•
Estimate for
\(|E_{2}|\)
. Employing the linearity of
\(T_{\pi b}\) and the atomic decomposition of
\(h_{1}\), we may get
$$\begin{aligned} &T_{\pi b}(h_{1}, g_{2}) (x) \\ &\quad= \int_{(\mathbb{R}^{n})^{m}}\prod_{j=1}^{2} \bigl(b_{j}(x)-b_{j}(y_{j}) \bigr)K(x, y_{1},y_{2})h_{1}(y_{1}) g_{2}(y_{2})\,dy_{1}\,dy_{2} \\ &\quad=\sum_{S_{1,l_{1}}}\sum_{Q_{1,k_{1}}\subset S_{1,l_{1}}} \lambda_{Q_{1,k_{1}}} \bigl(b_{1}(x)b_{2}(x)T(a_{Q_{1,k_{1}}},g_{2}) (x)-b_{2}(x)T(b_{1}a_{Q_{1,k_{1}}},g_{2}) (x) \\ &\qquad{} -b_{1}(x)T(a_{Q_{1,k_{1}}}, b_{2}g_{2}) (x)+T(b_{1}a_{Q_{1,k_{1}}}, b_{2}g_{2}) (x) \bigr) \\ &\quad=\sum_{S_{1,l_{1}}}\sum_{Q_{1,k_{1}}\subset S_{1,l_{1}}} \lambda_{Q_{1,k_{1}}} \bigl(b_{1}(x)-b_{1,Q_{1,k_{1}}} \bigr) \bigl(b_{2}(x)-b_{2,Q_{1,k_{1}}} \bigr)T(a_{Q_{1,k_{1}}},g_{2}) (x) \\ &\qquad{} -\sum_{S_{1,l_{1}}}\sum _{Q_{1,k_{1}}\subset S_{1,l_{1}}}\lambda_{Q_{1,k_{1}}} \bigl(b_{2}(x)-b_{2,Q_{1,k_{1}}} \bigr)T\bigl((b_{1}-b_{1,Q_{1,k_{1}}})a_{Q_{1,k_{1}}},g_{2} \bigr) (x) \\ &\qquad{} -\sum_{S_{1,l_{1}}}\sum _{Q_{1,k_{1}}\subset S_{1,l_{1}}}\lambda_{Q_{1,k_{1}}} \bigl(b_{1}(x)-b_{1,Q_{1,k_{1}}} \bigr)T\bigl(a_{Q_{1,k_{1}}}, (b_{2}-b_{2,Q_{1,k_{1}}})g_{2} \bigr) (x) \\ &\qquad{} +\sum_{S_{1,l_{1}}}\sum _{Q_{1,k_{1}}\subset S_{1,l_{1}}}\lambda_{Q_{1,k_{1}}}T\bigl((b_{1}-b_{1,Q_{1,k_{1}}})a_{Q_{1,k_{1}}}, (b_{2}-b_{2,Q_{1,k_{1}}})g_{2}\bigr) (x) \\ &\quad=:I_{2,1}(x)+I_{2,2}(x)+I_{2,3}(x)+I_{2,4}(x). \end{aligned}$$
Thus
$$\begin{aligned} |E_{2}|={}&\bigl\vert \bigl\{ x\in \mathbb{R}^{n}:\bigl\vert T_{\pi b}(g_{1},h_{2}) (x)\bigr\vert >\lambda/4 \bigr\} \bigr\vert \\ \leq{}&\bigl\vert \bigl\{ x\in \mathbb{R}^{n}:\bigl\vert I_{2,1}(x)\bigr\vert >\lambda/16\bigr\} \bigr\vert +\bigl\vert \bigl\{ x\in \mathbb{R}^{n}:\bigl\vert I_{2,2}(x)\bigr\vert > \lambda/16\bigr\} \bigr\vert \\ &{} +\bigl\vert \bigl\{ x\in \mathbb{R}^{n}:\bigl\vert I_{2,3}(x)\bigr\vert >\lambda/16\bigr\} \bigr\vert +\bigl\vert \bigl\{ x\in \mathbb{R}^{n}:\bigl\vert I_{2,4}(x)\bigr\vert > \lambda/16\bigr\} \bigr\vert \\ =:{}&|E_{2,1}|+|E_{2,2}|+|E_{2,3}|+|E_{2,4}|. \end{aligned}$$
By the definition of
\(I_{2,1}\) and the moment condition of
\(H^{1}\)-atoms, we have
$$\begin{aligned} I_{2,1}(x)={}&\sum_{S_{1,l_{1}}}\sum _{Q_{1,k_{1}}\subset S_{1,l_{1}}}\lambda_{Q_{1,k_{1}}} \bigl(b_{1}(x)-b_{1,Q_{1,k_{1}}} \bigr) \bigl(b_{2}(x)-b_{2,Q_{1,k_{1}}} \bigr) \\ &{} \times \iint_{(\mathbb{R}^{n})^{2}} \bigl(K(x,y_{1},y_{2})-K(x,c_{1,k_{1}},y_{2}) \bigr)a_{Q_{1,k_{1}}}(y_{1})g_{2}(y_{2}) \,dy_{1}\,dy_{2}. \end{aligned}$$
Putting the above identity into the definition of
\(|E_{2,1}|\) and noting that
\(\Vert g_{2} \Vert _{L^{\infty}(\mathbb{R}^{n})}\leq (\gamma\lambda )^{1/2}\),
\(\mathbb{R}^{n}\backslash S^{*}\subset\bigcup_{i=1}^{\infty}\mathscr {R}_{1, k_{1}}^{i}\), together with the Chebyshev inequality and condition (
1.3), we have
$$\begin{aligned} |E_{2,1}|\leq{}&\frac{16}{\lambda}\sum _{S_{1,l_{1}}}\sum_{Q_{1,k_{1}}\subset S_{1,l_{1}}}| \lambda_{Q_{1,k_{1}}}| \int _{(S^{*})^{c}} \iint_{(\mathbb{R}^{n})^{2}}\bigl|b_{1}(x)-b_{1,Q_{1,k_{1}}}\bigr|\bigl|b_{2}(x)-b_{2,Q_{1,k_{1}}}\bigr| \\ &{} \times \bigl|K(x,y_{1},y_{2})-K(x,c_{1,k_{1}},y_{2})\bigr| \bigl|a_{Q_{1,k_{1}}}(y_{1})\bigr|\bigl|g_{2}(y_{2})\bigr| \,dy_{1}\,dy_{2}\,dx \\ \leq{}& CC_{0}\lambda^{1/2}\gamma^{-1/2}\sum _{S_{1,l_{1}}}\sum_{Q_{1,k_{1}}\subset S_{1,l_{1}}}| \lambda_{Q_{1,k_{1}}}|\sum_{i=1}^{\infty} \int_{\mathscr{R}_{1, k_{1}}^{i}} \iint_{(\mathbb{R}^{n})^{2}}\bigl|b_{1}(x)-b_{1,Q_{1,k_{1}}}\bigr| \\ &{} \times\bigl|b_{2}(x)-b_{2,Q_{1,k_{1}}}\bigr|\frac{|a_{1, k_{1}}(y_{1})|}{(|x-y_{1}|+|x-y_{2}|)^{2n}}\omega \\ &{}\times \biggl(\frac {|y_{1}-c_{1,k_{1}}|}{|x-y_{1}|+|x-y_{2}|} \biggr)\,dy_{1}\,dy_{2}\,dx. \end{aligned}$$
(3.4)
By (
2.6) and the non-decreasing property of
ω, we have
$$\begin{aligned} |E_{2,1}|\leq{}& CC_{0}\gamma^{1/2} \lambda^{-1/2}\sum_{S_{1,l_{1}}}\sum _{Q_{1,k_{1}}\subset S_{1,l_{1}}}|\lambda _{Q_{1,k_{1}}}|\sum _{i=1}^{\infty} \int_{\mathscr{R}_{1, k_{1}}^{i}} \iint_{(\mathbb{R}^{n})^{2}}\bigl|b_{1}(x)-b_{1,Q_{1,k_{1}}}\bigr| \\ &{} \times\bigl|b_{2}(x)-b_{2,Q_{1,k_{1}}}\bigr|\frac{|a_{1, k_{1}}(y_{1})|}{(|x-y_{1}|+|x-y_{2}|)^{2n}}\omega \bigl(2^{-i}\bigr)\,dy_{1}\,dy_{2}\,dx \\ \leq{}& CC_{0}\gamma^{1/2}\lambda^{-1/2}\sum _{S_{1,l_{1}}}\sum_{Q_{1,k_{1}}\subset S_{1,l_{1}}}| \lambda_{Q_{1,k_{1}}}|\sum_{i=1}^{\infty} \int_{(S^{*})^{c}\cap\mathscr{R}_{1, k_{1}}^{i}} \iint _{(\mathbb{R}^{n})^{2}}\bigl|b_{1}(x)-b_{1,Q_{1,k_{1}}}\bigr| \\ &{} \times\bigl|b_{2}(x)-b_{2,Q_{1,k_{1}}}\bigr|\frac{|a_{1, k_{1}}(y_{1})|}{|x-y_{1}|^{n}}\omega \bigl(2^{-i}\bigr)\,dy_{1}\,dx \\ \leq{}& CC_{0}\gamma^{1/2}\lambda^{-1/2}\sum _{S_{1,l_{1}}}\sum_{Q_{1,k_{1}}\subset S_{1,l_{1}}}| \lambda_{Q_{1,k_{1}}}|\sum_{i=1}^{\infty} \int_{(S^{*})^{c}\cap\mathscr{R}_{1, k_{1}}^{i}} \int _{\mathbb{R}^{n}}\bigl|b_{1}(x)-b_{1,Q_{1,k_{1}}}\bigr| \\ &{} \times\bigl|b_{2}(x)-b_{2,Q_{1,k_{1}}}\bigr|\frac{|a_{1, k_{1}}(y_{1})|}{|2^{i+2}\sqrt{n}Q_{1,k_{1}}|}\omega \bigl(2^{-i}\bigr)\,dy_{1}\,dx \\ \leq{}& CC_{0}\gamma^{1/2}\lambda^{-1/2}\sum _{S_{1,l_{1}}}\sum_{Q_{1,k_{1}}\subset S_{1,l_{1}}}| \lambda_{Q_{1,k_{1}}}|\sum_{i=1}^{\infty} \omega\bigl(2^{-i}\bigr)\frac{1}{|2^{i+2}\sqrt{n}Q_{1,k_{1}}|} \\ &{} \times \int_{\mathscr{R}_{1, k_{1}}^{i}}\bigl|b_{1}(x)-b_{1,Q_{1,k_{1}}}\bigr|\bigl|b_{2}(x)-b_{2,Q_{1,k_{1}}}\bigr| \,dx. \end{aligned}$$
(3.5)
By the Hölder inequality, one obtains
$$\begin{aligned} &\frac{1}{|2^{i+2}\sqrt{n}Q_{1,k_{1}}|} \int_{\mathscr {R}_{1,k_{1}}^{i}}\bigl|b_{1}(x)-b_{1,Q_{1,k_{1}}}\bigr|\bigl|b_{2}(x)-b_{2,Q_{1,k_{1}}}\bigr| \,dx \\ &\quad\leq \biggl(\frac{1}{|2^{i+2}\sqrt{n}Q_{1,k_{1}}|} \int_{2^{i+2}\sqrt {n}Q_{1,k_{1}}}\bigl|b_{1}(x)-b_{1,Q_{1,k_{1}}}\bigr|^{2} \,dx \biggr)^{1/2} \\ &\qquad{} \times \biggl(\frac{1}{|2^{i+2}\sqrt{n}Q_{1,k_{1}}|} \int _{2^{i+2}\sqrt{n}Q_{1,k_{1}}}\bigl|b_{2}(x)-b_{2,Q_{1,k_{1}}}\bigr|^{2} \,dx \biggr)^{1/2} \\ &\quad\leq Ci\Vert b \Vert _{*}. \end{aligned}$$
(3.6)
Combining (
3.5) and (
3.6), we get
$$|E_{2,1}|\leq CC_{0}\gamma^{1/2} \lambda^{-1/2}\sum_{S_{1,l_{1}}}\sum _{Q_{1,k_{1}}\subset S_{1,l_{1}}}|\lambda _{Q_{1,k_{1}}}|\sum _{i=1}^{\infty}\omega\bigl(2^{-i}\bigr)i\leq CC_{0}\gamma^{1/2}\lambda^{-1/2}. $$
Now we begin to estimate
\(|E_{2,2}|\).
Similarly to our dealing with
\(|E_{2,1}|\), and together with the size condition of
\(H^{1}\)-atoms, it follows that
$$\begin{aligned} |E_{2,2}|\leq{}& CC_{0}\gamma^{1/2} \lambda^{-1/2}\sum_{S_{1,l_{1}}}\sum _{Q_{1,k_{1}}\subset S_{1,l_{1}}}|\lambda _{Q_{1,k_{1}}}|\sum _{i=1}^{\infty} \int_{(S^{*})^{c}\cap\mathscr {R}_{1, k_{1}}^{i}} \iint_{(\mathbb{R}^{n})^{2}}\bigl|b_{1}(y_{1})-b_{1,Q_{1,k_{1}}}\bigr| \\ &{} \times\bigl|b_{2}(x)-b_{2,Q_{1,k_{1}}}\bigr|\frac{|a_{Q_{1, k_{1}}}(y_{1})|}{(|x-y_{1}|+|x-y_{2}|)^{2n}}\omega \biggl(\frac {|y_{1}-c_{1,k_{1}}|}{|x-y_{1}|+|x-y_{2}|} \biggr)\,dy_{1}\,dy_{2}\,dx \\ \leq{}& CC_{0}\gamma^{1/2}\lambda^{-1/2}\sum _{S_{1,l_{1}}}\sum_{Q_{1,k_{1}}\subset S_{1,l_{1}}}| \lambda_{Q_{1,k_{1}}}|\sum_{i=1}^{\infty} \int_{(S^{*})^{c}\cap\mathscr{R}_{1, k_{1}}^{i}} \int _{\mathbb{R}^{n}}\bigl|b_{1}(y_{1})-b_{1,Q_{1,k_{1}}}\bigr| \\ &{} \times\bigl|b_{2}(x)-b_{2,Q_{1,k_{1}}}\bigr|\frac{1}{(|x-y_{1}|)^{n}|Q_{1, k_{1}}|}\omega \bigl(2^{-i}\bigr)\,dy_{1}\,dx \\ \leq{}& CC_{0}\gamma^{1/2}\lambda^{-1/2}\Vert b_{1} \Vert _{*}\sum_{S_{1,l_{1}}}\sum _{Q_{1,k_{1}}\subset S_{1,l_{1}}}|\lambda _{Q_{1,k_{1}}}|\sum _{i=1}^{\infty}\omega\bigl(2^{-i}\bigr) \frac {1}{(|2^{i+2}Q_{1,k_{1}}|)^{n}} \\ &{} \times \int_{(S^{*})^{c}\cap\mathscr{R}_{1, k_{1}}^{i}}\bigl|b_{2}(x)-b_{2,Q_{1,k_{1}}}\bigr|\,dx \\ \leq{}& CC_{0}\gamma^{1/2}\lambda^{-1/2}\Vert b_{1} \Vert _{*}\Vert b_{2} \Vert _{*}\sum _{S_{1,l_{1}}}\sum_{Q_{1,k_{1}}\subset S_{1,l_{1}}}|\lambda _{Q_{1,k_{1}}}|\sum_{i=1}^{\infty}\omega \bigl(2^{-i}\bigr)i \\ \leq{}& CC_{0}\gamma^{1/2}\lambda^{-1/2}. \end{aligned}$$
The estimate for
\(|E_{2,3}|\) is more complicated, and we need to split the domain of the variable
\(y_{2}\). First, similar to our dealing with
\(|E_{2,1}|\) in (
3.4) and (
3.5), we may get
$$\begin{aligned}[b] |E_{2,3}|\leq{}& CC_{0}\gamma^{1/2} \lambda^{-1/2}\sum_{S_{1,l_{1}}}\sum _{Q_{1,k_{1}}\subset S_{1,l_{1}}}|\lambda _{Q_{1,k_{1}}}|\sum _{i=1}^{\infty} \int_{(S^{*})^{c}\cap\mathscr {R}_{1, k_{1}}^{i}} \iint_{(\mathbb{R}^{n})^{2}}\bigl|b_{1}(x)-b_{1,Q_{1,k_{1}}}\bigr| \\ &{} \times\bigl|b_{2}(y_{2})-b_{2,Q_{1,k_{1}}}\bigr| \frac{|a_{Q_{1, k_{1}}}(y_{1})|}{(|x-y_{1}|+|x-y_{2}|)^{2n}}\omega \biggl(\frac {|y_{1}-c_{1,k_{1}}|}{|x-y_{1}|+|x-y_{2}|} \biggr)\,dy_{1} \,dy_{2}\,dx. \end{aligned} $$
Denote
\(\mathscr{R}_{1, k_{1}}^{h}=(2^{h+2}\sqrt{n}Q_{1, k_{1}})\backslash (2^{h+1}\sqrt{n}Q_{1, k_{1}})\) and recall that
\(Q_{1, k_{1}}^{*}=4\sqrt {n}Q_{1, k_{1}}\), then
$$y_{2}\in \mathbb{R}^{n}\subset \Biggl(\bigcup_{h=1}^{\infty} \mathscr{R}_{1, k_{1}}^{h} \Biggr)\cup Q_{1, k_{1}}^{*}. $$
Thus
\(|E_{2,3}|\) can be controlled by
$$\begin{aligned} &CC_{0}\gamma^{1/2}\lambda^{-1/2}\sum _{S_{1,l_{1}}}\sum_{Q_{1,k_{1}}\subset S_{1,l_{1}}}| \lambda_{Q_{1,k_{1}}}|\sum_{i=1}^{\infty} \int_{(S^{*})^{c}\cap\mathscr{R}_{1, k_{1}}^{i}} \int_{\bigcup _{i=1}^{\infty}\mathscr{R}_{1, k_{1}}^{h}} \int_{\mathbb{R}^{n}}\bigl|b_{1}(x)-b_{1,Q_{1,k_{1}}}\bigr| \\ &\qquad{} \times\bigl|b_{2}(y_{2})-b_{2,Q_{1,k_{1}}}\bigr| \frac{|a_{Q_{1, k_{1}}}(y_{1})|}{(|x-y_{1}|+|x-y_{2}|)^{2n}}\omega \biggl(\frac {|y_{1}-c_{1,k_{1}}|}{|x-y_{1}|+|x-y_{2}|} \biggr)\,dy_{1} \,dy_{2}\,dx \\ &\qquad{} +CC_{0}\gamma^{1/2}\lambda^{-1/2}\sum _{S_{1,l_{1}}}\sum_{Q_{1,k_{1}}\subset S_{1,l_{1}}}| \lambda_{Q_{1,k_{1}}}|\sum_{i=1}^{\infty} \int_{(S^{*})^{c}\cap\mathscr{R}_{1, k_{1}}^{i}} \int_{Q_{1, k_{1}}^{*}} \int_{\mathbb{R}^{n}}\bigl|b_{1}(x)-b_{1,Q_{1,k_{1}}}\bigr| \\ &\qquad{} \times\bigl|b_{2}(y_{2})-b_{2,Q_{1,k_{1}}}\bigr| \frac{|a_{Q_{1, k_{1}}}(y_{1})||}{(|x-y_{1}|+|x-y_{2}|)^{2n}}\omega \biggl(\frac {|y_{1}-c_{1,k_{1}}|}{|x-y_{1}|+|x-y_{2}|} \biggr)\,dy_{1} \,dy_{2}\,dx \\ &\quad=:\bigl|E_{2,3}^{1}\bigr|+\bigl|E_{2,3}^{2}\bigr|. \end{aligned}$$
For any
\(h\in\mathbb{N}\), if
\(y_{2}\in\mathscr{R}_{1, k_{1}}^{h}\), note that
\(y_{1}\in Q_{1,k_{1}}\), then
$$|x-y_{1}|+|x-y_{2}|\geq|y_{1}-y_{2}| \sim|y_{2}-c_{1,k_{1}}|\sim l_{2^{h+2}Q_{1,k_{1}}}. $$
On the other hand, for any
\(i\in\mathbb{N}\), if
\(x\in\mathscr {R}_{1, k_{1}}^{i}\) and
\(y_{1}\in Q_{1,k_{1}}\), then
$$ |x-y_{1}|+|x-y_{2}|\geq|x-y_{1}|\sim l_{2^{i+2}Q_{1,k_{1}}}. $$
(3.7)
By the geometric properties of
\(y_{1}\),
\(y_{2}\),
x above, we may obtain
$$\begin{aligned} &\bigl|E_{2,3}^{1}\bigr| \\ &\quad\leq CC_{0}\gamma^{1/2}\lambda^{-1/2}\sum _{S_{1,l_{1}}}\sum_{Q_{1,k_{1}}\subset S_{1,l_{1}}}|\lambda _{Q_{1,k_{1}}}|\sum_{i=1}^{\infty}\sum _{h=1}^{\infty } \int_{(S^{*})^{c}\cap\mathscr{R}_{1, k_{1}}^{i}} \int_{\mathscr{R}_{1, k_{1}}^{h}} \int_{\mathbb{R}^{n}}\bigl|b_{1}(x)-b_{1,Q_{1,k_{1}}}\bigr| \\ & \qquad{}\times\bigl|b_{2}(y_{2})-b_{2,Q_{1,k_{1}}}\bigr| \frac{|a_{Q_{1, k_{1}}}(y_{1})|}{(|x-y_{1}|+|x-y_{2}|)^{2n}}\omega \biggl(\frac {|y_{1}-c_{1,k_{1}}|}{|x-y_{1}|+|x-y_{2}|} \biggr)\,dy_{1} \,dy_{2}\,dx \\ &\quad\leq CC_{0}\gamma^{1/2}\lambda^{-1/2} \sum_{S_{1,l_{1}}}\sum_{Q_{1,k_{1}}\subset S_{1,l_{1}}}| \lambda_{Q_{1,k_{1}}}|\sum_{i=1}^{\infty}\sum _{h=1}^{\infty} \int_{(S^{*})^{c}\cap \mathscr{R}_{1, k_{1}}^{i}} \int_{\mathscr{R}_{1, k_{1}}^{h}} \int_{\mathbb{R}^{n}}\bigl|b_{1}(x)-b_{1,Q_{1,k_{1}}}\bigr| \\ &\qquad{} \times\bigl|b_{2}(y_{2})-b_{2,Q_{1,k_{1}}}\bigr| \frac{|a_{Q_{1, k_{1}}}(y_{1})|}{|2^{i+2}Q_{1,k_{1}}||2^{h+2}Q_{1,k_{1}}|}\omega \bigl(2^{-i}\bigr)^{1/2}\omega \bigl(2^{-h}\bigr)^{1/2}\,dy_{1}\,dy_{2} \,dx. \end{aligned}$$
(3.8)
It is easy to see that
$$\begin{aligned} \sum_{h=1}^{\infty}\omega \bigl(2^{-h}\bigr)^{1/2} \int_{\mathscr {R}_{1, k_{1}}^{h}}\frac {|b_{2}(y_{2})-b_{2,Q_{1,k_{1}}}|}{|2^{h+2}Q_{1,k_{1}}|}\,dy_{2}\leq C\sum _{h=1}^{\infty}\omega\bigl(2^{-h} \bigr)^{1/2}h\Vert b_{2} \Vert _{*}\leq C. \end{aligned}$$
(3.9)
Since
\(a(y_{1})\in L^{1}(\mathbb{R}^{n})\), putting the above estimate into (
3.8), we have
$$\begin{aligned} \bigl|E_{2,3}^{1}\bigr|&\leq CC_{0}\gamma^{1/2} \lambda^{-1/2}\sum_{S_{1,l_{1}}}\sum _{Q_{1,k_{1}}\subset S_{1,l_{1}}}|\lambda _{Q_{1,k_{1}}}|\sum _{i=1}^{\infty}\omega\bigl(2^{-i} \bigr)^{1/2} \int _{2^{i+2}Q_{1,k_{1}}}\frac {|b_{1}(x)-b_{1,Q_{1,k_{1}}}|}{|2^{i+2}Q_{1,k_{1}}|}\,dx \\ &\leq CC_{0}\gamma^{1/2}\lambda^{-1/2}\sum _{S_{1,l_{1}}}\sum_{Q_{1,k_{1}}\subset S_{1,l_{1}}}| \lambda_{Q_{1,k_{1}}}|\sum_{i=1}^{\infty} \omega\bigl(2^{-i}\bigr)^{1/2}i\Vert b_{1} \Vert _{*} \\ &\leq CC_{0}\gamma^{1/2}\lambda^{-1/2}. \end{aligned}$$
If
\(y_{2}\in Q_{1, k_{1}}^{*}\), note that
\(x\in(8\sqrt{n}Q_{1, k_{1}})^{c}\), then
$$|x-y_{1}|+|x-y_{2}|\geq|x-y_{2}|\geq Cl_{Q_{1,k_{1}}}. $$
By the definition of
\(|E_{2,3}^{2}|\) and (
3.7), we have
$$\begin{aligned} \bigl|E_{2,3}^{2}\bigr|\leq{}& CC_{0}\gamma^{1/2} \lambda^{-1/2}\sum_{S_{1,l_{1}}}\sum _{Q_{1,k_{1}}\subset S_{1,l_{1}}}|\lambda _{Q_{1,k_{1}}}|\sum _{i=1}^{\infty} \int_{(S^{*})^{c}\cap\mathscr {R}_{1, k_{1}}^{i}} \int_{Q_{1, k_{1}}^{*}} \int_{\mathbb{R}^{n}}\bigl|b_{1}(x)-b_{1,Q_{1,k_{1}}}\bigr| \\ &{} \times\bigl|b_{2}(y_{2})-b_{2,Q_{1,k_{1}}}\bigr|\frac{|a_{Q_{1, k_{1}}}(y_{1})|}{|2^{i+2}Q_{1,k_{1}}||Q_{1, k_{1}}^{*}|} \omega \bigl(2^{-i}\bigr)\,dy_{1}\,dy_{2}\,dx \\ \leq{}& CC_{0}\gamma^{1/2}\lambda^{-1/2}\sum _{S_{1,l_{1}}}\sum_{Q_{1,k_{1}}\subset S_{1,l_{1}}}| \lambda_{Q_{1,k_{1}}}|\sum_{i=1}^{\infty} \int_{2^{i+2}Q_{1,k_{1}}} \int_{\mathbb{R}^{n}}\bigl|b_{1}(x)-b_{1,Q_{1,k_{1}}}\bigr| \\ &{} \times\frac{|a_{Q_{1, k_{1}}}(y_{1})|}{|2^{i+2}Q_{1,k_{1}}|}\omega \bigl(2^{-i}\bigr) \,dy_{1}\,dx \\ \leq{}& CC_{0}\gamma^{1/2}\lambda^{-1/2}\sum _{S_{1,l_{1}}}\sum_{Q_{1,k_{1}}\subset S_{1,l_{1}}}| \lambda_{Q_{1,k_{1}}}|\sum_{i=1}^{\infty} \omega\bigl(2^{-i}\bigr)\frac{1}{|2^{i+2}Q_{1,k_{1}}|} \\ &{} \times \int_{2^{i+2}Q_{1,k_{1}}}\bigl|b_{1}(x)-b_{1,Q_{1,k_{1}}}\bigr|\,dx \\ \leq{}& CC_{0}\gamma^{1/2}\lambda^{-1/2}\sum _{S_{1,l_{1}}}\sum_{Q_{1,k_{1}}\subset S_{1,l_{1}}}| \lambda_{Q_{1,k_{1}}}|\sum_{i=1}^{\infty} \omega\bigl(2^{-i}\bigr)i\Vert b_{1} \Vert _{*} \\ \leq{}& CC_{0}\gamma^{1/2}\lambda^{-1/2}. \end{aligned}$$
Hence, we obtain
$$|E_{2,3}|\leq\bigl|E_{2,3}^{1}\bigr|+\bigl|E_{2,3}^{2}\bigr| \leq CC_{0}\gamma^{1/2}\lambda^{-1/2}. $$
Now we begin to consider
\(|E_{2,4}|\). Similarly,
$$\begin{aligned} |E_{2,4}|\leq{}& CC_{0}\gamma^{1/2} \lambda^{-1/2}\sum_{S_{1,l_{1}}}\sum _{Q_{1,k_{1}}\subset S_{1,l_{1}}}|\lambda _{Q_{1,k_{1}}}|\sum _{i=1}^{\infty} \int_{(S^{*})^{c}\cap\mathscr {R}_{1, k_{1}}^{i}} \iint_{(\mathbb{R}^{n})^{2}}\bigl|b_{1}(y_{1})-b_{1,Q_{1,k_{1}}}\bigr| \\ &{} \times\bigl|b_{2}(y_{2})-b_{2,Q_{1,k_{1}}}\bigr| \frac{|a_{Q_{1, k_{1}}}(y_{1})|}{(|x-y_{1}|+|x-y_{2}|)^{2n}}\omega \biggl(\frac {|y_{1}-c_{1,k_{1}}|}{|x-y_{1}|+|x-y_{2}|} \biggr)\,dy_{1} \,dy_{2}\,dx. \end{aligned}$$
Repeating the same steps as in the estimate of
\(|E_{2,3}|\), we have
$$\begin{aligned} |E_{2,4}|\leq{}& CC_{0}\gamma^{1/2} \lambda^{-1/2}\sum_{S_{1,l_{1}}}\sum _{Q_{1,k_{1}}\subset S_{1,l_{1}}}|\lambda _{Q_{1,k_{1}}}|\sum _{i=1}^{\infty} \int_{(S^{*})^{c}\cap\mathscr {R}_{1, k_{1}}^{i}} \int_{\bigcup_{i=1}^{\infty}\mathscr{R}_{1, k_{1}}^{h}} \int _{\mathbb{R}^{n}}\bigl|b_{1}(y_{1})-b_{1,Q_{1,k_{1}}}\bigr| \\ &{} \times\bigl|b_{2}(y_{2})-b_{2,Q_{1,k_{1}}}\bigr| \frac{|a_{Q_{1, k_{1}}}(y_{1})|}{(|x-y_{1}|+|x-y_{2}|)^{2n}}\omega \biggl(\frac {|y_{1}-c_{1,k_{1}}|}{|x-y_{1}|+|x-y_{2}|} \biggr)\,dy_{1} \,dy_{2}\,dx \\ &{} +CC_{0}\gamma^{1/2}\lambda^{-1/2}\sum _{S_{1,l_{1}}}\sum_{Q_{1,k_{1}}\subset S_{1,l_{1}}}| \lambda_{Q_{1,k_{1}}}|\sum_{i=1}^{\infty} \int_{(S^{*})^{c}\cap\mathscr{R}_{1, k_{1}}^{i}} \int_{Q_{1, k_{1}}^{*}} \int_{\mathbb{R}^{n}}\bigl|b_{1}(y_{1})-b_{1,Q_{1,k_{1}}}\bigr| \\ &{} \times\bigl|b_{2}(y_{2})-b_{2,Q_{1,k_{1}}}\bigr| \frac{|a_{Q_{1, k_{1}}}(y_{1})|}{(|x-y_{1}|+|x-y_{2}|)^{2n}}\omega \biggl(\frac {|y_{1}-c_{1,k_{1}}|}{|x-y_{1}|+|x-y_{2}|} \biggr)\,dy_{1} \,dy_{2}\,dx \\ =:{}&\bigl|E_{2,4}^{1}\bigr|+\bigl|E_{2,4}^{2}\bigr|. \end{aligned}$$
By the definition of
\(|E_{2,4}^{1}|\), one may obtain
$$\begin{aligned} \bigl|E_{2,4}^{1}\bigr|\leq{}& CC_{0}\gamma^{1/2} \lambda^{-1/2}\sum_{S_{1,l_{1}}}\sum _{Q_{1,k_{1}}\subset S_{1,l_{1}}}|\lambda _{Q_{1,k_{1}}}|\sum _{i=1}^{\infty}\sum_{h=1}^{\infty } \int_{(S^{*})^{c}\cap\mathscr{R}_{1, k_{1}}^{i}} \int_{\mathscr{R}_{1, k_{1}}^{h}} \int_{\mathbb{R}^{n}}\bigl|b_{1}(x)-b_{1,Q_{1,k_{1}}}\bigr| \\ &{} \times\bigl|b_{2}(y_{2})-b_{2,Q_{1,k_{1}}}\bigr| \frac{|a_{Q_{1, k_{1}}}(y_{1})|}{|x-y_{1}|^{n}|2^{h+2}Q_{1,k_{1}}|}\omega \biggl(\frac {y_{1}-c_{1,k_{1}}}{|x-y_{1}|} \biggr)^{1/2}\omega \bigl(2^{-h}\bigr)^{1/2}\,dy_{1}\,dy_{2} \,dx. \end{aligned}$$
By (
3.9), and taking the integral for
x first, we have
$$\begin{aligned} \bigl|E_{2,4}^{1}\bigr| \leq{}& CC_{0}\gamma^{1/2} \lambda^{-1/2}\sum_{S_{1,l_{1}}}\sum _{Q_{1,k_{1}}\subset S_{1,l_{1}}}|\lambda_{Q_{1,k_{1}}}|\sum _{i=1}^{\infty} \int_{\mathscr{R}_{1, k_{1}}^{i}} \int_{Q_{1,k_{1}}}\frac {|b_{1}(y_{1})-b_{1,Q_{1,k_{1}}}|}{ |Q_{1,k_{1}}||x-y_{1}|^{n}} \\ &{} \times\omega \biggl(\frac{y_{1}-c_{1,k_{1}}}{|x-y_{1}|} \biggr)^{1/2} \,dy_{1}\,dx \\ \leq{}& CC_{0}\gamma^{1/2}\lambda^{-1/2}\sum _{S_{1,l_{1}}}\sum_{Q_{1,k_{1}}\subset S_{1,l_{1}}}| \lambda_{Q_{1,k_{1}}}| \int _{Q_{1,k_{1}}}\frac{|b_{1}(y_{1})-b_{1,Q_{1,k_{1}}}|}{ |Q_{1,k_{1}}|}\,dy_{1} \\ \leq{}& CC_{0}\gamma^{1/2}\lambda^{-1/2}\sum _{S_{1,l_{1}}}\sum_{Q_{1,k_{1}}\subset S_{1,l_{1}}}| \lambda_{Q_{1,k_{1}}}|\Vert b_{1} \Vert _{*} \\ \leq{}& CC_{0}\gamma^{1/2}\lambda^{-1/2}. \end{aligned}$$
The estimate for
\(|E_{2,4}^{2}|\) is quite similar to
\(|E_{2,3}^{2}|\), we may get
\(|E_{2,4}^{2}|\leq CC_{0}\gamma^{1/2}\lambda^{-1/2}\).
•
Estimate for
\(|E_{4}|\)
.
$$\begin{aligned} T_{\Pi b}(h_{1},h_{2})={}&\bigl[b_{1},[b_{2}, T]_{2},\bigr]_{1}(h_{1},h_{2}) \\ ={}& \int_{(\mathbb{R}^{n})^{m}}\prod_{j=1}^{2} \bigl(b_{j}(x)-b_{j}(y_{j}) \bigr)K(x, y_{1},y_{2})h_{1}(y_{1}) h_{2}(y_{2})\,dy_{1}\,dy_{2} \\ ={}&\sum_{S_{1,l_{1}}}\sum_{Q_{1,k_{1}}\subset S_{1,l_{1}}} \sum_{S_{2,l_{2}}}\sum_{Q_{2,k_{2}}\subset S_{2,l_{2}}} \lambda_{Q_{1,k_{1}}}\lambda_{Q_{2,k_{2}}} \bigl(b_{1}(x)-b_{1,Q_{1,k_{1}}} \bigr) \bigl(b_{2}(x)-b_{2,Q_{1,k_{1}}} \bigr) \\ &{}\times T(a_{Q_{1,k_{1}}},a_{Q_{2,k_{2}}}) (x) \\ &{} -\sum_{S_{1,l_{1}}}\sum_{Q_{1,k_{1}}\subset S_{1,l_{1}}} \sum_{S_{2,l_{2}}}\sum_{Q_{2,k_{2}}\subset S_{2,l_{2}}} \lambda_{Q_{1,k_{1}}}\lambda_{Q_{2,k_{2}}} \bigl(b_{2}(x)-b_{2,Q_{1,k_{1}}} \bigr) \\ &{} \times T\bigl((b_{1}-b_{1,Q_{1,k_{1}}})a_{Q_{1,k_{1}}},a_{Q_{2,k_{2}}} \bigr) (x) \\ &{} -\sum_{S_{1,l_{1}}}\sum_{Q_{1,k_{1}}\subset S_{1,l_{1}}} \sum_{S_{2,l_{2}}}\sum_{Q_{2,k_{2}}\subset S_{2,l_{2}}} \lambda_{Q_{1,k_{1}}}\lambda_{Q_{2,k_{2}}} \bigl(b_{1}(x)-b_{1,Q_{1,k_{1}}} \bigr) \\ &{}\times T\bigl(a_{Q_{1,k_{1}}}, (b_{2}-b_{2,Q_{1,k_{1}}})a_{Q_{2,k_{2}}} \bigr) (x) \\ &{} +\sum_{S_{1,l_{1}}}\sum_{Q_{1,k_{1}}\subset S_{1,l_{1}}} \sum_{S_{2,l_{2}}}\sum_{Q_{2,k_{2}}\subset S_{2,l_{2}}} \lambda_{Q_{1,k_{1}}}\lambda_{Q_{2,k_{2}}} \\ &{} \times T\bigl((b_{1}-b_{1,Q_{1,k_{1}}})a_{Q_{1,k_{1}}}, (b_{2}-b_{2,Q_{1,k_{1}}})a_{Q_{2,k_{2}}}\bigr) (x) \\ =:{}&I_{4,1}(x)+I_{4,2}(x)+I_{4,3}(x)+I_{4,4}(x). \end{aligned}$$
Thus, we obtain
$$\begin{aligned} |E_{4}|={}&\bigl\vert \bigl\{ x\in \mathbb{R}^{n}/S^{*}:\bigl\vert T_{\pi b}(h_{1},h_{2}) (x)\bigr\vert > \lambda/4\bigr\} \bigr\vert \\ \leq{}&\bigl\vert \bigl\{ x\in \mathbb{R}^{n}/S^{*}:\bigl\vert I_{4,1}(x)\bigr\vert >\lambda/16\bigr\} \bigr\vert +\bigl\vert \bigl\{ x\in \mathbb{R}^{n}/S^{*}:\bigl\vert I_{4,2}(x)\bigr\vert >\lambda/16\bigr\} \bigr\vert \\ &{} +\bigl\vert \bigl\{ x\in \mathbb{R}^{n}/S^{*}:\bigl\vert I_{4,3}(x)\bigr\vert >\lambda/16\bigr\} \bigr\vert +\bigl\vert \bigl\{ x\in \mathbb{R}^{n}/S^{*}:\bigl\vert I_{4,4}(x)\bigr\vert >\lambda/16\bigr\} \bigr\vert \\ =:{}&|E_{4,1}|+|E_{4,2}|+|E_{4,3}|+|E_{4,4}|. \end{aligned}$$
Now we begin considering
\(|E_{4,1}|\). By the definition of
\(I_{4,1}(x)\), we can write
$$\begin{aligned} \bigl\vert I_{4,1}(x)\bigr\vert \leq{}&\sum _{S_{1,l_{1}}}\sum_{Q_{1,k_{1}}\subset S_{1,l_{1}}}\sum _{S_{2,l_{2}}}\sum_{Q_{2,k_{2}}\subset S_{2,l_{2}}}| \lambda_{Q_{1,k_{1}}}||\lambda _{Q_{2,k_{2}}}| \biggl| \iint_{(\mathbb{R}^{n})^{2}} \bigl(b_{1}(x)-b_{1,Q_{1,k_{1}}} \bigr) \\ &{} \times \bigl(b_{2}(x)-b_{2,Q_{1,k_{1}}} \bigr)K(x,y_{1},y_{2})a_{Q_{1,k_{1}}}(y_{1}) a_{Q_{2,k_{2}}}(y_{2})\,dy_{1}\,dy_{2} \biggr|. \end{aligned}$$
Fix for a moment
\(k_{1}\),
\(k_{2}\) and assume, without loss of generality, that
\(l(Q_{1,k_{1}})\leq l(Q_{2,k_{2}})\). By the moment condition of
\(H^{1}\)-atoms and the regularity condition (
1.3) of the kernel
K, we have
$$\begin{aligned} & \biggl| \int_{\mathbb{R}^{n}}K(x,y_{1},y_{2})a_{1,k_{1}}(y_{1}) \,dy_{1} \biggr| \\ &\quad= \biggl| \int_{\mathbb{R}^{n}} \bigl(K(x,y_{1},y_{2})-K(x,c_{1,k_{1}},y_{2}) \bigr)a_{1,k_{1}}(y_{1})\,dy_{1} \biggr| \\ &\quad\leq \biggl| \int_{\mathbb{R}^{n}}\frac{C_{0}}{(|x-y_{1}|+|x-y_{2}|)^{2n}} \omega \biggl(\frac{|y_{1}-c_{1,k_{1}}|}{|x-y_{1}|+|x-y_{2}|} \biggr)a_{Q_{1,k_{1}}}(y_{1})\,dy_{1} \biggr|. \end{aligned}$$
Recalling the definition of
\(\mathscr{R}_{1, k_{1}}^{i}\),
\(\mathscr {R}_{2, k_{2}}^{h}\), and note that
\(y_{1}\in Q_{1,k_{1}}\),
\(y_{2}\in Q_{2,k_{2}}\), it is obvious that, for any fixed
\(i, h, k_{1}, k_{2}\), if
\(x\in (S^{*})^{c}\cap\mathscr{R}_{1, k_{1}}^{i}\cap\mathscr{R}_{2, k_{2}}^{h}\), then we have
$$|x-y_{1}|\sim2^{i}l_{Q_{1,k_{1}}},\qquad |x-y_{2}| \sim2^{h}l_{Q_{2,k_{2}}}. $$
This and the non-decreasing property of
ω give
$$\begin{aligned} \frac{\omega (\frac{|y_{1}-c_{1,k_{1}}|}{|x-y_{1}|+|x-y_{2}|} )^{\frac{1}{2}}}{(|x-y_{1}|+|x-y_{2}|)^{n}}\leq \frac{\omega (\frac{l_{Q_{1,k_{1}}}}{|x-y_{1}|+|x-y_{2}|} )^{\frac {1}{2}}}{(|x-y_{1}|+|x-y_{2}|)^{n}}\lesssim \prod _{i=1}^{2}\frac{\omega (\frac{l_{Q_{i,k_{i}}}}{|x-y_{i}|} )^{\frac{1}{4}}}{|x-y_{i}|^{\frac{n}{2}}}\lesssim \frac{\omega (2^{-i})^{\frac{1}{4}}\omega(2^{-h})^{\frac {1}{4}}}{(2^{i}l_{Q_{1,k_{1}}}2^{h}l_{Q_{2,k_{2}}})^{\frac{n}{2}}}. \end{aligned}$$
By (
2.11), the Chebychev inequality and the estimate above, we control
\(|E_{4,1}|\) by
$$\begin{aligned} &\frac{CC_{0}}{\lambda}\sum_{S_{1,l_{1}}}\sum _{Q_{1,k_{1}}\subset S_{1,l_{1}}}\sum_{S_{2,l_{2}}}\sum _{Q_{2,k_{2}}\subset S_{2,l_{2}}}\sum_{i=1}^{\infty} \sum_{h=1}^{\infty}|\lambda _{Q_{1,k_{1}}}|| \lambda_{Q_{2,k_{2}}}| \int_{(S^{*})^{c}\cap\mathscr{R}_{1, k_{1}}^{i}\cap\mathscr{R}_{2, k_{2}}^{h}} \\ &\qquad{} \times \iint_{(\mathbb{R}^{n})^{2}}\bigl|b_{1}(x)-b_{1,Q_{1,k_{1}}}\bigr|\bigl|b_{2}(x)-b_{2,Q_{1,k_{1}}}\bigr| \frac {|a_{Q_{1,k_{1}}}(y_{1})||a_{Q_{2,k_{2}}}(y_{2})|}{(|x-y_{1}|+|x-y_{2}|)^{2n}} \\ &\qquad{} \times\omega \biggl(\frac{|y_{1}-c_{1,k_{1}}|}{|x-y_{1}|+|x-y_{2}|} \biggr) \,dy_{1}\,dy_{2}\,dx \\ &\quad\leq\frac{CC_{0}}{\lambda}\sum_{S_{1,l_{1}}}\sum _{Q_{1,k_{1}}\subset S_{1,l_{1}}}\sum_{S_{2,l_{2}}}\sum _{Q_{2,k_{2}}\subset S_{2,l_{2}}}\sum_{i=1}^{\infty} \sum_{h=1}^{\infty}\omega\bigl(2^{-i} \bigr)^{\frac{1}{4}}\omega\bigl(2^{-h}\bigr)^{\frac{1}{4}}| \lambda_{Q_{1,k_{1}}}||\lambda_{Q_{2,k_{2}}}| \\ &\qquad{} \times \int_{(S^{*})^{c}\cap\mathscr{R}_{1, k_{1}}^{i}\cap\mathscr {R}_{2, k_{2}}^{h}}\frac {|b_{1}(x)-b_{1,Q_{1,k_{1}}}|}{(2^{i}l_{Q_{1,k_{1}}}2^{h}l_{Q_{2,k_{2}}})^{\frac {n}{2}}} \biggl( \iint_{(\mathbb{R}^{n})^{2}} \bigl|b_{2}(x)-b_{2,Q_{1,k_{1}}}\bigr| \\ &\qquad{} \times\frac {|a_{Q_{1,k_{1}}}(y_{1})||a_{Q_{2,k_{2}}}(y_{2})|}{(|x-y_{1}|+|x-y_{2}|)^{n}}\omega \biggl(\frac{|y_{1}-c_{1,k_{1}}|}{|x-y_{1}|+|x-y_{2}|} \biggr)^{\frac{1}{2}}\,dy_{1}\,dy_{2} \biggr)\,dx. \end{aligned}$$
(3.10)
Let us first consider the inside integrals, by the Hölder inequality, we may have
$$\begin{aligned} &\int_{(S^{*})^{c}\cap\mathscr{R}_{1, k_{1}}^{i}\cap\mathscr{R}_{2, k_{2}}^{h}}\frac {|b_{1}(x)-b_{1,Q_{1,k_{1}}}|}{(2^{i}l_{Q_{1,k_{1}}}2^{h}l_{Q_{2,k_{2}}})^{\frac {n}{2}}} \biggl( \iint_{(\mathbb{R}^{n})^{2}} \bigl|b_{2}(x)-b_{2,Q_{1,k_{1}}}\bigr| \\ &\qquad{} \times\frac {|a_{Q_{1,k_{1}}}(y_{1})||a_{Q_{2,k_{2}}}(y_{2})|}{(|x-y_{1}|+|x-y_{2}|)^{n}} \omega \biggl(\frac{|y_{1}-c_{1,k_{1}}|}{|x-y_{1}|+|x-y_{2}|} \biggr)^{\frac{1}{2}}\,dy_{1}\,dy_{2} \biggr)\,dx \\ &\quad\leq \biggl(\frac{1}{(2^{h}l_{Q_{2,k_{2}}})^{n}} \int_{\mathscr{R}_{2, k_{2}}^{h}}\bigl|b_{1}(x)-b_{1,Q_{1,k_{1}}}\bigr|^{2} \,dx \biggr)^{\frac{1}{2}} \\ &\qquad{} \times \biggl(\frac{1}{(2^{i}l_{Q_{1,k_{1}}})^{n}} \int _{(S^{*})^{c}\cap\mathscr{R}_{1, k_{1}}^{i}} \biggl| \iint_{(\mathbb{R}^{n})^{2}} \bigl|b_{2}(x)-b_{2,Q_{1,k_{1}}}\bigr| \\ &\qquad{}\times\frac {|a_{Q_{1,k_{1}}}(y_{1})||a_{Q_{2,k_{2}}}(y_{2})|}{(|x-y_{1}|+|x-y_{2}|)^{n}} \omega \biggl(\frac{|y_{1}-c_{1,k_{1}}|}{|x-y_{1}|+|x-y_{2}|} \biggr)^{\frac{1}{2}}\,dy_{1}\,dy_{2} \biggr|^{2}\,dx \biggr)^{\frac{1}{2}}. \end{aligned}$$
(3.11)
Note that
\(a_{2,k_{2}}(y_{2})\in L^{1}(\mathbb{R}^{n})\), a similar argument to (
2.15) yields
$$\begin{aligned} (3.11)\leq{}& h^{\frac{1}{2}}\Vert b_{2} \Vert _{*}^{\frac {1}{2}} \biggl[\frac{1}{(2^{i}l_{Q_{1,k_{1}}})^{n}} \int_{(S^{*})^{c}\cap\mathscr{R}_{1, k_{1}}^{i}} \biggl| \int_{\mathbb{R}^{n}} \bigl|b_{2}(x)-b_{2,Q_{1,k_{1}}}\bigr| \\ &{} \times\sup_{y_{1}, y_{2}\in S} \biggl(\frac {1}{(|x-y_{1}|+|x-y_{2}|)^{n}}\omega \biggl( \frac{|y_{1}-c_{1,k_{1}}|}{|x-y_{1}|+|x-y_{2}|} \biggr)^{\frac{1}{2}} \biggr) \bigl|a_{Q_{1,k_{1}}}(y_{1})\bigr| \,dy_{1} \biggr|^{2}\,dx \biggr]^{\frac{1}{2}}. \end{aligned}$$
Note that the integrals in the above inequality are independent of
\(S_{2,l_{2}}\) and
\(Q_{2,k_{2}}\) and
ω is doubling, similar to what we have done with (
2.14), for fixed
\(x\in(S^{*})^{c}\) and any
\(y_{1}, y_{2}\in S\), we have
$$\begin{aligned} &\sup_{y_{1}, y_{2}\in S} \biggl(\frac {1}{(|x-y_{1}|+|x-y_{2}|)^{n}}\omega \biggl(\frac{|y_{1}-c_{1,k_{1}}|}{|x-y_{1}|+|x-y_{2}|} \biggr)^{\frac{1}{2}} \biggr) \\ &\quad\approx\frac{1}{(|x-y_{1}|+|x-y_{2}|)^{n}}\omega \biggl(\frac {|y_{1}-c_{1,k_{1}}|}{|x-y_{1}|+|x-y_{2}|} \biggr)^{\frac{1}{2}}. \end{aligned}$$
(3.12)
Recalling (I) in Theorem
1.1 and putting the inequality above into (
3.10), we may get
$$\begin{aligned} |E_{4,1}|\leq{}&\frac{CC_{0}}{\lambda}\sum_{S_{1,l_{1}}} \sum_{Q_{1,k_{1}}\subset S_{1,l_{1}}}\sum_{i=1}^{\infty} \sum_{h=1}^{\infty}\omega\bigl(2^{-i} \bigr)^{\frac{1}{4}} \omega\bigl(2^{-h}\bigr)^{\frac{1}{4}}h^{\frac{1}{2}}| \lambda_{Q_{1,k_{1}}}| \biggl(\frac{1}{(2^{i}l_{Q_{1,k_{1}}})^{n}} \\ &{}\times \int_{(S^{*})^{c}\cap\mathscr{R}_{1, k_{1}}^{i}} \biggl| \int _{\mathbb{R}^{n}} \bigl|b_{2}(x)-b_{2,Q_{1,k_{1}}}\bigr| \biggl(\sum _{S_{2,l_{2}}}\sum_{Q_{2,k_{2}}\subset S_{2,l_{2}}}| \lambda_{Q_{2,k_{2}}}| \biggr) \\ &{} \times\sup_{y_{1}, y_{2}\in S} \biggl(\frac {1}{(|x-y_{1}|+|x-y_{2}|)^{n}}\omega \biggl( \frac{|y_{1}-c_{1,k_{1}}|}{|x-y_{1}|+|x-y_{2}|} \biggr)^{\frac{1}{2}} \biggr) \bigl|a_{Q_{1,k_{1}}}(y_{1})\bigr| \,dy_{1} \biggr|^{2}\,dx \biggr)^{\frac{1}{2}} \\ \leq{}& CC_{0}\gamma^{\frac{1}{2}}\lambda^{-\frac{1}{2}}\sum _{S_{1,l_{1}}}\sum_{Q_{1,k_{1}}\subset S_{1,l_{1}}}\sum _{i=1}^{\infty}\sum _{h=1}^{\infty}\omega\bigl(2^{-i} \bigr)^{\frac{1}{4}} \omega\bigl(2^{-h}\bigr)^{\frac{1}{4}}h^{\frac{1}{2}}| \lambda_{Q_{1,k_{1}}}| \biggl(\frac{1}{(2^{i}l_{Q_{1,k_{1}}})^{n}} \\ &{} \times \int_{(S^{*})^{c}\cap\mathscr{R}_{1, k_{1}}^{i}} \biggl| \int _{\mathbb{R}^{n}} \bigl|b_{2}(x)-b_{2,Q_{1,k_{1}}}\bigr| \biggl(\sum _{S_{2,l_{2}}} \int _{S_{2,l_{2}}}\frac{1}{(|x-y_{1}|+|x-y_{2}|)^{n}} \\ &{} \times\omega \biggl(\frac{|y_{1}-c_{1,k_{1}}|}{|x-y_{1}|+|x-y_{2}|} \biggr)^{\frac{1}{2}} \,dy_{2} \biggr)\bigl|a_{Q_{1,k_{1}}}(y_{1})\bigr|\,dy_{1} \biggr|^{2}\,dx \biggr)^{\frac{1}{2}} \\ \leq{}& CC_{0}\gamma^{\frac{1}{2}}\lambda^{-\frac{1}{2}}\sum _{S_{1,l_{1}}}\sum_{Q_{1,k_{1}}\subset S_{1,l_{1}}}\sum _{i=1}^{\infty}\sum _{h=1}^{\infty}\omega\bigl(2^{-i} \bigr)^{\frac{1}{4}} \omega\bigl(2^{-h}\bigr)^{\frac{1}{4}}h^{\frac{1}{2}}| \lambda_{Q_{1,k_{1}}}| \\ &{} \times \biggl(\frac{1}{(2^{i}l_{Q_{1,k_{1}}})^{n}} \int _{(S^{*})^{c}\cap\mathscr{R}_{1, k_{1}}^{i}}\bigl|b_{2}(x)-b_{2,Q_{1,k_{1}}}\bigr|^{2} \biggl( \int_{\mathbb{R}^{n}} \bigl|a_{Q_{1,k_{1}}}(y_{1})\bigr| \,dy_{1} \biggr)^{2}\,dx \biggr)^{\frac{1}{2}} \\ \leq{}& CC_{0}\gamma^{\frac{1}{2}}\lambda^{-\frac {1}{2}}\sum _{i=1}^{\infty}\sum _{h=1}^{\infty}\omega \bigl(2^{-i} \bigr)^{\frac{1}{4}} \omega\bigl(2^{-h}\bigr)^{\frac{1}{4}}h^{\frac{1}{2}}i^{\frac{1}{2}} \\ \leq{}& CC_{0}\gamma^{\frac{1}{2}}\lambda^{-\frac{1}{2}}. \end{aligned}$$
Now we begin with the estimate for
\(|E_{4,2}|\).
Recalling the definition of
\(I_{4,2}(x)\), the moment condition of
\(H^{1}\)-atoms and smoothness condition (
1.3). Similar to the estimates in (
3.10), we may obtain
$$\begin{aligned} |E_{4,2}|\leq{}&\frac{CC_{0}}{\lambda}\sum _{S_{1,l_{1}}}\sum_{Q_{1,k_{1}}\subset S_{1,l_{1}}}\sum _{S_{2,l_{2}}}\sum_{Q_{2,k_{2}}\subset S_{2,l_{2}}}\sum _{i=1}^{\infty }|\lambda_{Q_{1,k_{1}}}|| \lambda_{Q_{2,k_{2}}}| \\ &{}\times \int_{(S^{*})^{c}\cap \mathscr{R}_{1, k_{1}}^{i}} \iint_{(\mathbb{R}^{n})^{2}}\bigl|b_{1}(x)-b_{1,Q_{1,k_{1}}}\bigr|\bigl|b_{2}(y_{2})-b_{2,Q_{1,k_{1}}}\bigr| \frac {|a_{Q_{1,k_{1}}}(y_{1})||a_{Q_{2,k_{2}}}(y_{2})|}{|x-y_{1}|^{n}} \\ &{} \times\frac{1}{(|x-y_{1}|+|x-y_{2}|)^{n}}\omega \biggl(\frac {|y_{1}-c_{1,k_{1}}|}{|x-y_{1}|+|x-y_{2}|} \biggr) \,dy_{1}\,dy_{2}\,dx. \end{aligned}$$
(3.13)
First, we consider the following summation.
$$\begin{aligned} &\sum_{S_{2,l_{2}}}\sum _{Q_{2,k_{2}}\subset S_{2,l_{2}}} \int _{\mathbb{R}^{n}}\bigl|b_{2}(y_{2})-b_{2,Q_{1,k_{1}}}\bigr| \frac{|\lambda _{Q_{2,k_{2}}}||a_{Q_{2,k_{2}}}(y_{2})|}{(|x-y_{1}|+|x-y_{2}|)^{n}} \\ &\quad{}\times \omega \biggl(\frac{|y_{1}-c_{1,k_{1}}|}{|x-y_{1}|+|x-y_{2}|} \biggr)\,dy_{2}. \end{aligned}$$
(3.14)
Property (I) in Theorem
1.1, inequality (
3.12), and the size condition of
\(H^{1}\)-atoms, that is,
\(\Vert a_{Q_{2,k_{2}}} \Vert _{L^{\infty}}\leq|Q_{2,k_{2}}|^{-1}\), together with the Hölder inequality, enable us to obtain
$$\begin{aligned} (3.14)\leq{}&\sum_{S_{2,l_{2}}}\sum _{Q_{2,k_{2}}\subset S_{2,l_{2}}}|\lambda_{Q_{2,k_{2}}}| \biggl( \int_{\mathbb{R}^{n}}\bigl|b_{2}(y_{2})-b_{2,Q_{1,k_{1}}}\bigr|^{2}\bigl|a_{Q_{2,k_{2}}}(y_{2})\bigr| \,dy_{2} \biggr)^{\frac {1}{2}} \\ &{} \times \biggl( \int_{\mathbb{R}^{n}}\frac{1}{(|x-y_{1}|+|x-y_{2}|)^{2n}} \omega \biggl(\frac{|y_{1}-c_{1,k_{1}}|}{|x-y_{1}|+|x-y_{2}|} \biggr)^{2}\bigl|a_{Q_{2,k_{2}}}(y_{2})\bigr|\,dy_{2} \biggr)^{\frac{1}{2}} \\ \leq{}&\omega\bigl(2^{-i}\bigr)\sum_{S_{2,l_{2}}} \sum_{Q_{2,k_{2}}\subset S_{2,l_{2}}}|\lambda_{Q_{2,k_{2}}}|\Vert b_{2} \Vert _{*}^{\frac {1}{2}}\sup_{y_{1}, y_{2}\in S} \biggl( \frac{1}{(|x-y_{1}|+|x-y_{2}|)^{n}} \\ &{} \times\omega \biggl(\frac{|y_{1}-c_{1,k_{1}}|}{|x-y_{1}|+|x-y_{2}|} \biggr)^{\frac{1}{2}} \biggr) \\ \leq{}& C(\gamma\lambda)^{\frac{1}{2}}\omega\bigl(2^{-i} \bigr)^{\frac {1}{2}}\sum_{S_{2,l_{2}}} \int_{S_{2,l_{2}}}\frac {1}{(|x-y_{1}|+|x-y_{2}|)^{n}} \omega \biggl(\frac{|y_{1}-c_{1,k_{1}}|}{|x-y_{1}|+|x-y_{2}|} \biggr)^{\frac {1}{2}}\,dy_{2} \\ \leq{}& C(\gamma\lambda)^{\frac{1}{2}}\omega\bigl(2^{-i} \bigr)^{\frac{1}{2}}. \end{aligned}$$
Therefore, by (
3.13) and noting that
\(a_{Q_{1,k_{1}}}(y_{2})\in L^{1}(\mathbb{R}^{n})\), we have
$$\begin{aligned} |E_{4,2}|\leq{}& CC_{0}\gamma^{\frac{1}{2}} \lambda^{-\frac{1}{2}}\sum_{S_{1,l_{1}}}\sum _{Q_{1,k_{1}}\subset S_{1,l_{1}}}\sum_{i=1}^{\infty} \omega\bigl(2^{-i}\bigr)^{\frac{1}{2}}|\lambda _{Q_{1,k_{1}}}| \int_{(S^{*})^{c}\cap\mathscr{R}_{1, k_{1}}^{i}} \int_{\mathbb{R}^{n}}\frac{1}{|x-y_{1}|^{2}} \\ &{} \times\bigl|b_{1}(x)-b_{1,Q_{1,k_{1}}}\bigr|\bigl|a_{Q_{1,k_{1}}}(y_{1})\bigr| \,dy_{1}\,dx \\ \leq{}& CC_{0}\Vert b_{1} \Vert _{*}\gamma^{\frac{1}{2}} \lambda ^{-\frac {1}{2}}\sum_{S_{1,l_{1}}}\sum _{Q_{1,k_{1}}\subset S_{1,l_{1}}}|\lambda_{Q_{1,k_{1}}}|\sum _{i=1}^{\infty}\omega \bigl(2^{-i} \bigr)^{\frac{1}{2}}i^{\frac{1}{2}}\leq CC_{0}\gamma^{\frac {1}{2}} \lambda^{-\frac{1}{2}}. \end{aligned}$$
Since
\(|E_{4,3}|\) is a symmetrical case of
\(|E_{4,2}|\) we may also obtain
$$|E_{4,3}|\leq CC_{0}\gamma^{\frac{1}{2}} \lambda^{-\frac{1}{2}}. $$
A similar argument still works as in (
2.9), we may have
$$|E_{4,4}|\leq CC_{2}^{\frac{1}{2}}\lambda^{-\frac{1}{2}}. $$
This completes the estimate for
\(|E_{4}|\). Thus, we have proved inequality (
3.3) and the proof of Theorem
1.2 is finished. □