=1}^l{\Omega }_i^{\ast }\right)}\frac{{\sum }_{i=1}^l{r}_{i,j_i}^{\delta }}{({\sum }_{i=1}^l|x-y_{i_{\ast }}|+{\sum }_{i=l+1}^m|x-y_i|{)}^{nm+\delta }}dx.$$

$$\frac{{\sum }_{i=1}^l|x-y_{i_{\ast }}|+{\sum }_{i=l+1}^m|x-y_i|}{{\sum }_{i=1}^l|x-y_i^{\ast }|+{\sum }_{i=l+1}^m|x-y_i|}\le \frac{{\sum }_{i=1}^l\sqrt{n}{r}_{i,j_i}}{{\sum }_{i=1}^l|x-y_i^{\ast }|+{\sum }_{i=l+1}^m|x-y_i|}+1\le 2\sqrt{n}+1.$$

$$\underset{\begin{array}{c}(y_1,\dots ,y_l)\\ \in {\prod }_{i=1}^l{Q}_{i,j_i}\end{array}}{\sup }{\int }_{{\mathbb{R}}^n\setminus \left({\bigcup }_{i=1}^l{\Omega }_i^{\ast }\right)}|K(x,\text{vv}{y}_{1,m})-K(x,\text{vv}{c}_{(1,j_1),(l,j_l)},\text{vv}{y}_{l+1,m})|dx$$

Full text

Keywords: singular integrals; multilinear operators; weak-type estimates; weighted estimates.

1. Introduction

The following weak-type $(1,1)$ estimate is essential to the theory of singular integrals.

• Theorem1.

Let $T$ be a Calderón-Zygmund operator. If $f\in L^{1}(\mathbb{R}^{n})$, then

[TABLE]

The original proof of Theorem 1 uses the Calderón-Zygmund decomposition of $f\in L^{1}(\mathbb{R}^{n})$, see [Grafakos1, Grafakos2, Stein]. The Calderón-Zygmund decomposition method has since become standard for proving endpoint weak-type results for related operators. The Calderón-Zygmund decomposition requires the underlying measure space to possess the doubling property: a Borel measure $\mu$ has the doubling property if

[TABLE]

for all $r>0$ and all $x$ in the space.

Extending the theory to more general settings, Nazarov, Treil, and Volberg gave a new proof of the weak-type $(1,1)$ estimate for Calderón-Zygmund operators on nonhomogeneous spaces in [NTV1998]. A nonhomogeneous space is a metric measure space where the underlying measure $\mu$ fails to possess the doubling property, but instead satisfies the polynomial growth condition

[TABLE]

for all $r>0$ and all $x$ in the space. Since Lebesgue measure on $\mathbb{R}^{n}$ satisfies the polynomial growth condition, the proof in [NTV1998] immediately gives a different proof of Theorem 1.

The Nazarov-Treil-Volberg technique has been studied further. The technique was extended to handle measures with the upper doubling growth condition in [HLYY2012]. It was shown in [S2018] that, if one again assumes the doubling condition, crucial steps in the Nazarov-Treil-Volberg proof of Theorem 1 may be bypassed. Also in [S2018], an adaptation of the Nazarov-Treil-Volberg argument was used to prove a weighted weak-type $(1,1)$ inequality; and in [SW2019], an adaptation of the argument was used to prove the weak-type $\left(1,\ldots,1;\frac{1}{m}\right)$ estimate for multilinear Calderón-Zygmund operators. The linear weighted estimate was previously proved in [OPR2016] using the Calderón-Zygmund decomposition, and the multilinear estimate was first proved in [GT2002], also using the Calderón-Zygmund decomposition.

In this paper, we combine both of the previously mentioned settings by proving a weighted weak-type $\left(1,\ldots,1;\frac{1}{m}\right)$ estimate for multilinear Calderón-Zygmund operators. Two proofs are given – one uses the Calderón-Zygmund decomposition and the other the Nazarov-Treil-Volberg method. See [S2018] for a comparison between the Calderón-Zygmund decomposition and Nazarov-Treil-Volberg proofs.

We describe the motivating results. For $1\leq p<\infty$, we say that $w$ is an $A_{p}$ weight if $w$ is locally integrable, positive almost everywhere, and satisfies the $A_{p}$ condition

[TABLE]

when $p=1$, the quantity $\left(\frac{1}{|Q|}\int_{Q}w(x)^{1-p^{\prime}}dx\right)^{p-1}$ is interpreted as $\left(\inf_{Q}w\right)^{-1}$.

• Theorem2.

Let $T$ be a Calderón-Zygmund operator. If $1\leq p<\infty$, $w\in A_{p}$, and $f\in L^{1}(w)$, then

[TABLE]

Theorem 2 was proved by Ombrosi, Pérez, and Recchi in [OPR2016] using the Calderón-Zygmund decomposition and by the author in [S2018] using the Nazarov-Treil-Volberg argument. See [CUMP2005, CRR2018, LOP2017, OP2016] for related mixed weak-type inequalities.

Much of the Calderón-Zygmund theory was extended to the multilinear setting by Grafakos and Torres in [GT2002]. In particular, they proved the following endpoint weak-type estimate.

• Theorem3.

Let $T$ be a multilinear Calderón-Zygmund operator. If $f_{1},\ldots,f_{m}\in L^{1}(\mathbb{R}^{n})$, then

[TABLE]

As in the classical theory, their proof uses the Calderón-Zygmund decomposition. Other proofs, also using the Calderón-Zygmund decomposition, were later given in the bilinear setting by Pérez and Torres in [PT2014], and by Maldonado and Naibo in [MN2009]. Another proof was given by the author and Wick in [SW2019] using a variation of the Nazarov-Treil-Volberg method.

Connecting the weighted theory and the multilinear theory, Lerner, Ombrosi, Pérez, Torres, and Trujillo-Gonzáles introduced the classes of multilinear weights in [LOPTTG2009]. We use the following notation for multilinear $A_{\vec{P}}$ weights: $1\leq p_{1},\ldots,p_{m}\leq\infty$, $\frac{1}{m}\leq p\leq\infty$ satisfies $\frac{1}{p}=\frac{1}{p_{1}}+\cdots+\frac{1}{p_{m}}$, $\vec{P}=(p_{1},\ldots,p_{m})$, $\vec{w}=(w_{1},\ldots,w_{m})$, and $v_{\vec{w}}=\prod_{i=1}^{m}w_{i}^{\frac{p}{p_{i}}}$. We say $\vec{w}\in A_{\vec{P}}$ if

[TABLE]

when $p_{i}=1$, the factor $\left(\frac{1}{|Q|}\int_{Q}w_{i}^{1-p_{i}^{\prime}}\right)^{\frac{1}{p_{i}^{\prime}}}$ is understood as $(\inf_{Q}w_{i})^{-1}$. Note that the quantities $[\vec{w}]_{A_{\vec{P}}}^{p}$ and $[w]_{A_{p}}$ coincide when $m=1$.

We give two proofs of the following theorem.

• Theorem4.

Let $T$ be a multilinear Calderón-Zygmund operator. If $\vec{w}\in A_{(1,\ldots,1)}$ and $f\in L^{1}(w_{i})$ for all $i\in\{1,\ldots,m\}$, then

[TABLE]

The first proof uses the Calderón-Zygmund decomposition and is a weighted version of the proof in [PT2014]; the second proof uses the Nazarov-Treil-Volberg method and is a weighted version of the proof in [SW2019]. See [LOP2019] for a related result that is deduced using multilinear extrapolation.

• Remark1.

The second proof is actually a weighted version of a simplification of the proof in [SW2019]. Referring to the contents of [SW2019], the current proof shows that the sets $E_{i,j}$ can be constructed as cubes, that the regularity of Lemma 1 is only required for collections of pairwise disjoint cubes, and that Theorem 2 is not necessary for the weak-type estimate.

Section 2 describes the definitions and preliminary results, including a weighted version of the regularity condition first described for bilinear kernels in [PT2014]. Section 3 contains two proofs of the main result, Theorem 4.

I would like to thank Brett Wick for his contributions to this article.

2. Preliminaries

We use the notation $A\lesssim B$ if there exists $C>0$, possibly depending on $n$, $T$, or $m$, such that $A\leq CB$. The Lebesgue measure of $A\subseteq\mathbb{R}^{n}$ is denoted by $|A|$, while for a weight $w$, $\int_{A}w(x)dx$ is denoted by $w(A)$. The cube with center $x\in\mathbb{R}^{n}$ and side length $r$ is denoted by $Q(x,r)$. If $Q$ is a cube, then $rQ$ denotes the cube with the same center as $Q$ and side length equal to $r$ times the side length of $Q$.

Let $m$ be a positive integer. We say $K:\mathbb{R}^{n(m+1)}\setminus\{x=y_{i}\text{ for all }i\in\{1,\ldots,m\}\}\rightarrow\mathbb{C}$ is a $m$-multilinear Calderón-Zygmund kernel if there exists $\delta>0$ such that the following conditions hold:

1. (1)

(size)

[TABLE]

for all $x,y_{1},\ldots,y_{m}\in\mathbb{R}^{n}$ with $x\neq y_{j}$ for some $j$, 2. (2)

(smoothness)

[TABLE]

whenever $\displaystyle|x-x^{\prime}|\leq\frac{1}{2}\max_{1\leq i\leq m}|x-y_{i}|$, and

[TABLE]

for each $j\in\{1,\ldots,m\}$ whenever $\displaystyle|y_{j}-y_{j}^{\prime}|\leq\frac{1}{2}\max_{1\leq i\leq m}|x-y_{i}|$.

Let $\mathscr{S}(\mathbb{R}^{n})$ denote the space of Schwartz functions on $\mathbb{R}^{n}$ and $\mathscr{S}^{\prime}(\mathbb{R}^{n})$ the space of tempered distributions on $\mathbb{R}^{n}$. We say that a $m$-multilinear operator $T:\mathscr{S}(\mathbb{R}^{n})\times\cdots\times\mathscr{S}(\mathbb{R}^{n})\rightarrow\mathscr{S}^{\prime}(\mathbb{R}^{n})$ is a multilinear Calderón-Zygmund operator associated to a kernel $K$ if $K$ is a $m$-multilinear Calderón-Zygmund kernel, if $T$ extends to a bounded operator from $L^{q_{1}}(\mathbb{R}^{n})\times\cdots\times L^{q_{m}}(\mathbb{R}^{n})$ to $L^{q}(\mathbb{R}^{n})$ for some $\frac{1}{m}<q,q_{1},\ldots,q_{m}<\infty$ satisfying $\frac{1}{q}=\frac{1}{q_{1}}+\cdots+\frac{1}{q_{m}}$, and if

[TABLE]

for compactly supported integrable functions $f_{i}$ and almost every $x\in\mathbb{R}^{n}\setminus\bigcap_{i=1}^{m}\text{supp }f_{i}$. In all instances that follow, $T$ will represent an $m$-multilinear Calderón-Zygmund operator.

The following theorem was proved by Grafakos and Torres in [GT2002].

• Theorem5.

If $1<p_{1},\ldots,p_{m}<\infty$, $p$ satisfies $\frac{1}{p}=\frac{1}{p_{1}}+\cdots+\frac{1}{p_{m}}$, and $f_{i}\in L^{p_{i}}(\mathbb{R}^{n})$ for $i\in\{1,\ldots,m\}$, then

[TABLE]

We will use Theorem 5 in the proofs of Theorem 4 when $p_{1}=\cdots=p_{m}=m$ and $p=1$.

A characterization of the multilinear $A_{\vec{P}}$ condition in terms of linear $A_{q}$ conditions was established in [LOPTTG2009].

• Theorem6.

The following conditions are equivalent:

1. (1)

$\vec{w}\in A_{\vec{P}}$. 2. (2)

$v_{\vec{w}}\in A_{mp}$ and $w_{i}^{1-p_{i}^{\prime}}\in A_{mp_{i}^{\prime}}$ for all $i\in\{1,\ldots,m\}$. When $p_{i}=1$, the condition $w_{i}^{1-p_{i}^{\prime}}\in A_{mp_{i}^{\prime}}$ is understood as $w_{i}^{\frac{1}{m}}\in A_{1}$.

• Remark2.

Tracking down the estimates in the proof of Theorem 6 gives the relationships

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $\left[w_{i}^{1-p_{i}^{\prime}}\right]_{A_{mp_{i}^{\prime}}}^{\frac{1}{p_{i}^{\prime}}}$ is interpreted as $\left[w_{i}^{\frac{1}{m}}\right]_{A_{1}}^{m}$ when $p_{i}=1$.

We will use the following property of $A_{1}$ weights.

• Lemma1.

If $w\in A_{1}$ and $0\leq\gamma\leq 1$, then $w^{\gamma}\in A_{1}$ with $[w^{\gamma}]_{A_{1}}\leq[w]_{A_{1}}^{\gamma}$.

Proof.

The cases when $\gamma=0$ and $\gamma=1$ are clear. If $0<\gamma<1$, then $\frac{1}{\gamma}>1$. Applying Hölder’s inequality and the $A_{1}$ condition of $w$ gives

[TABLE]

∎

We will use the following maximal function in the second proof of the main theorem in Section 3. Given a weight $w$, define the uncentered maximal function associated to $w$ by

[TABLE]

• Lemma2.

If $w\in A_{p}$ and $f\in L^{1}(w)$, then

[TABLE]

The operator norm of $M_{w}$ does not depend on the $A_{p}$ characteristic of $w$.

The following lemma is well-known and proved in [Grafakos1, Grafakos2, Stein].

• Lemma3.

Let $k:[0,\infty)\rightarrow[0,\infty)$ be decreasing and continuous except at a finite number of points. If $K(x)=k(|x|)$ is in $L^{1}(\mathbb{R}^{n})$, then for all $f\in L_{\text{loc}}^{1}(\mathbb{R}^{n})$,

[TABLE]

where $M$ denotes the classical Hardy-Littlewood maximal operator.

The following lemma is a weighted version of the multilinear geometric Hörmander condition first introduced in the bilinear setting in [PT2014] and generalized in the multilinear setting in [SW2019]. We use the following vector notations $\vv{y}_{i,k}=(y_{i},y_{i+1},\ldots,y_{k})$ and $\vv{c}_{(i,j_{i}),(k,j_{k})}=(c_{i,j_{i}},c_{i+1,j_{i+1}},\ldots,c_{k,j_{k}})$.

• Lemma4.

If $w\in A_{1}$, $l\in\{1,\ldots,m\},$ and each of $\mathcal{Q}_{1},\ldots,\mathcal{Q}_{l}$ consists of pairwise disjoint cubes $\mathcal{Q}_{i}=\{Q_{i,1},Q_{i,2},\ldots\}$ where $Q_{i,j}=Q(c_{i,j},r_{i,j})$, then

[TABLE]

where $\Omega_{i}:=\bigcup_{j=1}^{\infty}Q_{i,j}$ and $\Omega_{i}^{*}:=\bigcup_{j=1}^{\infty}2\sqrt{n}Q_{i,j}$.

It is not important that the indices of the $\mathcal{Q}_{i}$ range from $1$ to $l$ – a symmetric proof yields the lemma whenever the set of indices is a nonempty subset of $\{1,\ldots,m\}$.

Proof.

For $i=1,\ldots,l$, fix $Q_{i,j_{i}}\in\mathcal{Q}_{i}$. Use the smoothness condition of $K$ to see

[TABLE]

Since for fixed $y_{i}\in\overline{Q_{i,j_{i}}}$, $i=l+1,\ldots,m$, the function

$\displaystyle\int_{\mathbb{R}^{n}\setminus\left(\bigcup_{i=1}^{l}\Omega_{i}^{*}\right)}\frac{\sum_{i=1}^{l}r_{i,j_{i}}^{\delta}}{(\sum_{i=1}^{m}|x-y_{i}|)^{nm+\delta}}dx$ is continuous in the variables $y_{i}\in\overline{Q_{i,j_{i}}}$, $i=1,\ldots,l$, we may write

[TABLE]

and

[TABLE]

Note that for $x\in\mathbb{R}^{n}\setminus\left(\bigcup_{i=1}^{l}\Omega_{i}^{*}\right)$, $|x-y_{i_{*}}|\leq\sqrt{n}r_{i,j_{i}}+|x-y_{i}^{*}|$ and $|x-y_{i}^{*}|\geq\frac{1}{2}r_{i,j_{i}}$, so

[TABLE]

Then

[TABLE]

Using the previous estimate, Fubini’s theorem, and trivial estimates, we get the bound

[TABLE]

We will control the term of the summation above with $k=1$; the other terms are handled similarly. Using trivial estimates, Fubini’s theorem, and the fact that the $Q_{i,j_{i}}$ have disjoint interiors, we obtain

[TABLE]

Repeatedly use Lemma 3 first with $K=\frac{1}{(\sum_{i=1}^{m-1}|x-y_{i}|+|\cdot|)^{nm}}$, second with

$K=\frac{1}{(\sum_{i=1}^{m-2}|x-y_{i}|+|\cdot|)^{n(m-1)}}$, etcetera, and the fact that $\displaystyle M\left(w^{\frac{1}{m}}\right)(x)\leq\left[w^{\frac{1}{m}}\right]_{A_{1}}w(x)^{\frac{1}{m}}$ (which is true by Lemma 1) to control the above expression by

[TABLE]

Use Lemma 3 with $K=\frac{r_{1,j_{1}}^{\delta}}{|\cdot|^{n+\delta}}\mathbbm{1}_{\left\{|\cdot|>\frac{1}{2}r_{1,j_{1}}\right\}}$, the fact that $\|K\|_{L^{1}(\mathbb{R}^{n})}\lesssim 1$, the fact that $M\left(w^{\frac{m-1}{m}}\right)(y_{1})\leq\left[w^{\frac{m-1}{m}}\right]_{A_{1}}w^{\frac{m-1}{m}}(y_{1})$ (which is true by Lemma 1), the estimates in Remark 2, and the pairwise disjointness of $Q_{1,j_{1}}$ to further estimate the previous expression by a constant multiplied by

[TABLE]

Similarly, for $k=2,\ldots,l$,

[TABLE]

This completes the proof. ∎

3. Main Results

We give two proofs of the main result. The first proof uses the Calderón-Zygmund decomposition and the second proof uses the Nazarov-Treil-Volberg method. Recall that, for a measure $\mu$, the quasinorm $\|\cdot\|_{L^{p,\infty}(\mu)}$ is given by $\|f\|_{L^{p,\infty}(\mu)}^{p}:=\sup_{t>0}t^{p}\mu(\{|f|>t\})$.

• Theorem4.

If $\vec{w}\in A_{(1,\ldots,1)}$ and $f\in L^{1}(w_{i})$ for all $i\in\{1,\ldots,m\}$, then

[TABLE]

Proof 1.

Let $t>0$ be given. We will show that

[TABLE]

Without loss of generality, assume that $f_{1},\ldots,f_{m}$ are continuous functions with compact support and that $\|f_{1}\|_{L^{1}(w_{1})}=\cdots=\|f_{m}\|_{L^{1}(w_{m})}=1$. Apply the Calderón-Zygmund decomposition to $\displaystyle f_{i}w_{i}v_{\vec{w}}^{-1}$ at height $t^{\frac{1}{m}}$ with respect to $v_{\vec{w}}dx$ to write

[TABLE]

where the following properties hold:

1. (1)

$\|g_{i}\|_{L^{\infty}(\mathbb{R}^{n})}\lesssim[v_{\vec{w}}]_{A_{1}}t^{\frac{1}{m}}$ and $\|g_{i}\|_{L^{1}(v_{\vec{w}})}\leq\|f_{i}\|_{L^{1}(w_{i})}$, 2. (2)

the $b_{i,j}$ are supported on pairwise disjoint cubes $Q_{i,j}$ satisfying

[TABLE] 3. (3)

$\int_{Q_{i,j}}b_{i,j}(x)v_{\vec{w}}(x)dx=0$, 4. (4)

$\|b_{i,j}\|_{L^{1}(v_{\vec{w}})}\lesssim[v_{\vec{w}}]_{A_{1}}t^{\frac{1}{m}}v_{\vec{w}}(Q_{i,j})$, and 5. (5)

$\|b_{i}\|_{L^{1}(v_{\vec{w}})}\lesssim\|f_{i}\|_{L^{1}(w_{i})}$.

Set

[TABLE]

where each $S_{s}=\left\{\left|T\left(h_{1}v_{\vec{w}}^{\frac{1}{m}},\ldots,h_{m}v_{\vec{w}}^{\frac{1}{m}}\right)\right|v_{\vec{w}}^{-1}>\frac{t}{2^{m}}\right\}$ with $h_{i}\in\left\{b_{i},g_{i}\right\}$ and all the sets $S_{s}$ are distinct. Since

[TABLE]

it suffices to control each $v_{\vec{w}}(S_{s})$.

Use Chebyshev’s inequality, the boundedness of $T$ from $(L^{m}(\mathbb{R}^{n}))^{m}$ to $L^{1}(\mathbb{R}^{n})$ (which holds by Theorem 5), and property (1) to see

[TABLE]

Consider the set $S_{s}$ for a fixed $2\leq s\leq 2^{m}$. Suppose that there are $l$ functions of the form $b_{i}$ and $m-l$ functions of the form $g_{i}$ appearing as entries in the $T\left(h_{1}v_{\vec{w}}^{\frac{1}{m}},\ldots,h_{m}v_{\vec{w}}^{\frac{1}{m}}\right)$ involved in the definition of $S_{s}$. By symmetry, we may assume that the $b_{i}$ are in the first $l$ entries and the $g_{i}$ are in the remaining $m-l$ entries. Let $Q_{i,j}^{*}:=2\sqrt{n}Q_{i,j}$, $\Omega_{i}^{*}:=\bigcup_{j=1}^{\infty}Q_{i,j}^{*}$, and $\Omega^{*}:=\bigcup_{i=1}^{m}\Omega_{i}^{*}$.

By the doubling property of $v_{\vec{w}}dx$, the fact that the $Q_{i,j}$ are pairwise disjoint, and property (3), we have

[TABLE]

Therefore

[TABLE]

Now use Chebyshev’s inequality, the fact that $\int_{Q_{i,j_{i}}}b_{i,j_{i}}(x)v_{\vec{w}}(x)dx=0$, and trivial bounds to estimate

[TABLE]

[TABLE]

Apply property (4), property (1), the fact that $\left(\sup_{Q_{i,j_{i}}}v_{\vec{w}}^{\frac{1-m}{m}}\right)=\left(\inf_{Q_{i,j_{i}}}v_{\vec{w}}\right)^{\frac{1-m}{m}}$ , the $A_{1}$ condition of $v_{\vec{w}}$, and trivial estimates to bound the above expression by a constant times

[TABLE]

[TABLE]

By Lemma 4 and property (2), the above expression is controlled by a constant times

[TABLE]

Therefore

[TABLE]

Putting the previous estimates together gives

[TABLE]

[TABLE]

∎

Proof 2.

Let $t>0$ be given. We will show that

[TABLE]

Assume that $f_{1},\ldots,f_{m}$ are nonnegative, continuous functions with compact support and that $\|f_{1}\|_{L^{1}(w_{1})}=\cdots=\|f_{m}\|_{L^{1}(w_{m})}=1$. Assume that $v_{\vec{w}}(\mathbb{R}^{n})>t^{-\frac{1}{m}}$ (otherwise there is nothing to prove). Set

[TABLE]

Apply a Whitney decomposition to write

[TABLE]

a disjoint union of dyadic cubes where

[TABLE]

Put

[TABLE]

Then

[TABLE]

where

1. (1)

$\|g_{i}\|_{L^{\infty}(\mathbb{R}^{n})}\leq t^{\frac{1}{m}}$ and $\|g_{i}\|_{L^{1}(v_{\vec{w}})}\leq\|f_{i}\|_{L^{1}(w_{i})}$, 2. (2)

the $b_{i,j}$ are supported on pairwise disjoint cubes $Q_{i,j}$ satisfying

[TABLE] 3. (3)

$\|b_{i,j}\|_{L^{1}(v_{\vec{w}})}\leq(17\sqrt{n})^{n}[v_{\vec{w}}]_{A_{1}}t^{\frac{1}{m}}v_{\vec{w}}(Q_{i,j})$, and 4. (4)

$\|b_{i}\|_{L^{1}(v_{\vec{w}})}\leq\|f_{i}\|_{L^{1}(w_{i})}$.

To justify the above properties, since

[TABLE]

for almost every $x\in\mathbb{R}^{n}\setminus\Omega_{i}$, it is true that $\|g_{i}\|_{L^{\infty}(\mathbb{R}^{n})}\leq t^{\frac{1}{m}}$. Noticing that $g_{i}$ is a restriction of $f_{i}w_{i}v_{\vec{w}}^{-1}$, we have $\|g_{i}\|_{L^{1}(v_{\vec{w}})}\leq\|f_{i}\|_{L^{1}(w_{i})}$, so property (1) holds. We obtain property (2) using Lemma 2 as follows

[TABLE]

Addressing (3), for a fixed $Q_{i,j}$, let $Q_{i,j}^{*}=17\sqrt{n}Q_{i,j}$. Then $Q_{i,j}^{*}\cap(\mathbb{R}^{n}\setminus G_{i})\neq\emptyset$, so there is a point $x\in Q_{i,j}^{*}$ such that $M_{v_{\vec{w}}}(f_{i}w_{i}v_{\vec{w}}^{-1})(x)\leq t^{\frac{1}{m}}$. In particular, $\int_{Q_{i,j}^{*}}f_{i}(y)w_{i}(y)dy\leq t^{\frac{1}{m}}v_{\vec{w}}(Q_{i,j}^{*})$. Since $v_{\vec{w}}(Q_{i,j}^{*})\leq(17\sqrt{n})^{n}[v_{\vec{w}}]_{A_{1}}v_{\vec{w}}(Q_{i,j})$, we have

[TABLE]

[TABLE]

proving (3). Property (4) follows since $b_{i}$ is a restriction of $f_{i}w_{i}v_{\vec{w}}^{-1}$.

Set

[TABLE]

where each $S_{s}=\left\{\left|T\left(h_{1}v_{\vec{w}}^{\frac{1}{m}},\ldots,h_{m}v_{\vec{w}}^{\frac{1}{m}}\right)\right|v_{\vec{w}}^{-1}>\frac{t}{2^{m}}\right\}$ with $h_{i}\in\left\{b_{i},g_{i}\right\}$ and all the sets $S_{s}$ are distinct. Since

[TABLE]

it suffices to control each $v_{\vec{w}}(S_{s})$.

Use Chebyshev’s inequality, the boundedness of $T$ from $(L^{m}(\mathbb{R}^{n}))^{m}$ to $L^{1}(\mathbb{R}^{n})$ (which holds by Theorem 5), and property (1) to see

[TABLE]

Consider the set $S_{s}$ for a fixed $2\leq s\leq 2^{m}$. Suppose that there are $l$ functions of the form $b_{i}$ and $m-l$ functions of the form $g_{i}$ appearing as entries in the $T\left(h_{1}w_{1}^{\frac{1}{m}},\ldots,h_{m}w_{m}^{\frac{1}{m}}\right)$ involved in the definition of $S_{s}$. By symmetry, we may assume that the $b_{i}$ are in the first $l$ entries and the $g_{i}$ are in the remaining $m-l$ entries.

Let $c_{i,j}$ denote the center of $Q_{i,j}$ and let $a_{i,j}=\|b_{i,j}\|_{L^{1}(v_{\vec{w}})}(17\sqrt{n})^{-n}[v_{\vec{w}}]_{A_{1}}^{-1}$. Set

[TABLE]

where $r_{i,j}>0$ is chosen so that $v_{\vec{w}}(E_{i,j})=a_{i,j}t^{-\frac{1}{m}}$. Note that such $E_{i,j}$ exist since the function $r\mapsto v_{\vec{w}}(Q(x,r))$ increases to $v_{\vec{w}}(\mathbb{R}^{n})>t^{-\frac{1}{m}}$ as $r\rightarrow\infty$, approaches [math] as $r\rightarrow 0$, and is continuous from the right for almost every $x\in\mathbb{R}^{n}$. Using property (3), we see

[TABLE]

Since $E_{i,j}$ is a cube with the same center as $Q_{i,j}$ and since $v_{\vec{w}}(E_{i,j})\leq v_{\vec{w}}(Q_{i,j}),$ it is true that $E_{i,j}\subseteq Q_{i,j}$. Define

[TABLE]

For $k=0,\ldots,l$, define

[TABLE]

Then, by adding and subtracting $\sigma_{k}$ for $1\leq k\leq l$, we have

[TABLE]

Using property (2), we have

[TABLE]

therefore

[TABLE]

where

[TABLE]

We will first estimate $v_{\vec{w}}(P_{k})$ for $k\in\{1,\ldots,l\}$. Notice that

[TABLE]

[TABLE]

Begin by using Chebyshev’s inequality, the fact that

[TABLE]

and trivial estimates to see

[TABLE]

Next use the fact that $v_{\vec{w}}(E_{i,j_{i}})\lesssim t^{-\frac{1}{m}}\|b_{i,j_{i}}\|_{L^{1}(v_{\vec{w}})}$, Fubini’s theorem, trivial estimates, the fact that

[TABLE]

and property (1) to control

[TABLE]

[TABLE]

Use property (3), the fact that $\left(\sup_{Q_{i,j_{i}}}v_{\vec{w}}^{\frac{1-m}{m}}\right)=\left(\inf_{Q_{i,j_{i}}}v_{\vec{w}}\right)^{\frac{1-m}{m}}$, the $A_{1}$ condition of $v_{\vec{w}}$, and trivial estimates to estimate $v_{\vec{w}}(P_{k})$

[TABLE]

[TABLE]

Use Lemma 4 and property (2) to finish the estimate

[TABLE]

The control of $v_{\vec{w}}(P)$ follows from Chebyshev’s inequality, construction of the sets $E_{i}$, property (1), and property (4):

[TABLE]

Put the estimates of $v_{\vec{w}}(P_{k})$ and $v_{\vec{w}}(P)$ together to get

[TABLE]

Finally, use the estimates of $v_{\vec{w}}(S_{s})$, $1\leq s\leq 2^{m}$ to complete the proof

[TABLE]

[TABLE]

∎