We borrow some ideas from [19]. Since the proof of Theorem 1.1 follows from similar steps in [22], we omit the proof. We just give three key lemmas.
Let \(u, v\in C^{\infty }([0,\infty ))\) be such that \(|u'(t)|\le Ct ^{-1}\), \(|v'(t)|\le Ct^{-1}\), and
$$ \chi _{[2,\infty )}(t)\le u(t)\le \chi _{[1,\infty )}(t),\qquad \chi _{[1,2]}(t)\le v(t)\le \chi _{[1/2,3]}(t). $$
For simplicity, we denote
$$\begin{aligned}& K_{u, \eta }(x,y_{1},\ldots ,y_{m})=K(x,y_{1}, \ldots ,y_{m})u\biggl(\frac{|x-y _{1}|+\cdots +|x-y_{m}|}{\eta }\biggr), \\& K_{v, \eta }(x,y_{1},\ldots ,y_{m})=K(x,y_{1}, \ldots ,y_{m})v\biggl(\frac{|x-y _{1}|+\cdots +|x-y_{m}|}{\eta }\biggr), \end{aligned}$$
and
$$\begin{aligned}& U_{\eta }(\vec{f}) (x)= \int _{({\mathbb{R}}^{n})^{m}}K_{u, \eta }(x,y _{1},\ldots ,y_{m})\prod_{i=1}^{m}f_{i}(y_{i}) \,dy_{1}\cdots \,dy_{m}, \\& V_{\eta }(\vec{f}) (x)= \int _{({\mathbb{R}}^{n})^{m}}K_{v, \eta }(x,y _{1},\ldots ,y_{m})\prod_{i=1}^{m}f_{i}(y_{i}) \,dy_{1}\cdots \,dy_{m}. \end{aligned}$$
Then we define the maximal operators
$$ U^{*}(\vec{f}) (x)=\sup_{\eta >0} \bigl\vert U_{\eta }(\vec{f}) (x) \bigr\vert \quad \mbox{and}\quad V^{*}(\vec{f}) (x)=\sup_{\eta >0} \bigl\vert V_{\eta }(\vec{f}) (x) \bigr\vert . $$
It is easy to get \(T^{*}(\vec{f})\le U^{*}(\vec{f})(x)+V^{*}(\vec{f})(x)\). Next, we show that the functions \(K_{u, \eta }\) and \(K_{v,\eta }\) satisfy some smoothness properties.
Lemma 3.1
For any
\(j=0,1,2,\ldots ,m\), we have
$$\begin{aligned} & \bigl\vert K_{u, \eta }(y_{0},\ldots ,y_{j}, \ldots ,y_{m})-K_{u, \eta }\bigl(y _{0},\ldots ,y'_{j},\ldots ,y_{m}\bigr) \bigr\vert \\ &\quad \lesssim \frac{\omega (\frac{|y_{j}-y_{j}'|}{|y_{0}-y_{1}|+ \cdots +|y_{0}-y_{m}|} )}{(|y_{0}-y_{1}|+\cdots +|y_{0}-y_{m}|)^{mn}} +\frac{|y_{j}-y_{j}'|}{(|y_{0}-y_{1}|+\cdots +|y_{0}-y_{m}|)^{mn+1}} \end{aligned}$$
and
$$\begin{aligned} & \bigl\vert K_{v, \eta }(y_{0},\ldots ,y_{j}, \ldots ,y_{m})-K_{v, \eta }\bigl(y _{0},\ldots ,y'_{j},\ldots ,y_{m}\bigr) \bigr\vert \\ &\quad \lesssim \frac{\omega (\frac{|y_{j}-y_{j}'|}{|y_{0}-y_{1}|+ \cdots +|y_{0}-y_{m}|} )}{(|y_{0}-y_{1}|+\cdots +|y_{0}-y_{m}|)^{mn}} +\frac{|y_{j}-y_{j}'|}{(|y_{0}-y_{1}|+\cdots +|y_{0}-y_{m}|)^{mn+1}} \end{aligned}$$
whenever
\(|y_{j}-y_{j}'|\leq \frac{1}{m+1}\max_{0\leq j \leq m} |y _{0}-y_{j}|\).
Proof
We just give the estimate for \(K_{u,\eta }\), since \(K_{v,\eta }\) can be estimated in a similar way with slight modifications. Without loss of generality, assuming that \(j=0\), we estimate
$$\begin{aligned}& \bigl\vert K_{u, \eta }(y_{0},y_{1},\ldots ,y_{m})-K_{u, \eta }\bigl(y'_{0},y _{1},\ldots ,y_{m}\bigr) \bigr\vert \\& \quad = \biggl\vert K(y_{0},y_{1},\ldots ,y_{m})u\biggl(\frac{|y_{0}-y_{1}|+\cdots +|y _{0}-y_{m}|}{\eta }\biggr) \\& \qquad {}-K\bigl(y_{0}',y_{1},\ldots ,y_{m}\bigr)u\biggl(\frac{|y'_{0}-y_{1}|+\cdots +|y'_{0}-y _{m}|}{\eta }\biggr) \biggr\vert \\& \quad = \biggl\vert \bigl[K(y_{0},y_{1},\ldots ,y_{m})-K\bigl(y'_{0},y_{1},\ldots ,y_{m}\bigr)\bigr]u\biggl(\frac{|y'_{0}-y _{1}|+\cdots +|y'_{0}-y_{m}|}{\eta }\biggr) \\& \qquad {}-K(y_{0},y_{1},\ldots ,y_{m}) \\& \qquad {}\times\biggl[u \biggl(\frac{|y'_{0}-y_{1}|+\cdots +|y'_{0}-y _{m}|}{\eta }\biggr)-u\biggl(\frac{|y_{0}-y_{1}|+\cdots +|y_{0}-y_{m}|}{\eta }\biggr)\biggr] \biggr\vert \\& \quad \lesssim \bigl\vert K(y_{0},y_{1},\ldots ,y_{m})-K\bigl(y'_{0},y_{1},\ldots ,y _{m}\bigr) \bigr\vert \\& \qquad {}+ \biggl\vert K(y_{0},y_{1},\ldots ,y_{m}) \\& \qquad {}\times\biggl[u\biggl(\frac{|y'_{0}-y_{1}|+ \cdots +|y'_{0}-y_{m}|}{\eta }\biggr)-u\biggl( \frac{|y_{0}-y_{1}|+\cdots +|y_{0}-y _{m}|}{\eta }\biggr)\biggr] \biggr\vert \\& \quad \doteq I +\mathit{II}. \end{aligned}$$
Since \(|y_{0}-y_{0}'|\leq \frac{1}{m+1}\max_{0\leq j \leq m} |y_{0}-y _{j}|\), by (1.3) we have
$$ I \lesssim \frac{1}{(|y_{0}-y_{1}|+\cdots +|y_{0}-y_{m}|)^{mn}} \omega \biggl(\frac{|y_{0}-y_{0}'|}{|y_{0}-y_{1}|+\cdots +|y_{0}-y _{m}|} \biggr). $$
It remains to estimate II. By the mean value theorem there is \(t_{0} \) between \(\frac{|y'_{0}-y_{1}|+\cdots +|y'_{0}-y_{m}|}{\eta }\) and \(\frac{|y_{0}-y_{1}|+\cdots +|y_{0}-y_{m}|}{\eta }\) such that
$$\begin{aligned}& \biggl\vert u\biggl(\frac{|y'_{0}-y_{1}|+\cdots +|y'_{0}-y_{m}|}{\eta }\biggr)-u\biggl(\frac{|y _{0}-y_{1}|+\cdots +|y_{0}-y_{m}|}{\eta } \biggr) \biggr\vert \\& \quad = \bigl\vert u'(t_{0}) \bigr\vert \biggl\vert \frac{|y'_{0}-y_{1}|+\cdots +|y'_{0}-y_{m}|}{ \eta }- \frac{|y_{0}-y_{1}|+\cdots +|y_{0}-y_{m}|}{\eta } \biggr\vert \\& \quad \leq \frac{1}{t_{0}} \frac{ || y'_{0}-y_{1}|-|y_{0}-y_{1}|| + \cdots + || y'_{0}-y_{m}|-|y_{0}-y_{m}| | }{\eta } \\& \quad \lesssim \frac{1}{t_{0}}\frac{m|y_{0}-y'_{0}|}{\eta }. \end{aligned}$$
Again, since \(|y_{0}-y_{0}'|\lesssim \frac{1}{m+1}\max_{0\leq j \leq m} |y_{0}-y_{j}|\), we have
$$\begin{aligned} \bigl\vert y'_{0}-y_{1} \bigr\vert +\cdots + \bigl\vert y'_{0}-y_{m} \bigr\vert &= \bigl\vert y_{0}-y_{1}+y'_{0}-y_{0} \bigr\vert + \cdots + \bigl\vert y_{0}-y_{m}+y'_{0}-y_{0} \bigr\vert \\ &\geq \vert y_{0}-y_{1} \vert +\cdots + \vert y_{0}-y_{m} \vert -m \bigl\vert y_{0}-y'_{0} \bigr\vert \\ &\geq \vert y_{0}-y_{1} \vert +\cdots + \vert y_{0}-y_{m} \vert -\frac{m}{m+1} \max _{0\leq j \leq m} \bigl\vert y_{0}-y'_{0} \bigr\vert \\ &\geq \frac{ \vert y_{0}-y_{1} \vert +\cdots + \vert y_{0}-y_{m} \vert }{m+1}. \end{aligned}$$
From this,
$$\begin{aligned} \frac{1}{t_{0}} &\lesssim \max \biggl\{ \frac{\eta }{|y'_{0}-y_{1}|+\cdots +|y'_{0}-y _{m}|}, \frac{\eta }{|y_{0}-y_{1}|+\cdots +|y_{0}-y_{m}|} \biggr\} \\ &\lesssim \frac{\eta }{|y_{0}-y_{1}|+\cdots +|y_{0}-y_{m}|}, \end{aligned}$$
and therefore
$$\begin{aligned} &\biggl|u\biggl(\frac{|y'_{0}-y_{1}|+\cdots +|y'_{0}-y_{m}|}{\eta }\biggr)-u\biggl(\frac{|y _{0}-y_{1}|+\cdots +|y_{0}-y_{m}|}{\eta }\biggr)\biggr| \\ &\quad \lesssim \frac{|y_{0}-y'_{0}|}{|y_{0}-y_{1}|+\cdots +|y_{0}-y_{m}|}. \end{aligned}$$
This, together with the size condition (1.2), implies that
$$ \mathit{II} \lesssim \frac{|y_{0}-y'_{0}|}{(|y_{0}-y_{1}|+\cdots +|y_{0}-y_{m}|)^{mn+1}}. $$
This ends the proof of Lemma 3.1. □
Lemma 3.2
Let
\(\frac{1}{p}=\frac{1}{p_{1}} +\cdots +\frac{1}{p_{2}}\)
and
\(\vec{\omega }\in A_{\vec{p}}\). Then we have:
-
(1)
If
\(1< p_{1}, \ldots, p_{m}<\infty \), then
$$ \bigl\Vert U^{*}\vec{f} \bigr\Vert _{L^{p}(\nu _{\vec{\omega }})}\lesssim \prod _{i=1}^{m}\|f_{i} \|_{L^{p_{i}}(\omega _{i})}. $$
-
(2)
If
\(1\leq p_{1}, \ldots, p_{m}<\infty \), then
$$ \bigl\Vert U^{*}\vec{f} \bigr\Vert _{L^{p,\infty }(\nu _{\vec{\omega }})}\lesssim \prod _{i=1}^{m}\|f_{i} \|_{L^{p_{i}}(\omega _{i})}. $$
Similar estimates hold for
\(V^{*}\).
Proof
Lemma 3.2 is a consequence of Lemma 2.3, Lemma 3.1, and Theorem 1.3 in [3]. □
For the maximal truncated operator \(T^{*}\) and a collection of locally integrable functions \(\vec{b}=(b_{1},\ldots ,b_{m})\), we define the commutator \(T^{*}_{\varSigma \vec{b}}\) by
$$ T^{*}_{\varSigma \vec{b}}(f_{1},\ldots ,f_{m})=\sum _{j=1}^{m}T_{\vec{b}} ^{*j}(\vec{f}), $$
where
$$ T_{\vec{b}}^{*j}(\vec{f}) (x)=\bigl[b_{j},T^{*} \bigr]_{j}(\vec{f}) (x)= \sup_{\delta >0} \bigl\vert b_{j}(x)T_{\delta }(f_{1},\ldots ,f_{m}) (x)-T_{ \delta }(f_{1},\ldots ,b_{j}f_{j}, \ldots ,f_{m}) (x) \bigr\vert . $$
Next, we give the key lemma, which plays important role in the proof of Theorem 1.1. We just consider the case \(m=2\) for simplicity.
Lemma 3.3
Let
T
be an
m-linear
ω-CZO with
\(\omega \in \operatorname{Dini}(1)\). Then we have:
-
(i)
If
\(b_{1}\in \operatorname{Lip}_{\beta _{1}} \)
and
\(b_{2}\in \operatorname{Lip}_{\beta _{2}} \)
with
\(0<\beta _{1}\), \(\beta _{2}<1\), \(0 <\delta <\epsilon < 1/ 2\), then
$$ \begin{aligned}[b] &M^{\sharp }_{\delta }T^{*}_{\varPi \vec{b}}(f_{1},f_{2}) (x) \\ &\quad \lesssim \Biggl\{ \prod_{i=1}^{2} \|b_{i}\|_{\operatorname{Lip}_{\beta _{i}}} M_{ \epsilon ,\beta } \bigl(T^{*}(f_{1},f_{2}) \bigr) (x)+\|b_{1}\|_{\operatorname{Lip}_{\beta _{1}}} M_{\epsilon ,\beta _{1}} \bigl(T_{\vec{b}}^{*2}(f_{1},f_{2})\bigr) (x) \\ &\qquad {}+\|b_{2}\|_{\operatorname{Lip}_{\beta _{1}}} M_{\epsilon ,\beta _{2}} \bigl(T_{\vec{b}} ^{*1}(f_{1},f_{2})\bigr) (x) \\ &\qquad {}+\prod_{i=1}^{2}\|b_{i} \|_{\operatorname{Lip}_{\beta _{i}}}M _{1,\beta _{1}}(f_{1}) (x) M_{1,\beta _{2}}(f_{2}) (x) \Biggr\} . \end{aligned} $$
(3.1)
-
(ii)
Suppose that
\(b_{j}\in \operatorname{Lip}_{\beta }\), \(j=1,2 \), \(0<\beta <1\), and
\(0 <\delta <\epsilon < 1/ 2 <1<n/ \beta \). Then
$$ \begin{aligned}[b] &M^{\sharp }_{\delta }T_{\varSigma \vec{b}}^{*}(f_{1},f_{2}) (x) \\ &\quad \lesssim \|b\|_{\operatorname{Lip}_{\beta }} \bigl\{ M_{\epsilon ,\beta } \bigl(T^{*}(f _{1},f_{2})\bigr) (x)+ M_{1,\beta }(f_{1}) (x)M (f_{2}) (x) \\ &\qquad {}+ M_{1,\beta }(f _{2}) (x)M (f_{1}) (x) \bigr\} . \end{aligned} $$
(3.2)
Proof
(i) We need two auxiliary maximal operators. The key role in the proof is played by the maximal operators \(U_{\varPi b}^{*}\) and \(V_{\varPi b}^{*}\) defined by
$$\begin{aligned}& \begin{aligned} U_{\varPi b}^{*}(\vec{f}) (x) &=\sup _{\eta >0} \bigl\vert \bigl[b_{1},[b_{2},U_{ \eta }]_{2} \bigr]_{1} (\vec{f}) (x) \bigr\vert \\ &=\sup_{\eta >0} \Biggl\vert \int _{({\mathbb{R}}^{n})^{m}}K_{u, \eta }(x,y_{1},y_{2}) \prod_{j=1} ^{2}\bigl(b_{j}(x)-b_{j}(y_{j}) \bigr)\prod_{i=1}^{2}f_{i}(y_{i}) \,dy_{1}\,dy_{2} \Biggr\vert , \end{aligned} \\& \begin{aligned} V_{\varPi b}^{*}(\vec{f}) (x)) &= \sup _{\eta >0} \bigl\vert \bigl[b_{1},[b_{2},V_{ \eta }]_{2} \bigr]_{1} (\vec{f}) (x) \bigr\vert \\ &=\sup_{\eta >0} \Biggl\vert \int _{({\mathbb{R}}^{n})^{2}}K_{v, \eta }(x,y_{1},y_{2}) \prod_{j=1} ^{2}b_{j}(x)-b_{j}(y_{j})) \prod_{i=1}^{2}f_{i}(y_{i}) \,dy_{1}\,dy_{2} \Biggr\vert . \end{aligned} \end{aligned}$$
It is easy to get that \(T^{*}_{\varPi b}(\vec{f})\le U_{\varPi b}^{*}( \vec{f})(x)+V_{\varPi b}^{*}(\vec{f})(x)\). We need to prove (3.1) for \(U^{*}_{\varPi \vec{b}}\) and \(V^{*}_{\varPi \vec{b}}\). We just give the proof for \(U^{*}_{\varPi \vec{b}}\), since the proof for \(V^{*}_{\varPi \vec{b}}\) is almost the same. Fix \(x\in \mathbb{R}^{n} \) and denote by \(Q=Q(x_{Q},l)\) the cube centered at \(x_{Q}\) and containing x with side length l. Denote \(c=\sup_{\eta >0}| c_{\eta }|\) and \(\lambda _{i}={(b_{i})}_{Q^{*}}=\frac{1}{|Q^{*}|}\int _{Q^{*}}b_{i}(y)\,dy\), where \(Q^{*}=8 \sqrt{n}Q\). For any \(z\in \mathbb{R}^{n}\), we have
$$\begin{aligned} \bigl\vert U^{*}_{\varPi \vec{b}}(f_{1},f_{2}) (z)- c \bigr\vert &\leq \bigl\vert \bigl(b_{1}(z)-\lambda _{1}\bigr) \bigl(b _{2}(z)-\lambda _{2} \bigr)U^{*}(f_{1},f_{2}) (z) \bigr\vert \\ &\quad {} +\sup_{\eta } \bigl\vert \bigl(b_{1}(z)- \lambda _{1}\bigr)[b_{2},U_{\eta }]_{2}(f _{1},f_{2}) (z) \bigr\vert \\ &\quad {} +\sup_{\eta } \bigl\vert \bigl(b_{2}(z)- \lambda _{2}\bigr)[b_{1},U_{\eta }]_{1}(f _{1},f_{2}) (z) \bigr\vert \\ &\quad {} + \Bigl\vert U^{*}\bigl((b_{1}-\lambda _{1})f_{1}, (b_{2}-\lambda _{2})f_{2} \bigr) (z)- \sup_{\eta >0}| c_{\eta }| \Bigr\vert . \end{aligned}$$
Thus we have
$$\begin{aligned} & \biggl(\frac{1}{|Q|} \int _{Q} \bigl\vert \bigl\vert U^{*}_{\varPi \vec{b}}(f_{1},f_{2}) (z) \bigr\vert ^{ \delta }- \vert c \vert ^{\delta } \bigr\vert \,dz \biggr)^{\frac{1}{\delta }} \\ &\quad \leq \biggl(\frac{1}{|Q|} \int _{Q} \Bigl\vert U^{*}_{\varPi \vec{b}}(f_{1},f_{2}) (z)- \sup_{\eta >0} \vert c_{\eta } \vert \Bigr\vert ^{\delta }\,dz \biggr)^{\frac{1}{\delta }} \\ &\quad \leq \biggl(\frac{1}{|Q|} \int _{Q} \bigl\vert \bigl(b_{1}(z)-\lambda _{1}\bigr) \bigl(b_{2}(z)- \lambda _{2} \bigr)U^{*}(f_{1},f_{2}) (z) \bigr\vert ^{\delta }\,dz \biggr)^{\frac{1}{ \delta }} \\ &\qquad {} + \biggl(\frac{1}{|Q|} \int _{Q} \bigl\vert \bigl(b_{1}(z)-\lambda _{1}\bigr)\bigl[b_{2},U ^{*}\bigr]_{2}(f_{1},f_{2}) (z) \bigr\vert ^{\delta }\,dz \biggr)^{\frac{1}{\delta }} \\ &\qquad {} + \biggl(\frac{1}{|Q|} \int _{Q} \bigl\vert \bigl(b_{2}(z)-\lambda _{2}\bigr)\bigl[b_{1},U ^{*}\bigr]_{1}(f_{1},f_{2}) (z) \bigr\vert ^{\delta }\,dz \biggr)^{\frac{1}{\delta }} \\ &\qquad {} + \biggl(\frac{1}{|Q|} \int _{Q}\sup_{\eta >0} \bigl\vert U_{\eta }\bigl((b_{1}- \lambda _{1})f_{1}, (b_{2}-\lambda _{2})f_{2}\bigr) (z)- c_{\eta } \bigr\vert ^{\delta }\,dz \biggr) ^{\frac{1}{\delta }} \\ &\quad \doteq T_{1}+T_{2}+T_{3}+T_{4}. \end{aligned}$$
By Hölder’s inequality,
$$\begin{aligned} T_{1} &\lesssim \prod_{i=1}^{2} \|b_{i}\|_{\operatorname{Lip}_{\beta _{i}}} \biggl(\frac{1}{|Q|^{1-\frac{ \delta \beta }{n}}} \int _{Q} \bigl\vert U^{*}(f_{1},f_{2}) (z) \bigr\vert ^{\delta }\,dz \biggr) ^{\frac{1}{\delta }} \\ &\lesssim \prod_{i=1}^{2} \|b_{i}\|_{\operatorname{Lip}_{\beta _{i}}} M_{\epsilon , \beta } \bigl(U^{*}(f_{1},f_{2}) \bigr) (x). \end{aligned}$$
In a similar way, we can prove that
$$ T_{2}+T_{3} \lesssim \|b_{1} \|_{\operatorname{Lip}_{\beta _{1}}} M_{\epsilon ,\beta _{1}} \bigl(\bigl[b_{2},U^{*} \bigr]_{2}(f_{1},f_{2})\bigr) (x)+ \|b_{2}\|_{\operatorname{Lip}_{\beta _{2}}} M_{\epsilon ,\beta _{2}} \bigl(\bigl[b_{1},U^{*} \bigr]_{1}(f_{1},f_{2})\bigr) (x). $$
It remains to estimate the last term \(T_{4}\). Take now \(c_{\eta }= U _{\eta }((b_{1}-\lambda _{1})f_{1}^{\infty },(b_{2}-\lambda _{2})f_{2} ^{\infty })(x)\). Then \(T_{4}\leq T_{41}+T_{42}+T_{43}+T_{44}\), where
$$\begin{aligned}& T_{41}= \biggl(\frac{1}{|Q|} \int _{Q} \bigl\vert U^{*}\bigl((b_{1}- \lambda _{1})f_{1} ^{0}, (b_{2}-\lambda _{2})f_{2}^{0}\bigr) (z) \bigr\vert ^{\delta }\,dx \biggr)^{\frac{1}{ \delta }}; \\& T_{42}= \biggl(\frac{1}{|Q|} \int _{Q}\sup_{\eta } \bigl\vert U_{\eta }\bigl((b_{1}- \lambda _{1})f_{1}^{0}, (b_{2}-\lambda _{2})f_{2}^{\infty }\bigr) (z) \bigr\vert ^{ \delta }\,dz \biggr)^{\frac{1}{\delta }}; \\& T_{43}= \biggl(\frac{1}{|Q|} \int _{Q} \sup_{\eta } \bigl\vert U_{\eta }\bigl((b_{1}- \lambda _{1})f_{1}^{\infty }, (b_{2}-\lambda _{2})f_{2}^{0}\bigr) (z) \bigr\vert ^{ \delta }\,dz \biggr)^{\frac{1}{\delta }}; \\& T_{44}= \biggl(\frac{1}{|Q|} \int _{Q} \sup_{\eta } \bigl\vert U_{\eta }\bigl((b_{1}- \lambda _{1})f_{1}^{\infty }, (b_{2}-\lambda _{2})f_{2}^{\infty }\bigr) (z) \\& \hphantom{T_{44}= {}}{}- U _{\eta }\bigl((b_{1}-\lambda _{1})f_{1}^{\infty }, (b_{2}-\lambda _{2})f_{2} ^{\infty }\bigr) (x) \bigr\vert ^{\delta }\,dz \biggr)^{\frac{1}{\delta }}. \end{aligned}$$
By the Kolmogorov inequality and by Lemma 3.2,
$$\begin{aligned} T_{41} & \lesssim \bigl\Vert U^{*} \bigl((b_{1}-\lambda _{1})f_{1}^{0}, (b_{2}-\lambda _{2})f_{2}^{0}\bigr) \bigr\Vert _{L^{1/2,\infty }(Q, \frac{dx}{|Q|})} \\ &\lesssim \frac{1}{|Q|} \int _{Q} \bigl\vert (b_{1}-\lambda _{1})f_{1}^{0}(z) \bigr\vert \,dz \frac{1}{|Q|} \int _{Q} \bigl\vert (b_{2}-\lambda _{2})f_{2}^{0}(z) \bigr\vert \,dz \\ &\lesssim \prod_{i=1}^{2} \|b_{i}\|_{\operatorname{Lip}_{\beta _{i}}}M_{1,\beta _{i}}(f _{i}) (x). \end{aligned}$$
Next, by Hölder’s inequality and by the size condition (1.2),
$$\begin{aligned} T_{42} &\leq \frac{1}{|Q|} \int _{Q}\sup_{\eta } \bigl\vert U_{\eta }\bigl((b_{1}-\lambda _{1})f_{1}^{0}, (b_{2}-\lambda _{2})f_{2}^{\infty }\bigr) (z) \bigr\vert \,dz \\ &\lesssim \frac{1}{|Q|} \int _{Q} \int _{Q^{*}} \int _{\mathbb{R}^{n}\setminus Q^{*}}\frac{|(b_{1}(y_{1})-\lambda _{1})f _{1}^{0}(y_{1})||(b_{2}(y_{2})-\lambda _{2})f_{2}^{\infty }(y_{2}) |\,dy _{1}\,dy_{2}}{(|z-y_{1}|+|z-y_{2}|)^{2n}}\,dz \\ &\lesssim \|b_{1}\|_{\operatorname{Lip}_{\beta _{1}}}M_{1,\beta _{1}}(f_{1}) (x)|Q| \sum_{k=1}^{\infty } \int _{2^{k+3 }\sqrt{n}Q\setminus 2^{k+2} \sqrt{n}Q}\frac{|f_{2}(y_{2})(b_{2}(y_{2})-\lambda _{2})|\,dy_{2}}{|y _{2}-x_{Q}|^{2n}} \\ &\lesssim \|b_{1}\|_{\operatorname{Lip}_{\beta _{1}}}M_{1,\beta _{1}}(f_{1}) (x) \|b _{2}\|_{\operatorname{Lip}_{\beta _{2}}}M_{1,\beta _{2}}(f_{2}) (x). \end{aligned}$$
The operator \(T_{43}\) can be estimated in the same way. Finally, we estimate \(T_{44}\). By Lemma 3.1 we have
$$\begin{aligned} T_{44} \lesssim& \frac{1}{|Q|} \int _{Q}\sup_{\eta } \bigl\vert U_{\eta }\bigl((b_{1}- \lambda _{1})f_{1}^{\infty }, (b_{2}-\lambda _{2})f_{2}^{\infty }\bigr) (z) \\ &{}- U _{\eta }\bigl((b_{1}-\lambda _{1})f_{1}^{\infty }, (b_{2}-\lambda _{2})f_{2} ^{\infty }\bigr) (x) \bigr\vert \,dz \\ \lesssim& \frac{1}{|Q|} \int _{Q} \int _{(\mathbb{R}^{n}\setminus Q^{*})^{2}}\sup_{\eta } \bigl\vert K_{\mu ,\eta }(z, \vec{y})-K_{\mu ,\eta }(x_{Q},\vec{y}) \bigr\vert \prod_{i=1}^{2} \bigl\vert \bigl(b_{i}(y_{i})- \lambda _{i} \bigr)f_{i}^{\infty }(y_{i}) \bigr\vert \,dy_{1}\,dy_{2}\,dz \\ \lesssim& \frac{1}{|Q|} \int _{Q} \int _{(\mathbb{R}^{n}\setminus Q^{*})^{2}}\frac{1}{(|x_{Q}-y_{1}|+|x _{Q}-y_{2}|)^{2n}}\omega \biggl( \frac{|z-x_{Q}|}{|x_{Q}-y_{1}|+|x_{Q}-y_{2}|} \biggr) \\ &{} \times \prod_{i=1}^{2} \bigl\vert \bigl(b_{i}(y_{i})-\lambda _{i} \bigr)f_{i}^{ \infty }(y_{i}) \bigr\vert \,dy_{1}\,dy_{2}\,dz \\ &{} + \frac{1}{|Q|} \int _{Q} \int _{(\mathbb{R}^{n}\setminus Q^{*})^{2}}\frac{|z-x _{Q}|}{(|x_{Q}-y_{1}|+|x_{Q}-y_{2}|)^{2n+1}} \prod _{i=1}^{2} \bigl\vert \bigl(b_{i}(y _{i})-\lambda _{i}\bigr)f_{i}^{\infty }(y_{i}) \bigr\vert \,dy_{1}\,dy_{2}\,dz \\ \lesssim& \frac{1}{|Q|} \int _{Q}\sum_{k=1}^{\infty } \int _{(2^{k+3 }\sqrt{n}Q\setminus 2^{k+2}\sqrt{n}Q)^{2}}\frac{1}{(|2^{k+3 } \sqrt{n}Q|)^{2}}\omega \bigl(2^{-k} \bigr) \\ &{}\times\prod_{i=1}^{2} \bigl\vert \bigl(b_{i}(y_{i})-\lambda _{i} \bigr)f_{i}^{\infty }(y_{i}) \bigr\vert \,dy_{1}\,dy_{2}\,dz \\ &{} + \frac{1}{|Q|} \int _{Q} \int _{\mathbb{R}^{n}\setminus Q^{*}}\frac{|z-x _{Q}|^{1/ 2}}{|x_{Q}-y_{1}|^{n+1/ 2}} \bigl\vert \bigl(b_{1}(y_{1})-\lambda _{1}\bigr)f _{1}^{\infty }(y_{1}) \bigr\vert \,dy_{1} \\ &{} \times \int _{\mathbb{R}^{n}\setminus Q^{*}}\frac{|z-x_{Q}|^{1/ 2}}{|x_{Q}-y_{2}|^{n+1/ 2}} \bigl\vert \bigl(b_{2}(y_{2})-\lambda _{2} \bigr)f_{2}^{\infty }(y_{2}) \bigr\vert \,dy_{2}\,dz \\ \lesssim& \sum_{k=1}^{\infty } \frac{1}{(|2^{k+3 }\sqrt{n}Q|)^{2}} \int _{(2^{k+3 }\sqrt{n}Q\setminus 2^{k+2}\sqrt{n}Q)^{2}}\omega \bigl(2^{-k}\bigr) \prod _{i=1}^{2} \bigl\vert \bigl(b_{i}(y_{i})- \lambda _{i}\bigr)f_{i}^{\infty }(y_{i}) \bigr\vert \, d \vec{y} \\ &{} + \sum_{k=1}^{\infty }2^{-\frac{k}{2}} \frac{1}{|2^{k+3 } \sqrt{n}Q| } \int _{2^{k+3 }\sqrt{n}Q\setminus 2^{k+2}\sqrt{n}Q} \bigl\vert \bigl(b _{1}(y_{1})- \lambda _{1}\bigr)f_{1}^{\infty }(y_{1}) \bigr\vert \,dy_{1} \\ &{} \times \sum_{k=1}^{\infty }2^{-\frac{k}{2}} \frac{1}{|2^{k+3 } \sqrt{n}Q| } \int _{2^{k+3 }\sqrt{n}Q\setminus 2^{k+2}\sqrt{n}Q} \bigl\vert \bigl(b _{2}(y_{2})- \lambda _{2}\bigr)f_{2}^{\infty }(y_{2}) \bigr\vert \,dy_{2} \\ \lesssim& \|b_{1}\|_{\operatorname{Lip}_{\beta _{1}}}M_{1,\beta _{1}}(f_{1}) (x) \|b _{2}\|_{\operatorname{Lip}_{\beta _{2}}}M_{1,\beta _{2}}(f_{2}) (x). \end{aligned}$$
Combining the obtained estimates proves (3.1).
(ii) It is sufficient to prove (3.2) for the operator with only one symbol. Set
$$\begin{aligned} U^{*1}_{\vec{b}}(\vec{f}) (x) &=\sup_{\eta >0} \bigl\vert b(x)U_{\eta }(f_{1},f _{2}) (x)-U_{\eta }(bf_{1},f_{2}) (x) \bigr\vert \\ &=\sup_{\eta >0} \bigl\vert \bigl(b(x)-\lambda \bigr)U_{\eta }(f_{1},f_{2}) (x)-U_{\eta } \bigl((b- \lambda )f_{1},f_{2}\bigr) (x) \bigr\vert , \end{aligned}$$
where \(\lambda =b_{Q^{*}}=\frac{1}{|Q^{*}|}\int _{Q^{*}}b(y)\,dy\). Let \(c=\sup_{\eta >0}|c_{\eta }|\). Then
$$\begin{aligned} & \biggl(\frac{1}{|Q|} \int _{Q} \bigl\vert \bigl\vert U^{*1}_{\vec{b}}(f_{1},f_{2}) (z) \bigr\vert ^{ \delta }-| c|^{\delta } \bigr\vert \,dz \biggr)^{\frac{1}{\delta }} \\ &\quad \lesssim \biggl(\frac{1}{|Q|} \int _{Q} \Bigl\vert U^{*1}_{\vec{b}}(f_{1},f_{2}) (z)-\sup_{\eta >0}| c_{\eta }| \Bigr\vert ^{ \delta }\,dz \biggr)^{\frac{1}{\delta }} \\ &\quad \lesssim \biggl(\frac{1}{|Q|} \int _{Q} \bigl\vert \bigl(b(z)-\lambda \bigr)U^{*}(f_{1},f _{2}) (z) \bigr\vert ^{\delta }\,dz \biggr)^{\frac{1}{\delta }} \\ &\qquad {}+ \biggl(\frac{1}{|Q|} \int _{Q}\sup_{\eta >0} \bigl\vert U_{\eta }\bigl((b-\lambda )f_{1}, f_{2}\bigr) (z)- c_{ \eta } \bigr\vert ^{\delta }\,dz \biggr)^{\frac{1}{\delta }} \\ &\quad =: (P_{1}+P_{2}). \end{aligned}$$
By Hölder’s inequality,
$$\begin{aligned} P_{1} &\lesssim \|b\|_{\operatorname{Lip}_{\beta }} \biggl( \frac{1}{|Q|^{1-\frac{ \delta \beta }{n}}} \int _{Q} \bigl\vert U^{*}(f_{1},f_{2}) (z) \bigr\vert ^{\delta }\,dz \biggr) ^{\frac{1}{\delta }} \\ &\lesssim \|b\|_{\operatorname{Lip}_{\beta }} M_{\epsilon ,\beta } \bigl(U^{*}(f_{1},f _{2})\bigr) (x). \end{aligned}$$
Set \(c_{\eta }= U_{\eta }((b-\lambda )f_{1}^{\infty },f_{2}^{\infty })(x)\). Then \(P_{2}\leq P_{21}+P_{22}+P_{23}+P_{24}\), where
$$\begin{aligned}& P_{21}= \biggl(\frac{1}{|Q|} \int _{Q} \bigl\vert U^{*}\bigl((b-\lambda )f_{1}^{0}, f _{2}^{0}\bigr) (z) \bigr\vert ^{\delta }\,dx \biggr)^{\frac{1}{\delta }}; \\& P_{22}= \biggl(\frac{1}{|Q|} \int _{Q} \sup_{\eta } \bigl\vert U_{\eta }\bigl((b-\lambda )f_{1}^{0}, f_{2}^{\infty }\bigr) (z) \bigr\vert ^{\delta }\,dz \biggr)^{\frac{1}{\delta }}; \\& P_{23}= \biggl(\frac{1}{|Q|} \int _{Q} \sup_{\eta } \bigl\vert U_{\eta }\bigl((b-\lambda )f_{1}^{\infty }, f_{2}^{0}\bigr) (z) \bigr\vert ^{\delta }\,dz \biggr)^{\frac{1}{\delta }}; \\& P_{24}= \biggl(\frac{1}{|Q|} \int _{Q} \sup_{\eta } \bigl\vert U_{\eta }\bigl((b-\lambda )f_{1}^{\infty }, f_{2}^{\infty }\bigr) (z)- U_{\eta }\bigl((b-\lambda )f_{1} ^{\infty }, f_{2}^{\infty }\bigr) (x) \bigr\vert ^{\delta }\,dz \biggr)^{\frac{1}{\delta }}. \end{aligned}$$
By the Kolmogorov inequality and by Lemma 3.2,
$$\begin{aligned} P_{21} & \lesssim \bigl\Vert U^{*}\bigl((b- \lambda )f_{1}^{0}, f_{2}^{0}\bigr) \bigr\Vert _{L^{1/2, \infty }(Q, \frac{dx}{|Q|})} \\ &\lesssim \frac{1}{|Q|} \int _{Q} \bigl\vert (b-\lambda )f_{1}^{0}(z) \bigr\vert \,dz \frac{1}{|Q|} \int _{Q} \bigl\vert f_{2}^{0}(z) \bigr\vert \,dz \\ &\lesssim \|b\|_{\operatorname{Lip}_{\beta }} \bigl\vert Q^{*} \bigr\vert ^{\beta / n}\frac{1}{|Q|} \int _{Q} \bigl\vert f_{1}^{0}(z) \bigr\vert \,dz \frac{1}{|Q|} \int _{Q} \bigl\vert f_{2}^{0}(z) \bigr\vert \,dz \\ &\lesssim \|b\|_{\operatorname{Lip}_{\beta }}M_{1,\beta }(f_{1}) (x)M (f_{2}) (x). \end{aligned}$$
Next, by the size condition (1.2),
$$\begin{aligned} P_{22} &\lesssim \frac{1}{|Q|} \int _{Q}\sup_{\eta } \bigl\vert U_{\eta }\bigl((b-\lambda )f_{1}^{0}, f_{2}^{\infty }\bigr) (z) \bigr\vert \,dz \\ &\lesssim \frac{1}{|Q|} \int _{Q} \int _{Q^{*}} \int _{(Q^{*})^{c}}\frac{1}{(|z-y _{1}|+|z-y_{2}|)^{2n}} \bigl\vert \bigl(b(y_{1})-\lambda \bigr)f_{1}(y_{1}) \bigr\vert \bigl\vert f_{2}(y_{2}) \bigr\vert \,dy _{2} \,dy_{1} \,dz \\ &\lesssim \int _{Q^{*}} \bigl\vert \bigl(b(y_{1})-\lambda \bigr)f_{1}(y_{1}) \bigr\vert \,dy_{1} \int _{\mathbb{R}^{n}\setminus Q^{*}}\frac{|f_{2}(y_{2})|\,dy_{2}}{|x _{Q}-y_{2}|^{2n}} \\ &\lesssim \|b\|_{\operatorname{Lip}_{\beta }} |Q|^{\frac{\beta }{n}} \int _{Q^{*}}f _{1}(y_{1})\,dy_{1} \sum_{k=1}^{\infty }\frac{1}{|2^{k+1}Q|^{2}} \int _{2^{k+1}Q^{*}\setminus 2^{k}Q^{*}} \bigl\vert f_{2}(y_{2}) \bigr\vert \,dy_{2} \\ &\lesssim \|b\|_{\operatorname{Lip}_{\beta }}M_{1,\beta }(f_{1}) (x) M(f_{2}) (x) . \end{aligned}$$
Similarly,
$$ P_{23} \lesssim \|b\|_{\operatorname{Lip}_{\beta }}M_{1,\beta }(f_{1}) (x) M(f_{2}) (x). $$
By Lemma 3.1 we obtain
$$\begin{aligned} P_{24} \lesssim& \frac{1}{|Q|} \int _{Q}\sup_{\eta } \bigl\vert U_{\eta }\bigl((b- \lambda )f_{1}^{\infty }, f_{2}^{\infty }\bigr) (z)- U_{\eta }\bigl((b-\lambda )f _{1}^{\infty }, f_{2}^{\infty }\bigr) (x) \bigr\vert \,dz \\ \lesssim& \frac{1}{|Q|} \int _{Q} \int _{(\mathbb{R}^{n}\setminus Q^{*})^{2}}\sup_{\eta } \bigl\vert K_{\mu , \eta }(z,\vec{y})-K_{\mu ,\eta }(x_{Q},\vec{y}) \bigr\vert \bigl\vert \bigl(b(y_{1})-\lambda \bigr) \bigr\vert \prod _{i=1}^{2} \bigl\vert f_{i}^{\infty }(y_{i}) \bigr\vert \,dy_{1}\,dy_{2}\,dz \\ \lesssim& \frac{1}{|Q|} \int _{Q} \int _{(\mathbb{R}^{n}\setminus Q^{*})^{2}}\frac{\omega (\frac{|z-x _{Q}|}{|z-y_{1}|+|z-y_{2}|})}{(|z-y_{1}|+|z-y_{2}|)^{2n}} \bigl\vert \bigl(b(y_{1})- \lambda \bigr) \bigr\vert \prod _{i=1}^{2} \bigl\vert f_{i}^{\infty }(y_{i}) \bigr\vert \,dy_{1}\,dy_{2}\,dz \\ &{} +\frac{1}{|Q|} \int _{Q} \int _{(\mathbb{R}^{n}\setminus Q^{*})^{2}}\frac{|z-x _{Q}|}{(|z-y_{1}|+|z-y_{2}|)^{2n+1}} \bigl\vert \bigl(b(y_{1})-\lambda \bigr) \bigr\vert \prod _{i=1} ^{2} \bigl\vert f_{i}^{\infty }(y_{i}) \bigr\vert \,dy_{1}\,dy_{2}\,dz \\ \lesssim& \frac{1}{|Q|} \int _{Q}\sum_{k=1}^{\infty } \int _{(2^{k+3 }\sqrt{n}Q\setminus 2^{k+2}\sqrt{n}Q)^{2}}\frac{1}{(|2^{k+3 } \sqrt{n}Q|)^{2}}\omega \bigl(2^{-k} \bigr) \bigl\vert \bigl(b(y_{1})-\lambda \bigr) \bigr\vert \\ &{} \times\prod _{i=1}^{2} \bigl\vert f _{i}^{\infty }(y_{i}) \bigr\vert \,dy_{1}\,dy_{2}\,dz+ \frac{1}{|Q|} \int _{Q} \int _{\mathbb{R}^{n}\setminus Q^{*}}\frac{|z-x _{Q}|^{1/ 2}}{|x_{Q}-y_{1}|^{n+1/ 2}} \bigl\vert \bigl(b(y_{1})-\lambda \bigr)f_{1}^{ \infty }(y_{1}) \bigr\vert \,dy_{1} \\ &{} \times \int _{\mathbb{R}^{n}\setminus Q^{*}}\frac{|z-x_{Q}|^{1/ 2}}{|x_{Q}-y_{2}|^{n+1/ 2}} \bigl\vert f_{2}^{\infty }(y_{2}) \bigr\vert \,dy_{2}\,dz \\ \lesssim& \|b\|_{\operatorname{Lip}_{\beta }} \sum_{k=1}^{\infty } \frac{\omega (2^{-k})}{(|2^{k+3 }\sqrt{n}Q|)^{1- \beta / n}} \int _{2^{k+3 }\sqrt{n}Q} \bigl\vert f_{1}^{\infty }(y_{1}) \bigr\vert \,dy_{1}\frac{1}{|2^{k}Q ^{*}|} \int _{2^{k+3 }\sqrt{n}Q} \bigl\vert f_{2}^{\infty }(y_{2}) \bigr\vert \,dy_{2} \\ &{} + \sum_{k=1}^{\infty }2^{-\frac{k}{2}} \frac{1}{|2^{k+3 } \sqrt{n}Q|^{1-\beta / n }} \int _{2^{k+3 }\sqrt{n}Q\setminus 2^{k+2}\sqrt{n}Q} \bigl\vert f_{1}^{\infty }(y_{1}) \bigr\vert \,dy_{1} \\ &{} \times \sum_{k=1}^{\infty }2^{-\frac{k}{2}} \frac{1}{|2^{k+3 } \sqrt{n}Q| } \int _{2^{k+3 }\sqrt{n}Q\setminus 2^{k+2}\sqrt{n}Q} \bigl\vert f _{2}^{\infty }(y_{2}) \bigr\vert \,dy_{2} \\ \lesssim& \|b\|_{\operatorname{Lip}_{\beta }}M_{1,\beta }(f_{1}) (x) M(f_{2}) (x). \end{aligned}$$
Thus we finish the proof of (3.2). Then Lemma 3.3 is proved. □