Weighted Lipschitz estimates for commutators of multilinear Calderón–Zygmund operators with Dini type kernels

Abstract

In this paper, we discuss the properties for commutators and iterated commutators generated by the multilinear ω-CZO and weighted Lipschitz functions on Lebesgue space.

1 Introduction and results

The singular integral operator theory plays an important role in many aspects of harmonic analysis. The Calderón–Zygmund operator theory is one of the most important achievements of classical analysis in the last century, which has many important applications in Fourier analysis, complex analysis, operator theory, and so on. The multilinear Calderón–Zygmund theory was introduced by Coifman and Meyer in [1, 2]. This theory was then further studied by Grafakos and Torres [7, 8], who considered the multilinear Calderón–Zygmund operator with classical standard kernels. And this topic keeps attracting many researchers.

In 1985, Yabuta [23] firstly considered Calderón–Zygmund operators with kernels of type ω as the generalizations of Calderón–Zygmund operators when studied pseudodifferential operator. In 2009, Maldonado and Naibo [15] studied the bilinear Calderón–Zygmund operators of type ω. In 2014, Lu and Zhang [14] considered the multilinear case. Assume that $\omega (t):[0,\infty )\rightarrow [0,\infty )$ is a nondecreasing function with $0<\omega (1)<\infty$. For $a>0$, we say $\omega \in \mathrm{{Dini}}\mathit{{(a)}}$ if

$$ \vert \omega \vert _{\mathrm{{Dini}}\mathit{{(a)}}}= \int _{0}^{1}\frac{\omega ^{a}(t)}{t} \,\mathrm{d}t< \infty .$$

Definition 1.1

([14])

A locally integrable function $K(x,y_{1},\ldots ,y_{m})$, defined away from the diagonal $x=y_{1}=\cdots =y_{m}$ in $(\mathbb{R}^{n})^{m+1}$, is called an m-linear Calderón–Zygmund kernel of type $\omega (t)$ if there exists a constant $A>0$ such that

$$ \bigl\vert K(x,y_{1},\ldots ,y_{m}) \bigr\vert \leq \frac{A}{( \vert x-y_{1} \vert +\cdots + \vert x-y_{m} \vert )^{mn}} $$

(1.1)

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

$$\begin{aligned}& \bigl|K(x,y_{1},\ldots ,y_{m})-K\bigl(x',y_{1},\ldots ,y_{m}\bigr)\bigr| \\& \quad \leq \frac{A}{( \vert x-y_{1} \vert +\cdots + \vert x-y_{m} \vert )^{mn}}\omega \biggl( \frac{ \vert x-x' \vert }{ \vert x-y_{1} \vert +\cdots + \vert x-y_{m} \vert } \biggr) \end{aligned}$$

(1.2)

whenever $|x-x'|\leq \frac{1}{2}\max_{1\leq j\leq m}|x-y_{j}|$, and

$$\begin{aligned}& \bigl|K(x,y_{1},\ldots ,y_{j},\ldots ,y_{m})-K\bigl(x,y_{1},\ldots ,y_{j}', \ldots ,y_{m}\bigr)\bigr| \\& \quad \leq \frac{A}{( \vert x-y_{1} \vert +\cdots + \vert x-y_{m} \vert )^{mn}}\omega \biggl( \frac{ \vert y_{j}-y_{j}' \vert }{ \vert x-y_{1} \vert +\cdots + \vert x-y_{m} \vert } \biggr) \end{aligned}$$

(1.3)

whenever $|y_{j}-y_{j}'|\leq \frac{1}{2}\max_{1\leq j\leq m}|x-y_{j}|$.

We say $T:\mathscr{S}(\mathbb{R}^{n})\times \cdots \times \mathscr{S}( \mathbb{R}^{n})\rightarrow \mathscr{S}'(\mathbb{R}^{n})$ is an m-linear operator with an m-linear Calderón–Zygmund kernel $K(x,y_{1},\ldots ,y_{m})$ of type $\omega (t)$ if

$$ T(f_{1},\ldots ,f_{m}) (x)= \int _{(\mathbb{R}^{n})^{m}}K(x,y_{1}, \ldots ,y_{m})f_{1}(y_{1}) \cdots f_{m}(y_{m})\,\mathrm{d}y_{1}\cdots \,\mathrm{d}y_{m}, $$

whenever $f_{1},\ldots ,f_{m}\in C_{c}^{\infty}(\mathbb{R}^{n})$ and $x\notin \cap _{j=1}^{m} \operatorname{supp}f_{j}$.

If T can be extended to a bounded multilinear operator from $L^{q_{1}}(\mathbb{R}^{n})\times \cdots \times L^{q_{m}}(\mathbb{R}^{n})$ to $L^{q,\infty}(\mathbb{R}^{n})$ for some $1< q, q_{1},\ldots ,q_{m}<\infty$ with ${1}/{q_{1}}+\cdots +{1}/{q_{m}}={1}/{q}$ or from $L^{q_{1}}(\mathbb{R}^{n})\times \cdots \times L^{q_{m}}(\mathbb{R}^{n})$ to $L^{1}(\mathbb{R}^{n})$ for some $1< q_{1},\ldots ,q_{m}<\infty$ with ${1}/{q_{1}}+\cdots +{1}/{q_{m}}=1$, then T is called an m-linear Calderón–Zygmund operator of type ω, abbreviated to m-linear ω-CZO.

Obviously, when $\omega (t)=t^{\varepsilon}$ for some $\varepsilon >0$, the m-linear ω-CZO is exactly the multilinear Calderón–Zygmund operator studied by Grafakos and Torres [7] and Lerner et al. [12].

To shorten the notation, we denote $\vec{f}=(f_{1},\ldots ,f_{m})$ and $\mathrm{d}\vec{y}=\mathrm{d}{y_{1}}\cdots \mathrm{d}{y_{m}}$ in the following.

In 2014, Lu and Zhang [14] gave the endpoint estimate for the m-linear ω-CZO under some weaker assumptions of $\omega (t)$ and also got the following multiple weighted estimates.

Theorem A

([14])

Let T be an m-linear ω-CZO with $\omega \in \mathrm{Dini}{(1)}$. Let ${1}/{p}={1}/{p_{1}}+\cdots +{1}/{p_{m}}$ and ${\mu}\in A_{\min \{p_{1},p_{2},\ldots ,p_{m}\}}(\mathbb{R}^{n})$. If $1< p_{j}<\infty$ for all $j=1,\ldots ,m$, then

$$ \bigl\Vert T(\vec{f}) \bigr\Vert _{L^{p}(\mu )}\leq C\prod _{j=1}^{m} \Vert f_{j} \Vert _{L^{p_{j}}( \mu )}.$$

Theorem B

([14])

Let T be an m-linear ω-CZO with $\omega \in \mathrm{{Dini}{(1)}}$. Then T can be extended to a bounded operator from $L^{1}(\mathbb{R}^{n})\times \cdots \times L^{1}(\mathbb{R}^{n})$ to $L^{1/m,\infty}(\mathbb{R}^{n})$.

Let $\vec{b}=(b_{1},\ldots ,b_{m})$ be a collection of locally integrable functions, the commutator generated by m-linear ω-CZO, and b⃗ is defined by

$$ T_{\Sigma \vec{b}}(f_{1},f_{2},\ldots ,f_{m}) (x)=\sum_{j=1}^{m}T_{b_{j}}^{j}(f_{1},f_{2}, \ldots ,f_{m}) (x),$$

where

$$\begin{aligned}& T_{b_{j}}^{j}(\vec{f}) (x)=[b_{j},T]_{j}( \vec{f}) (x)=b_{j}(x)T(f_{1}, \ldots ,f_{j}, \ldots ,f_{m}) (x)-T(f_{1},\ldots ,f_{j}b_{j}, \ldots ,f_{m}) (x),\\& \quad j=1, \ldots ,m. \end{aligned}$$

Let $\vec{b}=(b_{1},\ldots ,b_{m})$ be a collection of locally integrable functions, the commutator generated by m-linear ω-CZO, and b⃗ is defined by

$$ T_{\Sigma \vec{b}}(f_{1},f_{2},\ldots ,f_{m}) (x)=\sum_{j=1}^{m}T_{b_{j}}^{j}(f_{1},f_{2}, \ldots ,f_{m}) (x),$$

where

$$\begin{aligned}& T_{b_{j}}^{j}(\vec{f}) (x)=[b_{j},T]_{j}( \vec{f}) (x)=b_{j}(x)T(f_{1}, \ldots ,f_{j}, \ldots ,f_{m})-T(f_{1},\ldots ,f_{j}b_{j}, \ldots ,f_{m}) (x),\\& \quad j=1, \ldots ,m. \end{aligned}$$

The iterated commutator $T_{\Pi \vec{b}}(\vec{f})$ is defined as follows:

$$ T_{\Pi \vec{b}}(\vec{f}) (x)=\bigl[b_{1},\bigl[b_{2}, \ldots \bigl[b_{m-1},[b_{m},T]_{m} \bigr]_{m-1}, \ldots \bigr]_{2}\bigr]_{1}( \vec{f}) (x),$$

which can also be given formally by

$$ T_{\Pi \vec{b}}(\vec{f}) (x)= \int _{(\mathbb{R}^{n})^{m}}\prod_{j=1}^{m} \bigl(b_{j}(x)-b_{j}(y_{j})\bigr)K(x, \vec{y})f_{1}(y_{1})\cdots f_{m}(y_{m}) \,\mathrm{d}\vec{y}.$$

When $m=1$, $T_{\Sigma \vec{b}}(\vec{f})=T_{\Pi \vec{b}}(\vec{f})=[b,T]f=bT(f)-T(bf)$, which is the well-known classical commutator studied in [3]. In 1995, Paluszynski [17] proved that the commutator $[b,T]$ generated by Calderón–Zygmund operators T with classical kernel and $b\in \mathrm{Lip}_{\beta}(\mathbb{R}^{n})$ is bounded from $L^{p}(\mathbb{R}^{n})$ to $L^{q}(\mathbb{R}^{n})$ whenever $0<\beta <1$, ${1}/{q}={1}/{p}-{\beta}/{n}$, and $1< p< q<\infty$, and from $L^{p}$ to homogenous Triebel–Lizorkin spaces $\dot{F}_{p}^{\beta ,\infty}(\mathbb{R}^{n})$ which is defined in [20].

For the weighted case, Hu and Gu [9] proved when $b\in \mathrm{Lip}_{\beta ,\mu}(\mathbb{R}^{n})$, the commutators $[b,T]$ is bounded from $L^{p}(\mu )$ to $L^{q}(\mu ^{1-q})$. In 2011, Lian, Ma, and Wu [13] studied the m-linear commutators generated by the multilinear Calderón–Zygmund operators with nonsmooth kernels and weighted Lipschitz functions bounded from the product of weighted Lebesgue spaces to the weighted Lebesgue space. For more articles about multilinear operators, see [1, 2, 4, 6, 7, 10–12, 14, 16, 24, 25], and [26].

In this paper, we will discuss the mapping properties of multilinear commutators generated by m-linear Dini’s type Calderón–Zygmund operators and weighted Lipschitz functions on some function spaces. We obtain the following results.

Theorem 1.1

Let T be an m-linear ω-CZO satisfying

$$ \int _{0}^{1}\frac{\omega (t)}{t}\biggl(1+\log \frac{1}{t}\biggr)\,\mathrm{d}t\leq \infty . $$

(1.4)

Suppose $0<\beta <1$ and $1/r=1/p-\beta /n$, $1< p< r<\infty$ for $1/{p_{1}}+\cdots +1/{p_{m}}=1/p$ with $1< p_{i}<\infty$, $i=1,\ldots ,m$. If $\mu \in A_{1}(\mathbb{R}^{n})$ and $b_{j}\in \mathrm{Lip}_{\beta ,\mu}(\mathbb{R}^{n})(1\leq j\leq m)$, then $T_{b_{j}}^{j}(\vec{f})$ is bounded from $L^{p_{1}}(\mu )\times \cdots \times L^{p_{m}}(\mu )$ to $L^{r}(\mu ^{1-r})$.

Furthermore, $T_{\sum \vec{b}}(\vec{f})$ is bounded from $L^{p_{1}}(\mu )\times \cdots \times L^{p_{m}}(\mu )$ to $L^{r}(\mu ^{1-r})$.

Theorem 1.2

Let T be an m-linear ω-CZO satisfying

$$ \int _{0}^{1}\frac{\omega (t)}{t}\biggl(1+\log \frac{1}{t}\biggr)^{m}\,\mathrm{d}t< \infty .$$

Suppose $0<\beta _{i}<1$, $i=1,\ldots ,m$, $1/{r_{i}}=1/{p_{i}}-{\beta _{i}}/n$, $1< p_{i}< r_{i}<\infty$ with $1/{p_{1}}+\cdots +1/{p_{m}}=1/p$, $1/{r_{1}}+\cdots +1/{r_{m}}=1/r$, ${\beta _{1}}+\cdots +{\beta _{m}}=\beta$, and $0<\beta <1$. If $\mu \in A_{1}(\mathbb{R}^{n})$, $b_{i}\in \mathrm{Lip}_{\beta _{i},\mu}(\mathbb{R}^{n})(1\leq i\leq m)$, then $T_{\Pi \vec{b}}(\vec{f})$ is bounded from $L^{p_{1}}(\mu )\times \cdots \times L^{p_{m}}(\mu )$ to $L^{r}(\mu ^{1-mr})$.

Remark 1.1

Theorem 1.1 and 1.2 are also valid for commutators of multilinear Calderón–Zygmund operator with standard kernels.

Remark 1.2

Theorem 1.2 extends the corresponding result in [19] and [22].

The rest of this paper is organized as follows. After recalling some notations and lemmas in Sect. 2, we prove our results in Sect. 3.

Throughout this paper, we denote by $p'$ the conjugate index of p, that is, $1/p+1/p'=1$. The letter C, sometimes with additional parameters, will stand for positive constants, not necessarily the same at each occurrence but independent of the main parameters.

2 Preliminaries and lemmas

A nonnegative locally integrable function is called a weight function.

Definition 2.1

([5])

Let μ be a weight function, $1< p<\infty$. If there is a constant $C>0$ such that, for every ball $B\subseteq \mathbb{R}^{n}$,

$$ \biggl(\frac{1}{ \vert B \vert } \int _{B}\mu (x)\,\mathrm{d}x \biggr) \biggl( \frac{1}{ \vert B \vert } \int _{B}\mu (x)^{-{1}/{(p-1)}}\,\mathrm{d}x \biggr)^{p-1} \leq C,$$

then we say $\mu \in A_{p}$. We say $\mu \in A_{1}$ if there is a constant $C>0$ such that, for every ball $B\subseteq \mathbb{R}^{n}$,

$$ \frac{1}{ \vert B \vert } \int _{B}\mu (x)\,\mathrm{d}x\leq C \mathop{\operatorname{essinf}} \limits _{x\in B}\mu (x).$$

A weight function $\mu \in A_{\infty}$ if it satisfies the $A_{p}$ condition for some $1 < p < \infty$. The smallest constant satisfying the formulas above is called $A_{p}$ constant of w, we denote it by $[\mu ]_{A_{p}}$.

For $1\leq p< q<\infty$, we have $A_{1}\subset A_{p}\subset A_{q}$. And $A_{\infty}=\cup _{1\leq p<\infty}A_{p}$.

If $\mu \in A_{r}$, $1< r<\infty$, then $\mu ^{1-r'}\in A_{r'}$.

For a function $f\in L_{loc}(\mathbb{R}^{n})$, the Hardy–Littlewood maximal and the sharp maximal functions are defined by

$$ Mf(x)=\sup_{Q\ni x}\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert f(y) \bigr\vert \,\mathrm{d}y$$

and

$$ M^{\sharp}f(x)=\sup_{Q\ni x}\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert f(y)-f_{B} \bigr\vert \,\mathrm{d}y \approx \sup_{ B\ni x}\inf_{C} \frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert f(y)-C \bigr\vert \,\mathrm{d}y,$$

where $f_{Q}$ denotes the average of f over cube Q, that is, $f_{Q}=\frac{1}{|Q|}\int _{Q}f(x)\,\mathrm{d}x$.

For $\delta >0$, we denote $M_{\delta}(f)$ and $M_{\delta}^{\sharp}(f)$ by $M_{\delta}(f)=M(|f|^{\delta})^{{1}/{\delta}}$ and $M_{\delta}^{\sharp}(f)=[M^{\sharp}(|f|^{\delta})]^{1/{\delta}}$. We denote the following fractional maximal operator:

$$ M_{\alpha ,\mu ,s}f(x)=\sup_{Q\ni x} \biggl( \frac{1}{\mu (Q)^{1-s\alpha /n}} \int _{Q} \bigl\vert f(y) \bigr\vert ^{s} \mu (y) \,\mathrm{d}y \biggr)^{{1}/{s}}.$$

Recall that $M_{\alpha}:=M_{\alpha ,1,1}$ is the fractional maximal operator

$$ M_{\alpha}(f) (x)=\sup_{Q\ni x}\frac{1}{ \vert Q \vert ^{1-\alpha /n}} \int _{Q} \bigl\vert f(y) \bigr\vert \,\mathrm{d}y.$$

Definition 2.2

([5])

Let $1\leq p\leq \infty$, $\ 0<\beta <1$, $\mu \in A_{\infty}$, the weighted Lipschitz space $\mathrm{Lip}_{ \beta ,\mu}^{p}$ contains all locally integrable functions f satisfying

$$ \Vert f \Vert _{\mathrm{Lip}_{\beta ,\mu}^{p}}=\sup_{B} \frac{1}{\mu (B)^{{\beta}/{n}}} \biggl[\frac{1}{\mu (B)} \int _{B} \bigl\vert f(x)-f_{B} \bigr\vert ^{p} \mu (x)^{1-p}\,\mathrm{d}x \biggr]^{1/p} \leq C< \infty ,$$

where $f_{B}=\frac{1}{|B|}\int _{B}f(y)\,\mathrm{d}y$, the supremum is taken over all balls B in $\mathbb{R}^{n}$.

The smallest number C satisfying the above inequality was denoted by $\|f\|_{\mathrm{Lip}_{\beta ,\mu}^{p}}$, and we also denote by $\|f\|_{\mathrm{Lip}_{\beta ,\mu}}=\|f\|_{\mathrm{Lip}_{\beta ,\mu}^{1}}$. Obviously, when $\mu =1$, $\mathrm{Lip}_{\beta ,\mu}=\mathrm{Lip}_{\beta}$.

$A\sim B$ means there exist $C_{1}>0$, $C_{2}>0$ such that $C_{1}A \leq B\leq C_{2}A$. When $\mu \in A_{1}$, García–Cuerva in [5] proved that for $1\leq p,q\leq \infty$, $\|f\|_{\mathrm{Lip}_{\beta ,\mu}^{p}}\sim \|f\|_{\mathrm{Lip}_{\beta ,\mu}^{q}}$.

As usual, we denote $\|f\|_{L^{p}(\mu )}=(\int _{\mathbb{R}^{n}}|f(x)|^{p}\mu (x) \,\mathrm{d}x)^{{1}/{p}}$ for $1< p<\infty$ and $p=\infty$, $\|f\|_{L^{\infty}(\mu )}=\|f\|_{L^{\infty}}$.

We will use the following Kolmogorov inequality:

$$ \Vert f \Vert _{L^{p}(Q,\frac{\,\mathrm{d}x}{ \vert Q \vert })}\leq C \Vert f \Vert _{L^{q,\infty}(Q, \frac{\,\mathrm{d}x}{ \vert Q \vert })},$$

where $0< p< q<\infty$. See [12, 21].

Lemma 2.1

([18])

Let $0< p,\delta <\infty$, $\mu \in A_{\infty}$, then there exists a constant C such that

$$ \int _{\mathbb{R}^{n}}M_{\delta}f(x)^{p}\mu (x) \,\mathrm{d}x \leq C \int _{\mathbb{R}^{n}}M_{\delta}^{\sharp}f(x)^{p} \mu (x)\,\mathrm{d}x $$

(2.1)

for any function f for which the left-hand side is finite.

Lemma 2.2

([13])

Suppose $\mu \in A_{1}$, $0<\beta <1$, $b\in \mathrm{Lip}_{\beta ,\mu}(\mathbb{R}^{n})$.

(i) For $k\geq 1$,

$$ \vert b_{B(x,2R)}-b_{B(x,2^{k+1}R)} \vert \leq Ck\mu (x)\mu \bigl(B\bigl(x,2^{k+1}R\bigr)\bigr)^{ \beta /n} \Vert b \Vert _{\mathrm{Lip}_{\beta ,\mu}}.$$

(ii) For any $1\leq s<\infty$ and any ball $B\ni x$, we have

$$ \frac{\mu (B)^{\beta /n}}{ \vert B \vert } \int _{B} \bigl\vert f(y) \bigr\vert \,\mathrm{d}y\leq CM_{ \beta ,\mu ,s}(f) (x).$$

(iii) For any $1< s<\infty$ and any ball $B\ni x$, we have

$$ \frac{1}{ \vert B \vert } \int _{B} \bigl\vert \bigl(b(y)-b_{B} \bigr)f(y) \bigr\vert \,\mathrm{d}y\leq C\mu (x) \Vert b \Vert _{\mathrm{Lip}_{ \beta ,\mu}}M_{\beta ,\mu ,s}(f) (x).$$

Lemma 2.3

([13])

Suppose that $0<\alpha <n$, $0< s< p< n/\alpha$, $1/q=1/p-\alpha /n$. If $\mu \in A_{\infty}(\mathbb{R}^{n})$, then

$$ \Vert M_{\alpha ,\mu ,s}f \Vert _{L^{q}(\mu )}\leq C \Vert f \Vert _{L^{p}(\mu )}.$$

Lemma 2.4

([13])

Suppose that $0<\alpha <n$, $1< p< n/\alpha$, and $1/q=1/p-\alpha /n$. If $\mu \in A_{1+q/p'}(\mathbb{R}^{n})$, then

$$ \Vert M_{\alpha}f \Vert _{L^{q}(\mu )}\leq C \Vert f \Vert _{L^{p}(\mu ^{p/q})}.$$

3 Proofs of theorems

For simplicity, we only prove for the case $m=2$. The argument for the case $m>2$ is similar. We first establish the following lemmas.

Lemma 3.1

Let T be a 2-linear ω-CZO satisfying (1.4). Suppose $\mu \in A_{1}(\mathbb{R}^{n})$ and $b_{j}\in \mathrm{Lip}_{\beta ,\mu}(\mathbb{R}^{n})$ with $0<\beta <1$, $j=1,2$. Let $0<\delta <1/2<1<s<n/\beta$. Then we have

$$\begin{aligned} M_{\delta}^{\sharp}\bigl[T_{b_{j}}^{j}(f_{1},f_{2}) \bigr](x)\leq{}& C\mu (x) \Vert b_{j} \Vert _{\mathrm{Lip}_{\beta ,\mu}} \bigl[M_{\beta ,\mu ,s}\bigl(T(f_{1},f_{2})\bigr) (x) \\ &{}+M_{\beta ,\mu ,s}(f_{1}) (x)M(f_{2}) (x) +M(f_{1}) (x)M_{\beta ,\mu ,s}(f_{2}) (x)\bigr] \end{aligned}$$

(3.1)

for $j=1,2$.

Proof

We only estimate $M_{\delta}^{\sharp} (T_{b_{1}}^{1}(f_{1},f_{2}) )$ and write $b_{1}=b$ for simplicity. A similar discussion also works for $M_{\delta}^{\sharp} (T_{b_{2}}^{2}(f_{1},f_{2}) )$.

Fix $x\in \mathbb{R}^{n}$ for any cube $Q(x_{Q},l_{Q})$ containing x with side-length $l_{Q}$, set $Q^{*}=8\sqrt{n}Q=Q(x_{Q},8\sqrt{n}l_{Q})$. We decompose $f_{j}=f_{j}^{0}+f_{j}^{\infty}$, where $f_{j}^{0}=f_{j}\chi _{Q^{*}}$ and $f_{j}^{\infty}=f\chi _{{\mathbb{R}^{n}}\setminus {Q^{*}}}$, $j=1,2$.

Since $0<\delta <1/2$, then for any constant c, we have

$$\begin{aligned}& \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert \bigl\vert T_{b}^{1}(f_{1},f_{2}) (z) \bigr\vert ^{ \delta}- \vert c \vert ^{\delta} \bigr\vert \,\mathrm{d}z \biggr)^{{1}/{\delta}} \\& \quad \leq \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert T_{b}^{1}(f_{1},f_{2}) (z)-c \bigr\vert ^{ \delta}\,\mathrm{d}z \biggr)^{{1}/{\delta}} \\& \quad \leq C \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert \bigl(b(z)-b_{Q^{*}} \bigr)T(f_{1},f_{2}) (z) \bigr\vert ^{ \delta} \,\mathrm{d}z \biggr)^{{1}/{\delta}}\\& \qquad {}+C \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert T\bigl((b-b_{Q^{*}})f_{1}^{0},f_{2}^{0} \bigr) (z) \bigr\vert ^{ \delta}\,\mathrm{d}z \biggr)^{{1}/{\delta}} \\& \qquad {}+C \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert T\bigl((b-b_{Q^{*}})f_{1}^{\infty},f_{2}^{0} \bigr) (z) \bigr\vert ^{ \delta}\,\mathrm{d}z \biggr)^{{1}/{\delta}}\\& \qquad {} +C \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert T\bigl((b-b_{Q^{*}})f_{1}^{0},f_{2}^{ \infty} \bigr) (z) \bigr\vert ^{\delta}\,\mathrm{d}z \biggr)^{{1}/{\delta}} \\& \qquad {}+C \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert T\bigl((b-b_{Q^{*}})f_{1}^{\infty},f_{2}^{ \infty} \bigr) (z)-c \bigr\vert ^{\delta}\,\mathrm{d}z \biggr)^{{1}/{\delta}} \\& \quad := I_{1}+I_{2}+I_{3}+I_{4}+I_{5}. \end{aligned}$$

Since $0<\delta <1$, $\mu \in A_{1}$, and $b\in \mathrm{Lip}_{\beta ,\mu}$, by Lemma 2.2(iii) and Hölder’s inequality, we get

$$ I_{1}\leq C\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert \bigl(b(z)-b_{Q^{*}} \bigr)T(f_{1},f_{2}) (z) \bigr\vert \,\mathrm{d}z\leq C \mu (x) \Vert b \Vert _{\mathrm{Lip}_{\beta ,\mu}}M_{\beta ,\mu ,s} \bigl(T(f_{1},f_{2})\bigr) (x).$$

For the second term $I_{2}$, since $0 <\delta < 1/2$, by Kolmogorov’s inequality, Theorem B, and Lemma 2.2(iii), we obtain

$$\begin{aligned} I_{2} =&C \vert Q \vert ^{-\delta} \bigl\Vert T \bigl((b-b_{Q^{*}})f_{1}^{0},f_{2}^{0} \bigr) \bigr\Vert _{L^{ \delta}(Q)} \\ \leq &C \vert Q \vert ^{-2} \bigl\Vert T\bigl((b-b_{Q^{*}})f_{1}^{0},f_{2}^{0} \bigr) \bigr\Vert _{L^{{1}/{2}, \infty}(Q)} \\ \leq &C \vert Q \vert ^{-2} \bigl\Vert T\bigl((b-b_{Q^{*}})f_{1}^{0},f_{2}^{0} \bigr) \bigr\Vert _{L^{{1}/{2}, \infty}(\mathbb{R}^{n})} \\ \leq &C \biggl(\frac{1}{ \vert Q^{*} \vert } \int _{Q^{*}} \bigl\vert \bigl(b(z)-b_{Q^{*}} \bigr)f_{1}(z) \bigr\vert \,\mathrm{d}z \biggr) \biggl( \frac{1}{ \vert Q^{*} \vert } \int _{Q^{*}} \bigl\vert f_{2}(z) \bigr\vert \,\mathrm{d}z \biggr) \\ \leq &C\mu (x) \Vert b \Vert _{\mathrm{Lip}_{\beta ,\mu}}M_{\beta ,\mu ,s}(f_{1}) (x)M(f_{2}) (x). \end{aligned}$$

For the term $I_{3}$, noting the fact that $|z-y_{1}|\sim |y_{1}-x_{Q}|$ for any $y_{1}\in {(Q^{*})}^{c}$ and $z\in Q$, then by (1.1) and Lemma 2.2(i)(ii), we obtain

$$\begin{aligned} I_{3} \leq &\frac{C}{ \vert Q \vert } \int _{Q} \bigl\vert T\bigl((b-b_{Q^{*}})f_{1}^{\infty},f_{2}^{0} \bigr) (z) \bigr\vert \,\mathrm{d}z \\ \leq &\frac{C}{ \vert Q \vert } \int _{Q} \int _{Q^{*}} \int _{{(Q^{*})}^{c}} \frac{A}{( \vert z-y_{1} \vert + \vert z-y_{2} \vert )^{2n}} \bigl\vert b(y_{1})-b_{Q^{*}} \bigr\vert \bigl\vert f_{1}(y_{1}) \bigr\vert \bigl\vert f_{2}(y_{2}) \bigr\vert \,\mathrm{d}y_{1} \,\mathrm{d}y_{2}\,\mathrm{d}z \\ \leq &C \int _{{(Q^{*})}^{c}} \frac{ \vert b(y_{1})-b_{Q^{*}} \vert }{ \vert y_{1}-x_{Q} \vert ^{2n}} \bigl\vert f_{1}(y_{1}) \bigr\vert \,\mathrm{d}y_{1} \int _{Q^{*}} \bigl\vert f_{2}(y_{2}) \bigr\vert \,\mathrm{d}y_{2} \\ \leq &C \bigl\vert Q^{*} \bigr\vert \sum _{k=1}^{\infty} \int _{(2^{k+3}\sqrt {n} Q)^{*} \setminus (2^{k+2}\sqrt {n} Q)^{*}} \frac{ \vert b(y_{1})-b_{Q^{*}} \vert }{ \vert y_{1}-x_{Q} \vert ^{2n}} \bigl\vert f_{1}(y_{1}) \bigr\vert \,\mathrm{d}y_{1} M(f_{2}) (x) \\ \leq &C\sum_{k=1}^{\infty} \frac{ \vert Q^{*} \vert }{ \vert 2^{k}Q^{*} \vert ^{2}} \int _{2^{k} Q^{*}} \bigl[ \bigl\vert b(y_{1})-b_{2^{k}Q^{*}} \bigr\vert + \vert b_{2^{k}Q^{*}}-b_{Q^{*}} \vert \bigr] \bigl\vert f_{1}(y_{1}) \bigr\vert \,\mathrm{d}y_{1} M(f_{2}) (x) \\ \leq &CM(f_{2}) (x)\sum_{k=1}^{\infty}2^{-k} \biggl[\mu (x) \Vert b \Vert _{\mathrm{Lip}_{ \beta ,\mu}}M_{\beta ,\mu ,s}(f_{1}) (x) \\ &{} +k\mu (x) \Vert b \Vert _{\mathrm{Lip}_{\beta ,\mu}} \frac{\mu (2^{k}Q^{*})^{\frac{\beta}{n}}}{ \vert 2^{k}Q^{*} \vert } \int _{2^{k} Q^{*}} \bigl\vert f_{1}(y_{1}) \bigr\vert \,\mathrm{d}y_{1} \biggr] \\ \leq &C\mu (x) \Vert b \Vert _{\mathrm{Lip}_{\beta ,\mu}}M_{\beta ,\mu ,s}(f_{1}) (x)M(f_{2}) (x). \end{aligned}$$

Similarly, we have

$$\begin{aligned} I_{4} \leq &\frac{C}{ \vert Q \vert } \int _{Q} \bigl\vert T\bigl((b-b_{Q^{*}})f_{1}^{0},f_{2}^{ \infty} \bigr) (z) \bigr\vert \,\mathrm{d}z \\ \leq &\frac{C}{ \vert Q \vert } \int _{Q} \int _{Q^{*}} \int _{{(Q^{*})}^{c}} \frac{A}{( \vert z-y_{1} \vert + \vert z-y_{2} \vert )^{2n}} \bigl\vert b(y_{1})-b_{Q^{*}} \bigr\vert \bigl\vert f_{1}(y_{1}) \bigr\vert \bigl\vert f_{2}(y_{2}) \bigr\vert \,\mathrm{d}y_{2} \,\mathrm{d}y_{1}\,\mathrm{d}z \\ \leq &C \int _{Q^{*}} \bigl\vert b(y_{1})-b_{Q^{*}} \bigr\vert \bigl\vert f_{1}(y_{1}) \bigr\vert \,\mathrm{d}y_{1} \int _{{(Q^{*})}^{c}}\frac{ \vert f_{2}(y_{2}) \vert }{ \vert y_{2}-x_{Q} \vert ^{2n}} \,\mathrm{d}y_{2} \\ \leq &C\mu (x) \Vert b \Vert _{{Lip}_{\beta ,\mu}}M_{\beta ,\mu ,s}(f_{1}) (x) \sum_{k=1}^{\infty}\frac{ \vert Q \vert }{ \vert 2^{k}Q^{*} \vert ^{2}} \int _{2^{k+3}\sqrt {n} Q \setminus 2^{k+2}\sqrt {n} Q} \bigl\vert f_{2}(y_{2}) \bigr\vert \,\mathrm{d}y_{2} \\ \leq &C\mu (x) \Vert b \Vert _{{Lip}_{\beta ,\mu}}M_{\beta ,\mu ,s}(f_{1}) (x)M(f_{2}) (x). \end{aligned}$$

For the last term $I_{5}$, since $(\mathbb{R}^{n}\setminus Q^{*})^{2}\subseteq \mathbb{R}^{2n} \setminus {(Q^{*})^{2}}\subseteq \cup _{k=1}^{\infty}(2^{k+3}\sqrt{n}Q)^{2} \setminus (2^{k+2}\sqrt{n}Q)^{2}$, making use of assumption (1.2), we have

$$\begin{aligned} I_{5} \leq &\frac{C}{ \vert Q \vert } \int _{Q} \bigl\vert T\bigl((b-b_{Q^{*}})f_{1}^{\infty},f_{2}^{ \infty} \bigr) (z)-T\bigl((b-b_{Q^{*}})f_{1}^{\infty},f_{2}^{\infty} \bigr) (x_{Q}) \bigr\vert \,\mathrm{d}z \\ \leq &\frac{C}{ \vert Q \vert } \int _{Q} \int _{(\mathbb{R}^{n}\setminus Q^{*})^{2}} \bigl\vert K(z,y_{1},y_{2})-K(x_{Q},y_{1},y_{2}) \bigr\vert \bigl\vert \bigl(b(y_{1})-b_{Q^{*}} \bigr)f_{1}^{ \infty}(y_{1})f_{2}^{\infty}(y_{2}) \bigr\vert \,\mathrm{d}y_{1}\,\mathrm{d}y_{2}dz \\ \leq &\frac{C}{ \vert Q \vert } \int _{Q} \int _{(\mathbb{R}^{n}\setminus Q^{*})^{2}} \frac{1}{( \vert x-y_{1} \vert + \vert x-y_{2} \vert )^{2n}}\omega \biggl( \frac{ \vert z-x_{Q} \vert }{ \vert x-y_{1} \vert + \vert x-y_{2} \vert } \biggr) \\ &{} \times \bigl\vert \bigl(b(y_{1})-b_{Q^{*}} \bigr)f_{1}(y_{1})f_{2}(y_{2}) \bigr\vert \,\mathrm{d}y_{1} \,\mathrm{d}y_{2}\,\mathrm{d}z \\ \leq &\frac{C}{ \vert Q \vert } \int _{Q}\sum_{k=1}^{\infty} \int _{(2^{k+3} \sqrt {n} Q)^{2}\setminus (2^{k+2}\sqrt {n} Q)^{2}} \frac{\omega (\frac{ \vert z-x_{Q} \vert }{ \vert x-y_{1} \vert + \vert x-y_{2} \vert })}{( \vert x-y_{1} \vert + \vert x-y_{2} \vert )^{2n}}\\ &{} \times \bigl\vert b(y_{1})-b_{Q^{*}} \bigr\vert \bigl\vert f_{1}(y_{1})f_{2}(y_{2}) \bigr\vert \,\mathrm{d}y_{1}\,\mathrm{d}y_{2}\,\mathrm{d}z \\ \leq &\frac{C}{ \vert Q \vert } \int _{Q}\sum_{k=1}^{\infty} \frac{\omega (2^{-k})}{ \vert 2^{k}Q^{*} \vert } \int _{2^{k}Q^{*}} \bigl\vert b(y_{1})-b_{Q^{*}} \bigr\vert \bigl\vert f_{1}(y_{1}) \bigr\vert \,\mathrm{d}y_{1} \times \frac{1}{ \vert 2^{k}Q^{*} \vert } \int _{2^{k}Q^{*}} \bigl\vert f_{2}(y_{2}) \bigr\vert \,\mathrm{d}y_{2}\,\mathrm{d}z \\ \leq &CM(f_{2}) (x)\sum_{k=1}^{\infty} \frac{\omega (2^{-k})}{ \vert 2^{k}Q^{*} \vert } \int _{2^{k}Q^{*}} \bigl[ \bigl\vert b(y_{1})-b_{2^{k}Q^{*}} \bigr\vert + \vert b_{2^{k}Q^{*}}-b_{Q^{*}} \vert \bigr] \bigl\vert f_{1}(y_{1}) \bigr\vert \,\mathrm{d}y_{1} \\ \leq &C\sum_{k=1}^{\infty}k\omega \bigl(2^{-k}\bigr)\mu (x) \Vert b \Vert _{\mathrm{Lip}_{ \beta ,\mu}}M_{\beta ,\mu ,s}(f_{1}) (x)M(f_{2}) (x) \\ \leq &C \int _{0}^{1}\frac{\omega (t)}{t}\biggl(1+\log \frac{1}{t}\biggr) \,\mathrm{d}t\mu (x) \Vert b \Vert _{{Lip}_{\beta ,\mu}}M_{\beta ,\mu ,s}(f_{1}) (x)M(f_{2}) (x) \\ \leq &C\mu (x) \Vert b \Vert _{\mathrm{Lip}_{\beta ,\mu}}M_{\beta ,\mu ,s}(f_{1}) (x)M(f_{2}) (x). \end{aligned}$$

 □

Proof of Theorem 1.1

Since $\mu \in A_{1}\subset A_{r'}$ and $\mu ^{1-r}\in A_{r}\subset A_{\infty}$, by Lemma 2.1 and Lemma 3.1, for any $j=1,2$, we get

$$\begin{aligned} \bigl\Vert T_{b_{j}}^{j}(f_{1},f_{2}) \bigr\Vert _{L^{r}(\mu ^{1-r})} \leq& \bigl\Vert M_{ \delta} \bigl[T_{b_{j}}^{j}(f_{1},f_{2}) \bigr] \bigr\Vert _{L^{r}(\mu ^{1-r})} \\ \leq & \bigl\Vert M_{\delta}^{\sharp}\bigl[T_{b_{j}}^{j}(f_{1},f_{2}) \bigr] \bigr\Vert _{L^{r}( \mu ^{1-r})} \\ \leq &C \bigl\Vert \mu M_{\beta ,\mu ,s}\bigl(T(f_{1},f_{2}) \bigr) \bigr\Vert _{L^{r}(\mu ^{1-r})} \\ &{} +C \bigl\Vert \mu M_{\beta ,\mu ,s}(f_{1})M(f_{2}) \bigr\Vert _{L^{r}(\mu ^{1-r})} +C \bigl\Vert \mu M(f_{1})M_{\beta ,\mu ,s}(f_{2}) \bigr\Vert _{L^{r}({\mu ^{1-r}})} \\ :=&U_{1}+U_{2}+U_{3}. \end{aligned}$$

For $U_{1}$, since $1/r=1/p-\beta /n$ and select s satisfying $1< s< p< n/\beta$, by Theorem A and Lemma 2.3, we have

$$\begin{aligned} \bigl\Vert \mu M_{\beta ,\mu ,s}\bigl(T(f_{1},f_{2}) \bigr) \bigr\Vert _{L^{r}(\mu ^{1-r})} =& \bigl\Vert M_{ \beta ,\mu ,s} \bigl(T(f_{1},f_{2})\bigr) \bigr\Vert _{L^{r}(\mu )} \\ \leq &C \bigl\Vert T(f_{1},f_{2}) \bigr\Vert _{L^{p}(\mu )} \\ \leq &C \Vert f_{1} \Vert _{L^{p_{1}}(\mu )} \Vert f_{2} \Vert _{L^{p_{2}}(\mu )}. \end{aligned}$$

For $U_{2}$, let $1/r=1/p_{2}+1/l$, then we have $1/l=1/p_{1}-\beta /n$. Then, by Hölder’s inequality and Lemma 2.3, we obtain

$$\begin{aligned} \bigl\Vert \mu M_{\beta ,\mu ,s}(f_{1})M(f_{2}) \bigr\Vert _{L^{r}(\mu ^{1-r})} =& \bigl\Vert M_{ \beta ,\mu ,s}(f_{1})M(f_{2}) \bigr\Vert _{L^{r}(\mu )} \\ \leq &C \bigl\Vert M_{\beta ,\mu ,s}(f_{1}) \bigr\Vert _{L^{l}(\mu )} \bigl\Vert M(f_{2}) \bigr\Vert _{L^{p_{2}}( \mu )} \\ \leq &C \Vert f_{1} \Vert _{L^{p_{1}}(\mu )} \Vert f_{2} \Vert _{L^{p_{2}}(\mu )}. \end{aligned}$$

Similarly as the estimate of $U_{2}$, we may get

$$ U_{3}\leq C \Vert f_{1} \Vert _{L^{p_{1}}(\mu )} \Vert f_{2} \Vert _{L^{p_{2}}(\mu )}.$$

Thus Theorem 1.1 is proved. □

Lemma 3.2

Let T be a 2-linear ω-CZO satisfying $\int _{0}^{1}\frac{\omega (t)}{t}(1+\log \frac{1}{t})^{2}dt<\infty$. Suppose $\mu \in A_{1}(\mathbb{R}^{n})$ and $b_{j}\in \mathrm{Lip}_{\beta _{j},\mu}(\mathbb{R}^{n})$, $j=1,2$. Let $\beta _{1}+\beta _{2}=\beta$, $0<\beta <1$, and $0<\delta <1/3<1<s<n/\beta$. Then we have

$$\begin{aligned} M_{\delta}^{\sharp}\bigl[T_{\Pi \vec{b}}(f_{1},f_{2}) \bigr](x) \leq& C\mu (x)^{2} \Vert b_{1} \Vert _{\mathrm{Lip}_{\beta _{1},\mu}} \Vert b_{2} \Vert _{\mathrm{Lip}_{\beta _{2},\mu}} \\ &{}\times\bigl[M_{ \beta ,\mu ,s}\bigl(T(f_{1},f_{2})\bigr) (x) +M_{\beta _{1},\mu ,s}(f_{1}) (x)M_{ \beta _{2},\mu ,s}(f_{2}) (x)\bigr]\\ &{}+ \Vert b_{1} \Vert _{\mathrm{Lip}_{\beta _{1},\mu}}\mu (x)^{1+{\beta _{1}}/{n}}M_{ \beta _{1}} \bigl(T_{b_{2}}^{2}(f_{1},f_{2}) \bigr) (x) \\ &{}+ \Vert b_{2} \Vert _{\mathrm{Lip}_{\beta _{2}, \mu}}\mu (x)^{1+{\beta _{2}}/{n}}M_{\beta _{2}}\bigl(T_{b_{1}}^{1}(f_{1},f_{2}) \bigr) (x). \end{aligned}$$

Proof

We fix $x\in \mathbb{R}^{n}$ for any cube $Q(x_{Q},l_{Q})$ containing x with side-length $l_{Q}$, $i=1,2$, $Q^{*}=8\sqrt {n} Q$. Set $\lambda _{i}=(b_{i})_{Q^{*}}$, let c be a constant to be fixed along the proof. Since $0<\delta <1/3$, we have

$$\begin{aligned}& \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert \bigl\vert T_{\Pi \mathbf{b}}(f_{1},f_{2}) (z) \bigr\vert ^{ \delta}- \vert c \vert ^{\delta} \bigr\vert \,\mathrm{d}z \biggr)^{{1}/{\delta}} \\& \quad \leq \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert T_{\Pi \mathbf{b}}(f_{1},f_{2}) (z)-c \bigr\vert ^{ \delta}\,\mathrm{d}z \biggr)^{{1}/{\delta}} \\& \quad \leq C \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert \bigl(b_{1}(z)-\lambda _{1}\bigr) \bigl(b_{2}(z)- \lambda _{2} \bigr)T(f_{1},f_{2}) (z) \bigr\vert ^{\delta} \,\mathrm{d}z \biggr)^{{1}/{ \delta}} \\& \qquad {}+ C \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert \bigl(b_{1}(z)-\lambda _{1}\bigr)T_{(b_{2}- \lambda _{2})}^{2}(f_{1},f_{2}) (z) \bigr\vert ^{\delta}\,\mathrm{d}z \biggr)^{{1}/{ \delta}} \\& \qquad {}+C \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert \bigl(b_{2}(z)-\lambda _{2}\bigr)T_{(b_{1}- \lambda _{1})}^{1}(f_{1},f_{2}) (z) \bigr\vert ^{\delta}\,\mathrm{d}z \biggr)^{{1}/{ \delta}} \\& \qquad {}+ C \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert T\bigl((b_{1}-\lambda _{1})f_{1},(b_{2}- \lambda _{2})f_{2}\bigr) (z)-c \bigr\vert ^{\delta} \,\mathrm{d}z \biggr)^{{1}/{\delta}} \\& \quad = K_{1}+K_{2}+K_{3}+K_{4}. \end{aligned}$$

Firstly we consider $K_{1}$. For $0<\delta <1/3$, it follows from Hölder’s inequality and Lemma 2.2(ii) that

$$\begin{aligned} K_{1} \leq &C \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert b_{1}(z)-\lambda _{1} \bigr\vert ^{3 \delta}\,\mathrm{d}z \biggr)^{1/{3\delta}} \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert b_{2}(z)- \lambda _{2} \bigr\vert ^{3\delta}\,\mathrm{d}z \biggr)^{1/{3\delta}} \\ &{} \times \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert T(f_{1},f_{2}) (z) \bigr\vert ^{3\delta} \,\mathrm{d}z \biggr)^{1/{3\delta}} \\ \leq &C \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert b_{1}(z)-\lambda _{1} \bigr\vert dz \biggr) \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert b_{2}(z)-\lambda _{2} \bigr\vert \,\mathrm{d}z \biggr) \\ &{} \times \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert T(f_{1},f_{2}) (z) \bigr\vert \,\mathrm{d}z \biggr) \\ \leq &C\frac{\mu (Q^{*})^{{\beta _{1}}/n+1}}{ \vert Q \vert } \Vert b_{1} \Vert _{\mathrm{Lip}_{ \beta _{1},\mu}}\frac{\mu (Q^{*})^{{\beta _{2}}/n+1}}{ \vert Q \vert } \Vert b_{2} \Vert _{\mathrm{Lip}_{ \beta _{2},\mu}} \frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert T(f_{1},f_{2}) (z) \bigr\vert \,\mathrm{d}z \\ \leq &C\mu (x)^{2} \Vert b_{1} \Vert _{\mathrm{Lip}_{\beta _{1},\mu}} \Vert b_{2} \Vert _{\mathrm{Lip}_{ \beta _{2},\mu}}\mu \bigl(Q^{*}\bigr)^{{\beta}/n}\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert T(f_{1},f_{2}) (z) \bigr\vert \,\mathrm{d}z \\ \leq &C\mu (x)^{2} \Vert b_{1} \Vert _{\mathrm{Lip}_{\beta _{1},\mu}} \Vert b_{2} \Vert _{\mathrm{Lip}_{ \beta _{2},\mu}}M_{\beta ,\mu ,s} \bigl(T(f_{1},f_{2})\bigr) (x). \end{aligned}$$

For the terms $K_{2}$, $K_{3}$, notice that $0<\delta <1/3$, we use the facts $1/\delta =1+(1-\delta )/\delta$ and $0<\frac{\delta}{1-\delta}<1/2$. By Hölder’s inequality, we get

$$\begin{aligned} K_{2} \leq &C \biggl(\frac{1}{ \vert Q^{*} \vert } \int _{Q^{*}} \bigl\vert b_{1}(z)-(b_{1})_{Q^{*}} \bigr\vert \,\mathrm{d}z \biggr) \biggl(\frac{1}{ \vert Q^{*} \vert } \int _{Q^{*}} \bigl\vert T^{2}_{b_{2}- \lambda _{2}}(f_{1},f_{2}) (z) \bigr\vert ^{\delta /{(1-\delta )}}\,\mathrm{d}z \biggr)^{(1-\delta )/\delta} \\ \leq &C\frac{\mu (Q^{*})^{{\beta _{1}}/n+1}}{ \vert Q^{*} \vert } \Vert b_{1} \Vert _{\mathrm{Lip}_{ \beta _{1},\mu}} \biggl(\frac{1}{ \vert Q^{*} \vert } \int _{Q^{*}} \bigl\vert T_{b_{2}- \lambda _{2}}^{2}(f_{1},f_{2}) (z) \bigr\vert \,\mathrm{d}z \biggr) \\ \leq &C \Vert b_{1} \Vert _{\mathrm{Lip}_{\beta _{1},\mu}}\mu (x)^{1+{\beta _{1}}/n} \biggl(\frac{1}{ \vert Q^{*} \vert ^{1-{\beta _{1}}/n}} \int _{Q^{*}} \bigl\vert T_{b_{2}- \lambda _{2}}^{2}(f_{1},f_{2}) (z) \bigr\vert \,\mathrm{d}z \biggr) \\ \leq &C \Vert b_{1} \Vert _{\mathrm{Lip}_{\beta _{1},\mu}}\mu (x)^{1+{\beta _{1}}/n}M_{ \beta _{1}}\bigl(T_{b_{2}-\lambda _{2}}^{2}(f_{1},f_{2}) \bigr) (x) \\ \leq &C \Vert b_{1} \Vert _{\mathrm{Lip}_{\beta _{1},\mu}}\mu (x)^{1+{\beta _{1}}/n}M_{ \beta _{1}}\bigl(T_{b_{2}}^{2}(f_{1},f_{2}) \bigr) (x). \end{aligned}$$

Similarly, we have

$$ K_{3}\leq C \Vert b_{2} \Vert _{\mathrm{Lip}_{\beta _{2},\mu}} \mu (x)^{1+{\beta _{2}}/n}M_{ \beta _{2}}\bigl(T_{b_{1}}^{1}(f_{1},f_{2}) \bigr) (x). $$

Now, we consider the last term $K_{4}$. For each $i = 1, 2$, we decompose $f_{i}=f_{i}^{0}+f_{i}^{\infty}$, where $f_{i}^{0}=f_{i}\chi _{Q^{*}}$, then

$$\begin{aligned} K_{4} \leq &C \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert T\bigl((b_{1}-\lambda _{1})f_{1}^{0}\bigr),(b_{2}- \lambda _{2})f_{2}^{0}\biggr) (z) \bigr\vert ^{\delta}\,\mathrm{d}z )^{{1}/{\delta}} \\ &{} +C \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert T\bigl((b_{1}-\lambda _{1})f_{1}^{0},(b_{2}- \lambda _{2})f_{2}^{\infty}\bigr) (z) \bigr\vert ^{\delta}\,\mathrm{d}z \biggr)^{{1}/{ \delta}} \\ &{} +C \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert T\bigl((b_{1}-\lambda _{1})f_{1}^{ \infty},(b_{2}-\lambda _{2})f_{2}^{0}\bigr) (z) \bigr\vert ^{\delta}\,\mathrm{d}z \biggr)^{{1}/{ \delta}} \\ &{} +C \biggl(\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert T\bigl((b_{1}-\lambda _{1})f_{1}^{ \infty},(b_{2}-\lambda _{2})f_{2}^{\infty}\bigr) (z)-c \bigr\vert ^{\delta}\,\mathrm{d}z \biggr)^{{1}/{\delta}} \\ :=&K_{41}+K_{42}+K_{43}+K_{44}. \end{aligned}$$

We first estimate $K_{41}$. Applying Kolmgorov’s inequality, Theorem B, and Lemma 2.2(iii), we have

$$\begin{aligned} K_{41} \leq &C \vert Q \vert ^{-2}\|T \bigl((b_{1}-\lambda _{1})f_{1}^{0} \bigr),(b_{2}- \lambda _{2})f_{2}^{0}) \|_{L^{\frac{1}{2},\infty}(Q)} \\ \leq &C \vert Q \vert ^{-2}\|T\bigl((b_{1}-\lambda _{1})f_{1}^{0}\bigr),(b_{2}- \lambda _{2})f_{2}^{0}) \|_{L^{\frac{1}{2},\infty}(\mathbb{R}^{n})} \\ \leq &C \biggl(\frac{1}{ \vert Q \vert } \int _{Q^{*}} \bigl\vert b_{1}(y_{1})-(b_{1})_{Q^{*}} \bigr\vert \bigl\vert f_{1}(y_{1}) \bigr\vert \,\mathrm{d}y_{1} \biggr) \biggl(\frac{1}{ \vert Q \vert } \int _{Q^{*}} \bigl\vert b_{2}(y_{2})-(b_{2})_{Q^{*}} \bigr\vert \bigl\vert f_{2}(y_{2}) \bigr\vert \,\mathrm{d}y_{2} \biggr) \\ \leq &C\mu (x)^{2} \Vert b_{1} \Vert _{\mathrm{Lip}_{\beta _{1},\mu}} \Vert b_{2} \Vert _{\mathrm{Lip}_{ \beta _{2},\mu}}M_{\beta _{1},\mu ,s}(f_{1}) (x)M_{\beta _{2},\mu ,s}(f_{2}) (x). \end{aligned}$$

Next, we consider the term $K_{42}$. Note that for any $z\in Q$, $y_{2}\in (Q^{*})^{c}$, $|z-y_{2}|\sim |y_{2}-x_{Q}|$, by (1.1) and Lemma 2.2(iii), we have

$$\begin{aligned} K_{42} \leq &C\frac{1}{ \vert Q \vert } \int _{Q} \bigl\vert T\bigl((b_{1}-\lambda _{1})f_{1}^{0},(b_{2}- \lambda _{2})f_{2}^{\infty}\bigr) (z) \bigr\vert \,\mathrm{d}z \\ \leq &C\frac{1}{ \vert Q \vert } \int _{Q} \int _{Q^{*}} \int _{\mathbb{R}^{n} \setminus Q^{*}}\frac{A}{( \vert z-y_{1} \vert + \vert z-y_{2} \vert )^{2n}}\\ &{} \times \bigl\vert b_{1}(y_{1})- \lambda _{1} \bigr\vert \bigl\vert f_{1}(y_{1}) \bigr\vert \bigl\vert b_{2}(y_{2})-\lambda _{2} \bigr\vert \bigl\vert f_{2}(y_{2}) \bigr\vert \,\mathrm{d}y_{2}\,\mathrm{d}y_{1}dz \\ \leq &C \int _{Q^{*}} \bigl\vert b_{1}(y_{1})-(b_{1})_{Q^{*}} \bigr\vert \bigl\vert f_{1}(y_{1}) \bigr\vert \,\mathrm{d}y_{1} \int _{\mathbb{R}^{n}\setminus Q^{*}} \frac{1}{ \vert y_{2}-x_{Q} \vert ^{2n}} \bigl\vert b_{2}(y_{2})-(b_{2})_{Q^{*}} \bigr\vert \bigl\vert f_{2}(y_{2}) \bigr\vert \,\mathrm{d}y_{2} \\ \leq &C\mu (x) \Vert b_{1} \Vert _{\mathrm{Lip}_{\beta _{1},\mu}}M_{\beta _{1},\mu ,s}(f_{1}) (x) \bigl\vert Q^{*} \bigr\vert \\ &{} \times \sum_{k=1}^{\infty} \int _{2^{k+3}\sqrt{n}Q\setminus 2^{k+2} \sqrt{n}Q}\frac{1}{ \vert y_{2}-x_{Q} \vert ^{2n}} \bigl\vert b_{2}(y_{2})-(b_{2})_{Q^{*}} \bigr\vert \bigl\vert f_{2}(y_{2}) \bigr\vert \,\mathrm{d}y_{2} \\ \leq &C\mu (x) \Vert b_{1} \Vert _{\mathrm{Lip}_{\beta _{1},\mu}}M_{\beta _{1},\mu ,s}(f_{1}) (x) \sum_{k=1}^{\infty}\frac{ \vert Q^{*} \vert }{ \vert 2^{k}Q \vert ^{2}} \int _{2^{k}Q^{*}} \bigl\vert b_{2}(y_{2})-(b_{2})_{Q^{*}} \bigr\vert \bigl\vert f_{2}(y_{2}) \bigr\vert \,\mathrm{d}y_{2} \\ \leq &C\mu (x)^{2} \Vert b_{1} \Vert _{\mathrm{Lip}_{\beta _{1},\mu}} \Vert b_{2} \Vert _{\mathrm{Lip}_{ \beta _{2},\mu}}M_{\beta _{1},\mu ,s}(f_{1}) (x)M_{\beta _{2},\mu ,s}(f_{2}) (x). \end{aligned}$$

Similarly,

$$ K_{43}\leq C\mu (x)^{2} \Vert b_{1} \Vert _{\mathrm{Lip}_{\beta _{1},\mu}} \Vert b_{2} \Vert _{\mathrm{Lip}_{ \beta _{2},\mu}}M_{\beta _{1},\mu ,s}(f_{1}) (x)M_{\beta _{2},\mu ,s}(f_{2}) (x).$$

Finally, we consider the term $K_{44}$. For any $z\in Q$ and $(y_{1},y_{2})\in (2^{k+3}\sqrt{n}Q)^{2}\setminus (2^{k+2}\sqrt{n}Q)^{2}$,

$$ \bigl\vert K(z,y_{1},y_{2})-K(x,y_{1},y_{2}) \bigr\vert \leq C \frac{\omega (2^{-k})}{ \vert 2^{k+3}\sqrt{n}Q \vert }. $$

(3.2)

Set $c=T((b_{1}-\lambda _{1})f_{1}^{\infty},(b_{2}-\lambda _{2})f_{2}^{ \infty})(x)$, then by (3.2) and Lemma 2.2(iii), we have

$$\begin{aligned} K_{44} \leq &\frac{C}{ \vert Q \vert } \int _{Q} \int _{(\mathbb{R}^{n}\setminus Q^{*})^{2}} \bigl\vert K(z,y_{1},y_{2})-K(x,y_{1},y_{2}) \bigr\vert \bigl(\Pi _{i=1}^{2} \bigl\vert b_{i}(y_{i})-\lambda _{i} \bigr\vert \bigl\vert f_{i}(y_{i}) \bigr\vert \bigr) \,\mathrm{d}y_{1}\,\mathrm{d}y_{2}\,\mathrm{d}z \\ \leq &C\sum_{k=1}^{\infty} \frac{\omega (2^{-k})}{ \vert 2^{k+3}\sqrt{n}Q \vert ^{2n}} \int _{(2^{k+3}\sqrt{n}Q)^{2}} \bigl(\Pi _{i=1}^{2} \bigl\vert b_{i}(y_{i})-\lambda _{i} \bigr\vert \bigl\vert f_{i}(y_{i}) \bigr\vert \bigr) \,\mathrm{d}y_{1}\,\mathrm{d}y_{2} \\ \leq &C\mu (x)^{2}\sum_{k=1}^{\infty}k^{2} \omega \bigl(2^{-k}\bigr) \Vert b_{1} \Vert _{\mathrm{Lip}_{ \beta _{1},\mu}} \Vert b_{2} \Vert _{\mathrm{Lip}_{\beta _{2},\mu}}M_{\beta _{1},\mu ,s}(f_{1}) (x)M_{ \beta _{2},\mu ,s}(f_{2}) (x) \\ \leq &C\mu (x)^{2} \int _{0}^{1}\frac{\omega (t)}{t}\biggl(1+\log{ \frac{1}{t}}\biggr)^{2}dt \Vert b_{1} \Vert _{\mathrm{Lip}_{\beta _{1},\mu}} \Vert b_{2} \Vert _{\mathrm{Lip}_{ \beta _{2},\mu}}M_{\beta _{1},\mu ,s}(f_{1}) (x)M_{\beta _{2},\mu ,s}(f_{2}) (x) \\ \leq &C\mu (x)^{2} \Vert b_{1} \Vert _{\mathrm{Lip}_{\beta _{1},\mu}} \Vert b_{2} \Vert _{\mathrm{Lip}_{ \beta _{2},\mu}}M_{\beta _{1},\mu ,s}(f_{1}) (x)M_{\beta _{2},\mu ,s}(f_{2}) (x). \end{aligned}$$

This, together with the estimates for $K_{1}$, $K_{2}$, $K_{3}$, $K_{4}$ gives

$$\begin{aligned} M_{\delta}^{\sharp}\bigl[T_{\Pi \vec{b}}(f_{1},f_{2}) \bigr](x) \leq& C\mu (x)^{2} \Vert b_{1} \Vert _{\mathrm{Lip}_{\beta _{1},\mu}} \Vert b_{2} \Vert _{\mathrm{Lip}_{\beta _{2},\mu}} \\ &{}\times \bigl[M_{ \beta ,\mu ,s}\bigl(T(f_{1},f_{2})\bigr) (x) +M_{\beta _{1},\mu ,s}(f_{1}) (x)M(f_{2}) (x) \bigr]\\ &{}+ \Vert b_{1} \Vert _{\mathrm{Lip}_{\beta _{1},\mu}}\mu (x)^{1+{\beta _{1}}/{n}}M_{ \beta _{1}} \bigl(T_{b_{2}}^{2}(f_{1},f_{2}) \bigr) (x) \\ &{}+ \Vert b_{2} \Vert _{\mathrm{Lip}_{\beta _{2}, \mu}}\mu (x)^{1+{\beta _{2}}/{n}}M_{\beta _{2}}\bigl(T_{b_{1}}^{1}(f_{1},f_{2}) \bigr) (x). \end{aligned}$$

Thus we finish the proof of Lemma 3.2. □

Proof of Theorem 1.2

Similarly as the proof of Theorem 1.1, since $\mu \in A_{1}$, by Lemma 2.1 and Lemma 3.2, for $0<\delta <1/2$,

$$\begin{aligned} \bigl\Vert T_{\Pi \vec{b}}(f_{1},f_{2}) \bigr\Vert _{L^{r}(\mu ^{1-2r})} \leq & C\|M_{ \delta}(T_{\Pi \vec{b}}(f_{1},f_{2}) \|_{L^{r}(\mu ^{1-2r})} \\ \leq & C \|M_{\delta}^{\sharp}(T_{\Pi \vec{b}}(f_{1},f_{2}) \|_{L^{r}( \mu ^{1-2r})} \\ \leq &C \bigl\Vert \mu ^{2}M_{\beta ,\mu ,s} \bigl(T(f_{1},f_{2})\bigr) \bigr\Vert _{L^{r}(\mu ^{1-2r})} \\ &{} +C \bigl\Vert \mu ^{2}M_{\beta _{1},\mu ,s}(f_{1})M_{\beta _{2},\mu ,s}(f_{2}) \bigr\Vert _{L^{r}(\mu ^{1-2r})} \\ &{} +C \bigl\Vert \mu ^{1+{\beta _{1}}/{n}}M_{\beta _{1}}\bigl(T_{b_{2}}^{2}(f_{1},f_{2}) \bigr) \bigr\Vert _{L^{r}(\mu ^{1-2r})} \\ &{} +C \bigl\Vert \mu ^{1+{\beta _{2}}/{n}}M_{\beta _{2}}\bigl(T_{b_{1}}^{1}(f_{1},f_{2}) \bigr) \bigr\Vert _{L^{r}(\mu ^{1-2r})} \\ :=&V_{1}+V_{2}+V_{3}+V_{4}. \end{aligned}$$

For $V_{1}$, by Lemma 2.3 and Theorem A, we have

$$\begin{aligned} \bigl\Vert \mu ^{2}M_{\beta ,\mu ,s}\bigl(T(f_{1},f_{2}) \bigr) \bigr\Vert _{L^{r}(\mu ^{1-2r})} =& \bigl\Vert M_{\beta ,\mu ,s} \bigl(T(f_{1},f_{2})\bigr) \bigr\Vert _{L^{r}(\mu )} \\ \leq &C \bigl\Vert T(f_{1},f_{2}) \bigr\Vert _{L^{p}(\mu )} \\ \leq &C \Vert f_{1} \Vert _{L^{p_{1}}(\mu )} \Vert f_{2} \Vert _{L^{p_{2}}(\mu )}. \end{aligned}$$

For $V_{2}$, since $1/{r_{1}}+1/{r_{2}}=1/r$, by Hölder’s inequality and Lemma 2.3, we get

$$\begin{aligned} \bigl\Vert \mu ^{2}M_{\beta _{1},\mu ,s}(f_{1})M_{\beta _{2},\mu ,s}(f_{2}) \bigr\Vert _{L^{r}( \mu ^{1-2r})} =& \bigl\Vert M_{\beta _{1},\mu ,s}(f_{1})M_{\beta _{2},\mu ,s}(f_{2}) \bigr\Vert _{L^{r}(\mu )} \\ \leq &C \bigl\Vert M_{\beta _{1},\mu ,s}(f_{1}) \bigr\Vert _{L^{r_{1}}(\mu )} \bigl\Vert M_{ \beta _{2},\mu ,s}(f_{2}) \bigr\Vert _{L^{r_{2}}(\mu )} \\ \leq &C \Vert f_{1} \Vert _{L^{p_{1}}(\mu )} \Vert f_{2} \Vert _{L^{p_{2}}(\mu )}. \end{aligned}$$

For the term $V_{3}$, by Theorem 1.1 and Lemma 2.4, let $1/r=1/{l_{1}}-{\beta _{1}}/n$, and then $1+r/{{l_{1}}'}=r-{r\beta _{1}}/n>1$, $1/{l_{1}}=1/p-{\beta _{2}}/n$. Since $\mu \in A_{1}$, then $\mu ^{1-r+{r\beta _{1}}/n}\in A_{1+r/{{l_{1}}'}}$, then we have

$$\begin{aligned} \bigl\Vert \mu ^{1+{\beta _{1}}/{n}}M_{\beta _{1}}\bigl(T_{b_{2}}^{2}(f_{1},f_{2}) \bigr) \bigr\Vert _{L^{r}(\mu ^{1-2r})} =& \bigl\Vert M_{\beta _{1}} \bigl(T_{b_{2}}^{2}(f_{1},f_{2}) \bigr) \bigr\Vert _{L^{r}(\mu ^{1-r+{r\beta _{1}}/n})} \\ \leq &C \bigl\Vert T_{b_{2}}^{2}(f_{1},f_{2}) \bigr\Vert _{L^{l_{1}}(\mu ^{1-l_{1}})} \\ \leq &C \Vert b_{2} \Vert _{\mathrm{Lip}_{\beta _{2},\mu}} \Vert f_{1} \Vert _{L^{p_{1}}(\mu )} \Vert f_{2} \Vert _{L^{p_{2}}(\mu )}. \end{aligned}$$

Similarly as the discussion of $V_{3}$, we have

$$ V_{4}\leq C \Vert b_{1} \Vert _{\mathrm{Lip}_{\beta _{1},\mu}} \Vert f_{1} \Vert _{L^{p_{1}}(\mu )} \Vert f_{2} \Vert _{L^{p_{2}}(\mu )}.$$

By combining the estimates of $V_{1}$, $V_{2}$, $V_{3}$, $V_{4}$, we finish the proof of Theorem 1.2. □