2.1 Hardy spaces

Let $\theta $ be a Schwartz function on ${{\mathbb R}^n}$ such that $\mbox {supp}(\widehat {\theta })\subset \{\xi \in {{\mathbb R}^n}: |\xi |\le 2\}$ and $\widehat {\theta }(\xi )=1$ for $|\xi |\le 1$ . Let $\psi :=\theta -2^{-n}\theta (2^{-1}\cdot )$ , and for each $j\in {\mathbb {Z}}$ we define $\theta _j:=2^{jn}\theta (2^j\cdot )$ and $\psi _j:=2^{jn}\psi (2^j\cdot )$ . Then $\{\psi _j\}_{j\in {\mathbb {Z}}}$ forms a Littlewood–Paley partition of unity, satisfying

$$ \begin{align*}\mbox{supp}(\widehat{\psi_j})\subset \big\{\xi\in{{\mathbb R}^n}: 2^{j-1}\le |\xi|\le 2^{j+1}\big\} \quad\text{ and }\quad \sum_{j\in{\mathbb{Z}}}\widehat{\psi_j}(\xi)=1, ~\xi\not= 0.\end{align*} $$

We define the convolution operators ${\Gamma }_j$ and ${\Lambda }_j$ by

$$ \begin{align*}{\Gamma}_jf:=\theta_j\ast f, \qquad {\Lambda}_jf:=\psi_j\ast f.\end{align*} $$

The Hardy space $H^p({{\mathbb R}^n})$ can be characterized with the (quasi-)norm equivalences

(2.1) $$ \begin{align} \Vert f\Vert_{H^p({{\mathbb R}^n})}\sim\big\Vert \big\{ {\Gamma}_jf \big\}_{j\in{\mathbb{Z}}}\big\Vert_{L^p(\ell^{\infty})}, \qquad 0<p\le \infty \end{align} $$

and

(2.2) $$ \begin{align} \Vert f\Vert_{H^p({{\mathbb R}^n})}\sim \big\Vert \big\{{\Lambda}_jf\big\}_{j\in{\mathbb{Z}}}\big\Vert_{L^p(\ell^{2})}, \qquad 0<p<\infty, \end{align} $$

which is the Littlewood–Paley theory for Hardy spaces. In addition, when $p\le 1$ , every $f\in H^p({{\mathbb R}^n})$ can be decomposed as

(2.3) $$ \begin{align} f=\sum_{k=1}^{\infty}\lambda_k a_k \quad \text{ in the sense of tempered distributions}, \end{align} $$

where $a_k$ ’s are $H^p$ -atoms having the properties that $\mbox {supp}(a_k)\subset Q_k$ , $\Vert a_k\Vert _{L^{\infty }({{\mathbb R}^n})}\le |Q_k|^{-1/p}$ for some cube $Q_k$ , $\int x^{\gamma }a_k(x)dx=0$ for all multi-indices $|\gamma |\le M$ , and $\big ( \sum _{k=1}^{\infty }|\lambda _k|^p\big )^{1/p}\lesssim \Vert f\Vert _{H^p({{\mathbb R}^n})},$ where M is a fixed integer satisfying $M\ge [n/p-n]_+$ , which may be actually arbitrarily large. Furthermore, each $H^p$ -atom $a_k$ satisfies

$$ \begin{align*} \Vert a_k\Vert_{H^1({{\mathbb R}^n})}\lesssim |Q_k|^{-1/p+1}. \end{align*} $$

2.2 Maximal inequalities

Let $\mathcal {M}$ denote the Hardy–Littlewood maximal operator, defined by

$$ \begin{align*}\mathcal{M}f(x):=\sup_{Q:x\in Q}\frac{1}{|Q|}\int_Q{|f(y)|}dy\end{align*} $$

for a locally integrable function f on ${{\mathbb R}^n}$ , where the supremum ranges over all cubes Q containing x. For given $0<r<\infty $ , we define $\mathcal {M}_rf:=\big ( \mathcal {M}\big (|f|^r\big )\big )^{1/r}$ . Then it is well-known that

(2.4) $$ \begin{align} \big\Vert \big\{ \mathcal{M}_rf_k \big\}_{k\in{\mathbb{Z}}}\big\Vert_{L^p(\ell^q)}\lesssim \Vert \{f_k\}_{k\in{\mathbb{Z}}}\Vert_{L^p(\ell^q)} \end{align} $$

whenever $r<p< \infty $ and $r<q\le \infty $ . We note that for $1\le r<\infty $

(2.5) $$ \begin{align} \bigg\Vert \frac{f(x-\cdot)}{\langle 2^j\cdot\rangle^t}\bigg\Vert_{L^r({{\mathbb R}^n})}\lesssim 2^{-{jn}/{r}}\mathcal{M}_rf(x)\qquad \text{ if }~t>n/r. \end{align} $$

For ${\boldsymbol {m}} \in {{\mathbb Z}^n}$ and any dyadic cubes $Q\in \mathcal {D}$ , we use the notation

$$ \begin{align*}Q({\boldsymbol{m}}):=Q+ \ell(Q)\,{\boldsymbol{m}}.\end{align*} $$

Then we define the dyadic shifted maximal operator $\mathcal {M}_{dyad}^{ {\boldsymbol {m}}}$ by

$$\begin{align*}\mathcal{M}_{dyad}^{ {\boldsymbol{m}}}f(x):=\sup_{Q\in\mathcal{D}: x\in Q} \frac{1}{|Q|} \int_{{Q({\boldsymbol{m}}})} |f(y)|\, dy,\end{align*}$$

where the supremum is taken over all dyadic cubes Q containing x. It is clear that $\mathcal {M}_{dyad}^{\mathbf {0}}f(x)\le \mathcal {M}f(x)$ and accordingly, $\mathcal {M}_{dyad}^{\mathbf {0}}$ is bounded on $L^p$ for $p>1$ . In general, the following maximal inequality holds: For $1<p<\infty $ and ${\boldsymbol {m}}\in {{\mathbb Z}^n}$ we have

(2.6) $$ \begin{align} \big\|\mathcal{M}_{dyad}^{{\boldsymbol{m}}}f\big\|_{L^p({{\mathbb R}^n})} \lesssim \big(\log{(10+|{\boldsymbol{m}}|)}\big)^{n/p}\, \|f\|_{L^p({{\mathbb R}^n})}. \end{align} $$

The inequality (2.6) follows from the repeated use of the inequality in one-dimensional setting that appears in [Reference Stein31, Chapter II, §5.10], and we omit the detailed proof here. Refer to [Reference Lee, Heo, Hong, Lee, Park, Park and Yang22, Appendix] for the argument.

2.3 Variants of $\varphi $ -transform

For a sequence of complex numbers ${\boldsymbol {b}}:=\{b_Q\}_{Q\in \mathcal {D}}$ , we define

$$ \begin{align*}\Vert {\boldsymbol{b}}\Vert_{\dot{f}^{p,q}}:=\big\Vert g^q({\boldsymbol{b}})\big\Vert_{L^p({{\mathbb R}^n})}\end{align*} $$

for $0<p<\infty $ , where

$$ \begin{align*}g^q({\boldsymbol{b}})(x):=\big\Vert \big\{ |b_Q||Q|^{-{1}/{2}}\chi_Q(x)\big\}_{Q\in\mathcal{D}}\big\Vert_{\ell^q}, \qquad 0<q\le\infty.\end{align*} $$

Let $\widetilde {\psi _j}:=\psi _{j-1}+\psi _j+\psi _{j+1}$ for $j\in {\mathbb {Z}}$ . Observe that $\widetilde {\psi _j}$ enjoys the properties that $\mbox {supp}(\widehat {\widetilde {\psi }})\subset \{\xi \in {{\mathbb R}^n}: 2^{j-2}\le |\xi |\le 2^{j+2}\}$ and $\psi _j=\psi _j\ast \widetilde {\psi _j}$ . Then we have the representation

(2.7) $$ \begin{align} {\Lambda}_jf(x)=\sum_{Q\in\mathcal{D}_j}b_Q\psi^{Q}(x), \end{align} $$

where $\psi ^Q(x):=|Q|^{{1}/{2}}\psi _j(x-{\mathbf {x}}_Q)$ , $\widetilde {\psi }^Q(x):=|Q|^{{1}/{2}}\widetilde {\psi _j}(x-{\mathbf {x}}_Q)$ for each $Q\in \mathcal {D}_j$ , and $b_Q=\langle f,\widetilde {\psi }^Q\rangle $ . This implies that

$$ \begin{align*}f=\sum_{j\in{\mathbb{Z}}}{\Lambda}_jf=\sum_{j\in{\mathbb{Z}}}\sum_{Q\in\mathcal{D}_j}b_Q\psi^Q \quad \text{ in }~\mathcal{S}'/\mathcal{P},\end{align*} $$

where $\mathcal {S}'/\mathcal {P}$ stands for a tempered distribution modulo polynomials. Moreover, in this case, we have

(2.8) $$ \begin{align} \Vert {\boldsymbol{b}}\Vert_{\dot{f}^{p,q}}\sim \big\Vert \{{\Lambda}_jf\}_{j\in{\mathbb{Z}}}\big\Vert_{L^p(\ell^q)}. \end{align} $$

Therefore, the Hardy space $H^p({{\mathbb R}^n})$ can be characterized by the discrete function space $\dot {f}^{p,2}$ , in view of the equivalence in Equation (2.2). We refer to [Reference Frazier and Jawerth8, Reference Frazier and Jawerth9, Reference Frazier and Jawerth10] for more details.

It is also known in [Reference Park26] that ${\Gamma }_jf$ has a representation analogous to (2.7) with an equivalence similar to (2.8), while $f\not = \sum _{j\in {\mathbb {Z}}}{\Gamma }_jf$ generally. Let $\widetilde {\theta }:=2^n\theta (2\cdot )$ and $\widetilde {\theta _j}:=2^{jn}\widetilde {\theta }(2^j\cdot )=\theta _{j+1}$ so that $\theta _j=\theta _j\ast \widetilde {\theta _{j}}$ . Let $\theta ^Q(x):=|Q|^{{1}/{2}}\theta _j(x-{\mathbf {x}}_Q)$ , $\widetilde {\theta }^Q(x):=|Q|^{{1}/{2}}\widetilde {\theta _{j}}(x-{\mathbf {x}}_Q)$ , and $b_Q=\langle f,\widetilde {\theta }^Q\rangle $ for each $Q\in \mathcal {D}_j$ . Then we have

(2.9) $$ \begin{align} {\Gamma}_jf(x)=\sum_{Q\in\mathcal{D}_j}b_Q\theta^Q(x) \end{align} $$

and for $0<p<\infty $ and $0<q\le \infty $

(2.10) $$ \begin{align} \big\Vert \{{\Gamma}_jf\}_{j\in{\mathbb{Z}}}\big\Vert_{L^p(\ell^q)}\sim \Vert {\boldsymbol{b}}\Vert_{\dot{f}^{p,q}}. \end{align} $$

We refer to [Reference Park26, Lemma 3.1] for more details.

3.1 Proof of Equation (1.18) when $(1/p_1,1/p_2,1/p_3)\in \mathscr {R}_4\cup \mathscr {R}_5\cup \mathscr {R}_6$

We need to work only with $(1/p_1,1/p_2,1/p_3)\in \mathscr {R}_4$ since the other cases are just symmetric versions. In this case, $2<p_3<\infty $ and as mentioned in Equation (1.19), the condition (1.17) is equivalent to

$$ \begin{align*}s>n/p_1+n/p_2.\end{align*} $$

Now, choose $\widetilde {p_1},\widetilde {p_2}<1$ such that

$$ \begin{align*}1/p_1+1/p_2=1/\widetilde{p_1}+1/2=1/2+1/\widetilde{p_2}\end{align*} $$

and thus

$$ \begin{align*}s>n/\widetilde{p_1}+n/2, \quad s>n/2+n/\widetilde{p_2}.\end{align*} $$

Let $\epsilon _1, \epsilon _2>0$ be numbers with

$$ \begin{align*}s>n/(\widetilde{p_1}-\epsilon_1)+n/2, \quad s>n/2+n/(\widetilde{p_2}-\epsilon_2)\end{align*} $$

and select $q_1,q_2>2$ such that

$$ \begin{align*}1/p=1/(\widetilde{p_1}-\epsilon_1)+1/q_1+1/p_3=1/q_2+1/(\widetilde{p_2}-\epsilon_2)+1/p_3.\end{align*} $$

Then we observe that

$$ \begin{align*}(1-\theta)\Big(\frac{1}{\widetilde{p_1}-\epsilon_1},\frac{1}{q_1},\frac{1}{p_3} \Big)+\theta \Big( \frac{1}{q_2},\frac{1}{\widetilde{p_2}-\epsilon_2},\frac{1}{p_3}\Big)=\Big(\frac{1}{p_1},\frac{1}{p_2},\frac{1}{p_3}\Big)\end{align*} $$

for some $0<\theta <1$ . Let $C_1:=(1/(\widetilde {p_1}-\epsilon _1),1/q_1,1/p_3)$ and $C_2:=(1/q_2,1/(\widetilde {p_2}-\epsilon _2),1/p_3)$ . It is obvious that $C_1\in \mathscr {R}_1$ , $C_2\in \mathscr {R}_2$ , and thus it follows from Proposition 3.1 that

$$ \begin{align*} \Vert T_{\sigma}\Vert_{H^{\widetilde{p_1}-\epsilon_1}\times H^{q_1}\times H^{p_3}\to H^p}\lesssim \mathcal{L}_s^2[\sigma] \quad &\text{ at }~ C_1=(1/(\widetilde{p_1}-\epsilon_1),1/q_1,1/p_3)\in \mathscr{R}_1, \\ \Vert T_{\sigma}\Vert_{H^{q_2}\times H^{\widetilde{p_2}-\epsilon_2}\times H^{p_3}\to H^p}\lesssim \mathcal{L}_s^2[\sigma] \quad &\text{ at }~ C_2=(1/q_2,1/(\widetilde{p_2}-\epsilon_2),1/p_3)\in \mathscr{R}_2. \end{align*} $$

Finally, the assertion (1.18) for $(1/p_1,1/p_2,1/p_3)\in \mathscr {R}_4$ is derived by means of interpolation in Lemma 3.1. See Figure 2 for the interpolation.

Figure 2

$(1-\theta )\big (\frac {1}{\widetilde {p_1}-\epsilon _1},\frac {1}{q_1},\frac {1}{p_3} \big )+\theta \big ( \frac {1}{q_2},\frac {1}{\widetilde {p_2}-\epsilon _2},\frac {1}{p_3}\big )=(\frac {1}{p_1},\frac {1}{p_2},\frac {1}{p_3})\in \mathscr {R}_4$ .

3.2 Proof of Equation (1.18) when $(1/p_1,1/p_2,1/p_3)\in \mathscr {R}_0$

We first fix $1/2<p<1$ such that $1/p_1+1/p_2+1/p_3=1/p$ and assume that, in view of Equation (1.19),

$$ \begin{align*}s>3n/2=n/1+n/2.\end{align*} $$

Then we choose $2<p_0<\infty $ such that $1+1/2+1/p_0=1/p$ . Then it is clear that $(1/p_1,1/p_2,1/p_3)$ is located inside the hexagon with the vertices $(1,1/p_0,1/2)$ , $(1,1/2,1/p_0)$ , $(1/2,1,1/p_0)$ , $(1/p_0,1,1/2)$ , $(1/p_0,1/2,1)$ and $(1/2,1/p_0,1)$ . Now, we choose a sufficiently small $\epsilon>0$ and $2<\widetilde {p_0}<\infty $ such that

$$ \begin{align*}\frac{1}{2+\epsilon}+\frac{1}{\widetilde{p_0}}=\frac{1}{2}+\frac{1}{p_0},\end{align*} $$

and the point $(1/p_1,1/p_2,1/p_3)$ is still inside the smaller hexagon with $D_1:=(1,1/\widetilde {p_0},1/(2+\epsilon ))$ , $D_2:=(1,1/(2+\epsilon ),1/\widetilde {p_0})$ , $D_3:=(1/(2+\epsilon ),1,1/\widetilde {p_0})$ , $D_4:=(1/\widetilde {p_0},1,1/ (2+\epsilon ))$ , $D_5:=(1/\widetilde {p_0},1/(2+\epsilon ),1)$ , and $D_6:=(1/(2+\epsilon ),1/\widetilde {p_0},1)$ . Now, Proposition 3.1 deduces that

$$ \begin{align*}\big\Vert T_{\sigma}\big\Vert_{H^{q_1}\times H^{q_2}\times H^{q_3}\to H^p}\lesssim \mathcal{L}_s^2[\sigma]\end{align*} $$

for $(1/q_1,1/q_2,1/q_3)\in \{D_1,D_2,D_3,D_4,D_5,D_6 \}$ , as $D_1,D_2\in \mathscr {R}_1$ , $D_3,D_4\in \mathscr {R}_2$ and $D_5,D_6\in \mathscr {R}_3$ . This implies, via interpolation in Lemma 3.1,

$$ \begin{align*}\big\Vert T_{\sigma}(f_1,f_2,f_3)\big\Vert_{H^p({{\mathbb R}^n})}\lesssim \mathcal{L}_s^2[\sigma]\Vert f_1\Vert_{H^{p_1}({{\mathbb R}^n})}\Vert f_2\Vert_{H^{p_2}({{\mathbb R}^n})}\Vert f_3\Vert_{H^{p_3}({{\mathbb R}^n})}.\end{align*} $$

See Figure 3 for the interpolation.

Figure 3

$\big (\frac {1}{p_1},\frac {1}{p_2},\frac {1}{p_3}\big )\in \mathscr {R}_0$ .

For the case $p=1$ , we interpolate the estimates in Proposition 3.2. To be specific, for any given $0<p_1,p_2,p_3<\infty $ with $1/p_1+1/p_2+1/p_3=1$ , the estimate (1.18) with $p=1$ follows from interpolating

$$ \begin{align*} \big\Vert T_{\sigma}(f_1,f_2,f_3)\big\Vert_{H^1({{\mathbb R}^n})}&\lesssim \mathcal{L}_s^2[\sigma]\Vert f_1\Vert_{H^1({{\mathbb R}^n})}\Vert f_2\Vert_{H^{\infty}({{\mathbb R}^n})}\Vert f_3\Vert_{H^{\infty}({{\mathbb R}^n})},\\ \big\Vert T_{\sigma}(f_1,f_2,f_3)\big\Vert_{H^1({{\mathbb R}^n})}&\lesssim \mathcal{L}_s^2[\sigma]\Vert f_1\Vert_{H^{\infty}({{\mathbb R}^n})}\Vert f_2\Vert_{H^{1}({{\mathbb R}^n})}\Vert f_3\Vert_{H^{\infty}({{\mathbb R}^n})},\\ \big\Vert T_{\sigma}(f_1,f_2,f_3)\big\Vert_{H^1({{\mathbb R}^n})}&\lesssim \mathcal{L}_s^2[\sigma]\Vert f_1\Vert_{H^{\infty}({{\mathbb R}^n})}\Vert f_2\Vert_{H^{\infty}({{\mathbb R}^n})}\Vert f_3\Vert_{H^{1}({{\mathbb R}^n})}. \end{align*} $$

3.3 Proof of Equation (1.18) when $(1/p_1,1/p_2,1/p_3)\in \mathscr {R}_7$

Let $0<p\le 1/2$ be such that $1/p=1/p_1+1/p_2+1/p_3$ , and assume that

$$ \begin{align*}s>n/p-n/2.\end{align*} $$

We choose $0<p_0\le 1$ , satisfying $1/p_0+1=1/p$ , so that

$$ \begin{align*}s>n/p_0+n/2.\end{align*} $$

Then there exist $\epsilon>0$ and $2<q<\infty $ so that $s>n/(p_0-\epsilon )+n/2$ and $1/p=1/(p_0-\epsilon )+2/q$ . Let $E_1:=\big (1/(p_0-\epsilon ),1/q,1/q\big )$ , $E_2:=\big (1/q,1/(p_0-\epsilon ),1/q\big )$ , and $E_3:=\big (1/q,1/q,1/(p_0-\epsilon )\big )$ . Then it is immediately verified that $E_1\in \mathscr {R}_1$ , $E_2\in \mathscr {R}_2$ , $E_3\in \mathscr {R}_3$ , and

$$ \begin{align*}\theta_1\Big(\frac{1}{(p_0-\epsilon)},\frac{1}{q},\frac{1}{q}\Big)+\theta_2\Big(\frac{1}{q},\frac{1}{(p_0-\epsilon)},\frac{1}{q}\Big)+\theta_3\Big(\frac{1}{q},\frac{1}{q},\frac{1}{(p_0-\epsilon)}\Big)=\Big(\frac{1}{p_1},\frac{1}{p_2},\frac{1}{p_3}\Big)\end{align*} $$

for some $0<\theta _1,\theta _2,\theta _3<1$ with $\theta _1+\theta _2+\theta _3=1$ . Therefore, Proposition 3.1 yields that

$$ \begin{align*} \Vert T_{\sigma}\Vert_{H^{p_0-\epsilon}\times H^{q}\times H^{q}\to H^p}\lesssim \mathcal{L}_s^2[\sigma] \quad &\text{ at }~ E_1=\big(1/(p_0-\epsilon),1/q,1/q\big)\in \mathscr{R}_1, \\ \Vert T_{\sigma}\Vert_{H^{q}\times H^{p_0-\epsilon}\times H^{q}\to H^p}\lesssim \mathcal{L}_s^2[\sigma] \quad &\text{ at }~ E_2=\big(1/q,1/(p_0-\epsilon),1/q\big)\in \mathscr{R}_2, \\ \Vert T_{\sigma}\Vert_{H^{q}\times H^{q}\times H^{p_0-\epsilon}\to H^p}\lesssim \mathcal{L}_s^2[\sigma] \quad &\text{ at }~ E_3=\big(1/q,1/q,1/(p_0-\epsilon)\big)\in \mathscr{R}_3, \end{align*} $$

and using the interpolation method in Lemma 3.1, we conclude the estimate (1.18) holds for $(1/p_1,1/p_2,1/p_3)\in \mathscr {R}_7$ . See Figure 4 for the interpolation.

Figure 4

$\theta _1\big (\frac {1}{p_0-\epsilon },\frac {1}{q},\frac {1}{q}\big )+\theta _2\big (\frac {1}{q},\frac {1}{p_0-\epsilon },\frac {1}{q}\big )+\theta _3\big (\frac {1}{q},\frac {1}{q},\frac {1}{p_0-\epsilon }\big )=\big (\frac {1}{p_1},\frac {1}{p_2},\frac {1}{p_3}\big )\in \mathscr {R}_7$ .

5.1 Reduction via paraproduct

Without loss of generality, we may assume

$$ \begin{align*}\Vert f_1\Vert_{H^{p_1}({{\mathbb R}^n})}=\Vert f_2\Vert_{H^{p_2}({{\mathbb R}^n})}=\Vert f_3\Vert_{H^{p_3}({{\mathbb R}^n})}=1 \quad \text{and}\quad \mathcal{L}_s^2[\sigma]=1.\end{align*} $$

We first note that $T_{\sigma }(f_1,f_2,f_3)$ can be written as

$$ \begin{align*} T_{\sigma}(f_1,f_2,f_3)=\sum_{j_1,j_2,j_3\in{\mathbb{Z}}}T_{\sigma}\big({\Lambda}_{j_1}f_1,{\Lambda}_{j_2}f_2,{\Lambda}_{j_3}f_3\big). \end{align*} $$

We shall work with only the case $j_1\ge j_2\ge j_3$ as other cases follow from a symmetric argument. When $j_1\ge j_2\ge j_3$ , it is easy to verify that

$$ \begin{align*}T_{\sigma}\big({\Lambda}_{j_1}f_1,{\Lambda}_{j_2}f_2,{\Lambda}_{j_3}f_3\big)=T_{\sigma_{j_1}}\big({\Lambda}_{j_1}f_1,{\Lambda}_{j_2}f_2,{\Lambda}_{j_3}f_3\big)\end{align*} $$

where $\sigma _j(\vec {\boldsymbol {\xi }}):=\sigma (\vec {\boldsymbol {\xi }}) \widehat {\Theta }(\vec {\boldsymbol {\xi }}/2^j)$ and $\widehat {\Theta }(\vec {\boldsymbol {\xi }}):=\sum _{l=-2}^2\widehat {\Psi }(2^{l}\vec {\boldsymbol {\xi }})$ so that $\widehat {\Theta }(\vec {\boldsymbol {\xi }})=1$ for $2^{-2}\le |\vec {\boldsymbol {\xi }}|\le 2^{2}$ and $\mbox {supp}(\widehat {\Theta })\subset \{\vec {\boldsymbol {\xi }}\in ({{\mathbb R}^n})^3:2^{-3}\le |\vec {\boldsymbol {\xi }}|\le 2^3\}$ . Then we observe that

(5.1) $$ \begin{align} \sup_{k\in{\mathbb{Z}}}\big\Vert \sigma_k(2^k{\vec{\cdot}\;})\big\Vert_{L_s^2(({{\mathbb R}^n})^3)}\lesssim \mathcal{L}_{s}^{2}[\sigma]=1 \end{align} $$

by virtue of the triangle inequality. Moreover, using the fact that $ {\Gamma }_jf=\sum _{k\le j}{\Lambda }_kf, $ we write

$$ \begin{align*} &\sum_{j_1\in{\mathbb{Z}}}\;\sum_{j_2,j_3\in{\mathbb{Z}}: j_3\le j_2\le j_1}T_{\sigma_{j_1}}\big({\Lambda}_{j_1}f_1,{\Lambda}_{j_2}f_2,{\Lambda}_{j_3}f_3\big)\\& \quad =\sum_{j\in{\mathbb{Z}}}T_{\sigma_{j}}\big({\Lambda}_{j}f_1,{\Gamma}_{j-10}f_2,{\Gamma}_{j-10}f_3\big)+\sum_{k=0}^{9}\sum_{j\in{\mathbb{Z}}}T_{\sigma_{j}}\big({\Lambda}_{j}f_1,{\Lambda}_{j-k}f_2,{\Gamma}_{j-k}f_3\big)\\& \quad =:T_{\sigma}^{(1)}(f_1,f_2,f_3)+\sum_{k=0}^{9}T_{\sigma}^{(2),k}(f_1,f_2,f_3), \end{align*} $$

and especially, let $T_{\sigma }^{(2)}:=T_{\sigma }^{(2),0}$ . Then it is enough to prove that

(5.2) $$ \begin{align} \big\Vert T_{\sigma}^{(\mu)}(f_1,f_2,f_3)\big\Vert_{H^p({{\mathbb R}^n})}\lesssim 1, \qquad \mu=1,2 \end{align} $$

since the operator $T_{\sigma }^{(2),k}$ , $1\le k\le 9$ , can be handled in the same way as $T_{\sigma }^{(2)}$ .

It should be remarked that the vanishing moment condition (1.16) now implies

(5.3) $$ \begin{align} \int_{{{\mathbb R}^n}}x^{\alpha}T_{\sigma_j}\big( f_1,f_2,f_3\big)(x) \; dx=0 \quad \text{ for all multi-indices ~}|\alpha|\le \frac{n}{p}-n. \end{align} $$

5.2 Proof of (5.2) for $\mu =1$

In this case, we may simply follow the arguments used in the proof of Theorems B and D. The proof is based on the fact that if $\widehat {g_k}$ is supported in $\{\xi \in {{\mathbb R}^n}: C_0^{-1} 2^{k}\le |\xi |\le C_02^{k}\}$ for $C_0>1$ , then

(5.4) $$ \begin{align} \bigg\Vert \Big\{ {\Lambda}_j \Big(\sum_{k\in{\mathbb{Z}}}{g_k}\Big)\Big\}_{j\in\mathbb{Z}}\bigg\Vert_{L^p(\ell^q)}\lesssim_{C_0} \big\Vert \big\{ g_j\big\}_{j\in\mathbb{Z}}\big\Vert_{L^p(\ell^q)}. \end{align} $$

The proof of Equation (5.4) is elementary and standard, simply using the estimate

$$ \begin{align*}\bigg| {\Lambda}_j \Big(\sum_{k\in{\mathbb{Z}}}{g_k}\Big)(x)\bigg|=\bigg| {\Lambda}_j\Big(\sum_{k=j-h}^{j+h}{g_k}\Big)(x)\bigg|\lesssim_{r,h}\mathcal{M}_rg_j(x)\end{align*} $$

for all $0<r<\infty $ and for some $h\in {\mathbb {N}}$ , depending on $C_0$ , and the maximal inequality (2.4). We refer to [Reference Yamazaki34, Theorem 3.6] for details.

By using the equivalence in Equation (2.2),

$$ \begin{align*} \big\Vert T_{\sigma}^{(1)}(f_1,f_2,f_3)\big\Vert_{H^p({{\mathbb R}^n})}\sim \bigg\Vert \bigg\{ {\Lambda}_j \Big(\sum_{k\in{\mathbb{Z}}}{ T_{\sigma_k}\big({\Lambda}_kf_1,{\Gamma}_{k-10}f_2,{\Gamma}_{k-10}f_3 \big)}\Big)\bigg\}_{j\in\mathbb{Z}}\bigg\Vert_{L^p(\ell^2)}. \end{align*} $$

We see that the Fourier transform of $T_{\sigma _k}\big ({\Lambda }_kf_1,{\Gamma }_{k-10}f_2,{\Gamma }_{k-10}f_3 \big )$ is supported in $\big \{\xi \in {{\mathbb R}^n} : 2^{k-2}\leq |\xi |\leq 2^{k+2} \big \}$ and thus the estimate (5.4) yields that

$$ \begin{align*} \big\Vert T_{\sigma}^{(1)}(f_1,f_2,f_3)\big\Vert_{H^p({{\mathbb R}^n})}\lesssim \Big\Vert \Big( \sum_{j\in{\mathbb{Z}}}\big| T_{\sigma_j}\big({\Lambda}_jf_1,{\Gamma}_{j-10}f_2,{\Gamma}_{j-10}f_3 \big) \big|^2\Big)^{1/2}\Big\Vert_{L^p({{\mathbb R}^n})}. \end{align*} $$

Then it is already proved in [Reference Grafakos, Miyachi, Nguyen and Tomita14, (3.14)] that the preceding expression is dominated by the right-hand side of Equation (5.2) for $s>n/\min {\{p_1,p_2,p_3\}}+n/2$ , where we remark that $\min {\{p_1,p_2,p_3\}}\le 1$ . This proves Equation (5.2) for $\mu =1$ .

5.3 Proof of Equation (5.2) for $\mu =2$

Recall that

(5.5) $$ \begin{align} T_{\sigma}^{(2)}(f_1,f_2,f_3)=\sum_{j\in{\mathbb{Z}}}T_{\sigma_j}\big({\Lambda}_jf_1,{\Lambda}_jf_2,{\Gamma}_jf_3\big) \end{align} $$

and observe that

$$ \begin{align*}T_{\sigma_j}\big({\Lambda}_jf_1,{\Lambda}_jf_2,{\Gamma}_jf_3\big)(x)=\sigma_j^{\vee}\ast_{3n}\big({\Lambda}_jf_1\otimes {\Lambda}_jf_2\otimes {\Gamma}_jf_3 \big)(x,x,x),\end{align*} $$

where $\ast _{3n}$ means the convolution on $\mathbb {R}^{3n}$ .

It suffices to consider the case when $(1/p_1,1/p_2,1/p_3)$ belongs to $\mathscr {R}_1$ or $\mathscr {R}_3$ , as the remaining case is symmetrical to the case $(1/p_1,1/p_2,1/p_3)\in \mathscr {R}_1$ , in view of Equation (5.5). We will mainly focus on the case $(1/p_1,1/p_2,1/p_3)\in \mathscr {R}_1$ , while simply providing a short description for the case $(1/p_1,1/p_2,1/p_3)\in \mathscr {R}_3$ in the remark below as almost same arguments will be applied in that case.

Therefore, we now assume $0<p_1\le 1$ and $2<p_2,p_3<\infty $ , and in turn, suppose that $s>n/p_1+n/2$ . By using the atomic decomposition in Equation (2.3), the function $f_1\in H^{p_1}({{\mathbb R}^n})$ can be written as $f_1=\sum _{k=1}^{\infty }\lambda _k a_k$ , where $a_k$ ’s are $H^{p_1}$ -atoms associated with cubes $Q_k$ , and

(5.6) $$ \begin{align} \Big(\sum_{k=1}^{\infty}|\lambda_k|^{p_1}\Big)^{1/{p_1}}\lesssim 1. \end{align} $$

As mentioned before, we may assume that M is sufficiently large and $\int {x^{\gamma }a_k(x)}dx=0$ holds for all multi-indices $|\gamma |\le M$ .

By the definition in Equation (1.2), we have

$$ \begin{align*} \big\Vert T_{\sigma}^{(2)}(f_1,f_2,f_3)\big\Vert_{H^p({{\mathbb R}^n})}\sim \bigg\Vert \sup_{l\in{\mathbb{Z}}} \Big| \sum_{k=0}^{\infty}\lambda _k\phi_l\ast \Big( \sum_{j\in{\mathbb{Z}}} T_{\sigma_j}({\Lambda}_ja_k,{\Lambda}_jf_2,{\Gamma}_jf_3) \Big) \Big|\bigg\Vert_{L^p({{\mathbb R}^n})} \end{align*} $$

and thus we need to prove that

(5.7) $$ \begin{align} \bigg\Vert \sup_{l\in{\mathbb{Z}}} \Big| \sum_{k=0}^{\infty}\lambda _k\phi_l\ast \Big( \sum_{j\in{\mathbb{Z}}} T_{\sigma_j}({\Lambda}_ja_k,{\Lambda}_jf_2,{\Gamma}_jf_3) \Big) \Big|\bigg\Vert_{L^p({{\mathbb R}^n})}\lesssim 1. \end{align} $$

The left-hand side is less than the sum of

$$ \begin{align*}\mathcal{I}:=\bigg\Vert \sup_{l\in{\mathbb{Z}}}\Big| \sum_{k=0}^{\infty}\lambda _k \chi_{Q_k^{***}} \phi_l\ast \Big( \sum_{j\in{\mathbb{Z}}} T_{\sigma_j}({\Lambda}_ja_k,{\Lambda}_jf_2,{\Gamma}_jf_3) \Big) \Big|\bigg\Vert_{L^p({{\mathbb R}^n})}\end{align*} $$

and

$$ \begin{align*}\mathcal{J}:=\bigg\Vert \sup_{l\in{\mathbb{Z}}}\Big| \sum_{k=0}^{\infty}\lambda _k \chi_{(Q_k^{***})^c}\phi_l\ast \Big( \sum_{j\in{\mathbb{Z}}} T_{\sigma_j}({\Lambda}_ja_k,{\Lambda}_jf_2,{\Gamma}_jf_3) \Big) \Big|\bigg\Vert_{L^p({{\mathbb R}^n})},\end{align*} $$

recalling that $Q_k^{***}$ is the dilate of $Q_k$ by a factor $(10\sqrt {n})^3$ . The two terms $\mathcal {I}$ and $\mathcal {J}$ will be treated separately in the next two sections.

Remark. When $(1/p_1,1/p_2,1/p_3)\in \mathscr {R}_3$ (that is, $0<p_3\le 1, ~2<p_1,p_2< \infty $ ), we need to prove

$$ \begin{align*}\bigg\Vert \sup_{l\in{\mathbb{Z}}} \Big| \sum_{k=0}^{\infty}\lambda _k\phi_l\ast \Big( \sum_{j\in{\mathbb{Z}}} T_{\sigma_j}({\Lambda}_jf_1,{\Lambda}_jf_2,{\Gamma}_j\widetilde{a_k}) \Big) \Big|\bigg\Vert_{L^p({{\mathbb R}^n})}\lesssim 1,\end{align*} $$

where $\widetilde {a_k}$ is the $H^{p_3}$ -atom associated with $f_3$ . This is actually, via symmetry, equivalent to the estimate that for $0<p_1\le 1$ and $2<p_2,p_3< \infty $ ,

(5.8) $$ \begin{align} \bigg\Vert \sup_{l\in{\mathbb{Z}}} \Big| \sum_{k=0}^{\infty}\lambda _k\phi_l\ast \Big( \sum_{j\in{\mathbb{Z}}} T_{\sigma_j}({\Gamma}_ja_k,{\Lambda}_jf_2,{\Lambda}_jf_3) \Big) \Big|\bigg\Vert_{L^p({{\mathbb R}^n})}\lesssim 1, \end{align} $$

where $a_k$ is the $H^{p_1}$ -atom for $f_1$ . The proof of Equation (5.8) is almost same as that of Equation (5.7) which will be discussed in Sections 6 and 7. So this will not be pursued in this paper, just saying that Equation (4.9) will be needed rather than Equation (4.8), and the estimate $\big \Vert \big \{ {\Gamma }_ja_k\big \}_{j\in {\mathbb {Z}}}\big \Vert _{L^r(\ell ^{\infty })}\sim \Vert a_k\Vert _{H^r({{\mathbb R}^n})}$ will be required in place of the equivalence $\big \Vert \big \{ {\Lambda }_ja_k\big \}_{j\in {\mathbb {Z}}}\big \Vert _{L^r(\ell ^{2})}\sim \Vert a_k\Vert _{H^r({{\mathbb R}^n})}$ .

7.1 Estimate for $\mathcal {J}_1$

Using the representations in Equations (2.7) and (2.9), we write

$$ \begin{align*}{\Lambda}_jf_2(x):=\sum_{P\in\mathcal{D}_j}b_P^2\psi^P(x),\qquad {\Gamma}_jf_3(x):=\sum_{R\in\mathcal{D}_j}b_R^3\theta^R(x),\end{align*} $$

where we recall $\psi ^P(x)=|P|^{{1}/{2}}\psi _j(x-{\mathbf {x}}_P)$ and $\theta ^R(x)=|R|^{{1}/{2}}\theta _j(x-{\mathbf {x}}_R)$ for $P,R\in \mathcal {D}_j$ . Then it follows from Equations (2.8), (2.10), (2.1) and (2.2) that

(7.1) $$ \begin{align} \big\Vert \{b_P^2\}_{P\in\mathcal{D}}\big\Vert_{\dot{f}^{p_2,2}}\sim \big\Vert \big\{{\Lambda}_jf_2\big\}_{j\in{\mathbb{Z}}}\big\Vert_{L^{p_2}(\ell^2)}\sim \Vert f_2\Vert_{H^{p_2}({{\mathbb R}^n})}= 1 \end{align} $$

and

(7.2) $$ \begin{align} \big\Vert \{b_R^3\}_{R\in\mathcal{D}}\big\Vert_{\dot{f}^{p_3,\infty}}\sim \big\Vert \big\{{\Gamma}_jf_3\big\}_{j\in{\mathbb{Z}}}\big\Vert_{L^{p_3}(\ell^{\infty})}\sim \Vert f_3\Vert_{H^{p_3}({{\mathbb R}^n})}=1. \end{align} $$

We write

$$ \begin{align*} \phi_l\ast \Big( \sum_{j\in{\mathbb{Z}}} T_{\sigma_j}({\Lambda}_ja_k,{\Lambda}_jf_2,{\Gamma}_jf_3) \Big)(x)= \sum_{\nu=1}^{4} \phi_l\ast \big( \mathcal{U}_{\nu}(x,\cdot)\big)(x), \end{align*} $$

where

(7.3) $$ \begin{align} \Omega_{\nu}(P,R):=\begin{cases} P\cap R & \nu=1\\ P^c\cap R & \nu=2\\ P\cap R^c & \nu=3\\ P^c \cap R^c & \nu=4 \end{cases} \end{align} $$

and

(7.4) $$ \begin{align} \mathcal{U}_{\nu}(x,y):= \sum_{j\in{\mathbb{Z}}} \sum_{P\in\mathcal{D}_j}\sum_{R\in\mathcal{D}_j} b_P^2 b_R^3 T_{\sigma_j}\big({\Lambda}_ja_k,\psi^P,\theta^R\big)(y) \chi_{\Omega_{\nu}(P,R)}(x), \quad \nu=1,2,3,4. \end{align} $$

Then we have

$$ \begin{align*}\mathcal{J}_1 \lesssim_p \mathcal{J}_{1}^1+\mathcal{J}_{1}^2+\mathcal{J}_{1}^3+\mathcal{J}_{1}^4,\end{align*} $$

where

$$ \begin{align*} \mathcal{J}_{1}^{\nu}:=\bigg\Vert \sup_{l\in{\mathbb{Z}}}\; \Big| \sum_{k=0}^{\infty}\lambda _k\chi_{(Q_k^{***})^c}(x)\chi_{(B_k^l)^c}(x) \phi_l\ast\big( \mathcal{U}_{\nu}(x,\cdot)\big)(x)\Big|\bigg\Vert_{L^p(x)},\qquad \nu=1,2,3,4. \end{align*} $$

Now, we will show that

(7.5) $$ \begin{align} \mathcal{J}_1^{\nu}\lesssim 1, \qquad \nu=1,2,3,4. \end{align} $$

7.1.1 Proof of Equation (7.5) for $\nu =1$

We further decompose $\mathcal {U}_1(x,y)$ as

$$ \begin{align*}\mathcal{U}_1(x,y)=\mathcal{U}_1^{\mathrm{in}}(x,y)+\mathcal{U}_1^{\mathrm{out}}(x,y),\end{align*} $$

where

(7.6) $$ \begin{align} \begin{aligned} \mathcal{U}_1^{\mathrm{in}}(x,y)&:=\sum_{j\in{\mathbb{Z}}} \sum_{P\in\mathcal{D}_j}\sum_{R\in\mathcal{D}_j} b_P^2 b_R^3 T_{\sigma_j}\big(\chi_{Q_k^*}{\Lambda}_ja_k,\psi^P,\theta^R\big)(y) \chi_{P\cap R}(x),\\ \mathcal{U}_1^{\mathrm{out}}(x,y)&:=\sum_{j\in{\mathbb{Z}}} \sum_{P\in\mathcal{D}_j}\sum_{R\in\mathcal{D}_j} b_P^2 b_R^3 T_{\sigma_j}\big(\chi_{(Q_k^*)^c}{\Lambda}_ja_k,\psi^P,\theta^R\big)(y) \chi_{P\cap R}(x), \end{aligned} \end{align} $$

and accordingly, we define

$$ \begin{align*} \mathcal{J}_1^{1,\mathrm{in}/\mathrm{out}}:=\bigg\Vert \sup_{l\in{\mathbb{Z}}}\; \Big| \sum_{k=0}^{\infty}\lambda _k\chi_{(Q_k^{***})^c}(x)\chi_{(B_k^l)^c}(x) \phi_l\ast\big( \mathcal{U}_{1}^{\mathrm{in}/\mathrm{out}}(x,\cdot)\big)(x)\Big|\bigg\Vert_{L^p(x)}. \end{align*} $$

Then we claim the following lemma.

Lemma 7.1. Let $0<p_1\le 1$ and $2<p_2,p_3<\infty $ , and let $\mathcal {U}_1^{\mathrm {in}/\mathrm {out}}$ be defined as in Equation (7.6). Suppose that $\Vert f_1\Vert _{H^{p_1}({{\mathbb R}^n})}=\Vert f_2\Vert _{H^{p_2}({{\mathbb R}^n})}=\Vert f_3\Vert _{H^{p_3}({{\mathbb R}^n})}=1$ and $\mathcal {L}_s^2[\sigma ]=1$ for $s>n/p_1+n/2$ . Then there exist nonnegative functions $u_1^{\mathrm {in}}$ , $u_1^{\mathrm {out}}$ , $u_2$ and $u_3$ on ${{\mathbb R}^n}$ such that

$$ \begin{align*}\big\Vert u_1^{\mathrm{in}/\mathrm{out}}\big\Vert_{L^{p_1}({{\mathbb R}^n})}\lesssim 1, \qquad \Vert u_{{\mathrm{i}}}\Vert_{L^{p_{{\mathrm{i}}}}({{\mathbb R}^n})}\lesssim 1 \quad \text{ for }~ {\mathrm{i}}=2,3,\end{align*} $$

and for $x\in {{\mathbb R}^n}$

(7.7) $$ \begin{align} \sup_{l\in{\mathbb{Z}}}\; \Big| \sum_{k=0}^{\infty}\lambda _k\chi_{(Q_k^{***})^c}(x)\chi_{(B_k^l)^c}(x) \phi_l\ast\big( \mathcal{U}_{1}^{\mathrm{in}/\mathrm{out}}(x,\cdot)\big)(x)\Big| \lesssim u_1^{\mathrm{in}/\mathrm{out}}(x) u_2(x) u_3(x). \end{align} $$

The proof of Lemma 7.1 will be given in Section 9. Taking the lemma for granted and using Hölder’s inequality, we can easily show that

$$ \begin{align*}\mathcal{J}_1^1\lesssim_p \mathcal{J}_1^{1,\mathrm{in}}+\mathcal{J}_1^{1,\mathrm{out}}\lesssim 1.\end{align*} $$

7.1.2 Proof of Equation (7.5) for $\nu =2$

For $P\in \mathcal {D}$ and $l\in {\mathbb {Z}}$ let $B_P^l:=B_{{\mathbf {x}}_P}^l=B({\mathbf {x}}_P,100n2^{-l})$ . By introducing

(7.8) $$ \begin{align} \begin{aligned} \mathcal{U}_2^{1,\mathrm{in}}(x,y)&:=\sum_{j\in{\mathbb{Z}}} \sum_{P\in\mathcal{D}_j}\sum_{R\in\mathcal{D}_j} b_P^2 b_R^3 T_{\sigma_j}\big(\chi_{Q_k^*}{\Lambda}_ja_k,\psi^P,\theta^R\big)(y) \chi_{P^c\cap (B_P^l)^c\cap R}(x),\\ \mathcal{U}_2^{1,\mathrm{out}}(x,y)&:=\sum_{j\in{\mathbb{Z}}} \sum_{P\in\mathcal{D}_j}\sum_{R\in\mathcal{D}_j} b_P^2 b_R^3 T_{\sigma_j}\big(\chi_{(Q_k^*)^c}{\Lambda}_ja_k,\psi^P,\theta^R\big)(y) \chi_{P^c\cap (B_P^l)^c\cap R}(x),\\ \mathcal{U}_2^{2,\mathrm{in}}(x,y)&:=\sum_{j\in{\mathbb{Z}}} \sum_{P\in\mathcal{D}_j}\sum_{R\in\mathcal{D}_j} b_P^2 b_R^3 T_{\sigma_j}\big(\chi_{Q_k^*}{\Lambda}_ja_k,\psi^P,\theta^R\big)(y) \chi_{P^c\cap B_P^l\cap R}(x),\\ \mathcal{U}_2^{2,\mathrm{out}}(x,y)&:=\sum_{j\in{\mathbb{Z}}} \sum_{P\in\mathcal{D}_j}\sum_{R\in\mathcal{D}_j} b_P^2 b_R^3 T_{\sigma_j}\big(\chi_{(Q_k^*)^c}{\Lambda}_ja_k,\psi^P,\theta^R\big)(y) \chi_{P^c\cap B_P^l\cap R}(x), \end{aligned} \end{align} $$

we write $\mathcal {U}_2=\mathcal {U}_2^{1,\mathrm {in}}+\mathcal {U}_2^{1,\mathrm {out}}+\mathcal {U}_2^{2,\mathrm {in}}+\mathcal {U}_2^{2,\mathrm {out}}$ and consequently,

$$ \begin{align*}\mathcal{J}_1^2 \lesssim_p \mathcal{J}_1^{2,1,\mathrm{in}}+\mathcal{J}_1^{2,1,\mathrm{out}}+\mathcal{J}_1^{2,2,\mathrm{in}}+\mathcal{J}_1^{2,2,\mathrm{out}},\end{align*} $$

where

$$ \begin{align*} \mathcal{J}_{1}^{2,\eta,\mathrm{in}/\mathrm{out}}:=\bigg\Vert \sup_{l\in{\mathbb{Z}}}\; \Big|\sum_{k=0}^{\infty}\lambda _k \chi_{(Q_k^{***})^c}(x)\chi_{(B_k^l)^c}(x)\; \phi_l\ast\big( \mathcal{U}_{2}^{\eta,\mathrm{in}/\mathrm{out}}(x,\cdot)\big)(x) \Big| \bigg\Vert_{L^p(x)},\quad \eta=1,2. \end{align*} $$

Then we apply the following lemma that will be proved in Section 9.

Lemma 7.2. Let $0<p_1\le 1$ and $2<p_2,p_3<\infty $ , and let $\mathcal {U}_2^{\eta ,\mathrm {in}/\mathrm {out}}$ be defined as in Equation (7.8). Suppose that $\Vert f_1\Vert _{H^{p_1}({{\mathbb R}^n})}=\Vert f_2\Vert _{H^{p_2}({{\mathbb R}^n})}=\Vert f_3\Vert _{H^{p_3}({{\mathbb R}^n})}=1$ and $\mathcal {L}_s^2[\sigma ]=1$ for $s>n/p_1+n/2$ . Then there exist nonnegative functions $u_1^{\mathrm {in}}$ , $u_1^{\mathrm {out}}$ , $u_2$ and $u_3$ on ${{\mathbb R}^n}$ such that

$$ \begin{align*}\big\Vert u_1^{\mathrm{in}/\mathrm{out}}\big\Vert_{L^{p_1}({{\mathbb R}^n})}\lesssim 1, \qquad \Vert u_{{\mathrm{i}}}\Vert_{L^{p_{{\mathrm{i}}}}({{\mathbb R}^n})}\lesssim 1 \quad \text{ for }~ {\mathrm{i}}=2,3,\end{align*} $$

and for each $\eta =1,2$

(7.9) $$ \begin{align} \sup_{l\in{\mathbb{Z}}}\; \Big| \sum_{k=0}^{\infty}\lambda _k\chi_{(Q_k^{***})^c}(x)\chi_{(B_k^l)^c}(x) \phi_l\ast\big( \mathcal{U}_{2}^{\eta,\mathrm{in}/\mathrm{out}}(x,\cdot)\big)(x)\Big| \lesssim u_1^{\mathrm{in}/\mathrm{out}}(x) u_2(x) u_3(x). \end{align} $$

Then Lemma 7.2 and Hölder’s inequality yield that $\mathcal {J}_1^2$ is controlled by the sum of four terms in the form

$$ \begin{align*} \big\Vert u_1^{\mathrm{in}/\mathrm{out}}\big\Vert_{L^{p_1}({{\mathbb R}^n})}\Vert u_2\Vert_{L^{p_2}({{\mathbb R}^n})} \Vert u_3\Vert_{L^{p_3}({{\mathbb R}^n})}, \end{align*} $$

which is obviously less than a constant. This proves Equation (7.5) for $\nu =2$ .

7.1.3 Proof of Equation (7.5) for $\nu =3$

This case is essentially symmetrical to the case $\nu =2$ . For $R\in \mathcal {D}$ and $l\in {\mathbb {Z}}$ , let $B_R^l:=B_{{\mathbf {x}}_R}^l=B({\mathbf {x}}_R,100n2^{-l})$ . Let

(7.10) $$ \begin{align} \begin{aligned} \mathcal{U}_3^{1,\mathrm{in}}(x,y)&:=\sum_{j\in{\mathbb{Z}}} \sum_{P\in\mathcal{D}_j}\sum_{R\in\mathcal{D}_j} b_P^2 b_R^3 T_{\sigma_j}\big(\chi_{Q_k^*}{\Lambda}_ja_k,\psi^P,\theta^R\big)(y) \chi_{P\cap R^c\cap (B_R^l)^c}(x),\\ \mathcal{U}_3^{1,\mathrm{out}}(x,y)&:=\sum_{j\in{\mathbb{Z}}} \sum_{P\in\mathcal{D}_j}\sum_{R\in\mathcal{D}_j} b_P^2 b_R^3 T_{\sigma_j}\big(\chi_{(Q_k^*)^c}{\Lambda}_ja_k,\psi^P,\theta^R\big)(y) \chi_{P\cap R^c\cap (B_R^l)^c}(x),\\ \mathcal{U}_3^{2,\mathrm{in}}(x,y)&:=\sum_{j\in{\mathbb{Z}}} \sum_{P\in\mathcal{D}_j}\sum_{R\in\mathcal{D}_j} b_P^2 b_R^3 T_{\sigma_j}\big(\chi_{Q_k^*}{\Lambda}_ja_k,\psi^P,\theta^R\big)(y) \chi_{P\cap R^c\cap B_R^l}(x),\\ \mathcal{U}_3^{2,\mathrm{out}}(x,y)&:=\sum_{j\in{\mathbb{Z}}} \sum_{P\in\mathcal{D}_j}\sum_{R\in\mathcal{D}_j} b_P^2 b_R^3 T_{\sigma_j}\big(\chi_{(Q_k^*)^c}{\Lambda}_ja_k,\psi^P,\theta^R\big)(y) \chi_{P\cap R^c\cap B_R^l}(x), \end{aligned} \end{align} $$

and then we write

$$ \begin{align*}\mathcal{J}_1^3\lesssim_p \mathcal{J}_1^{3,1,\mathrm{in}}+\mathcal{J}_1^{3,1,\mathrm{out}}+\mathcal{J}_1^{3,2,\mathrm{in}}+\mathcal{J}_1^{3,2,\mathrm{out}},\end{align*} $$

where

$$ \begin{align*} \mathcal{J}_{1}^{3,\eta,\mathrm{in}/\mathrm{out}}:=\bigg\Vert \sup_{l\in{\mathbb{Z}}}\; \Big| \sum_{k=0}^{\infty} \lambda _k \chi_{(Q_k^{***})^c}(x)\chi_{(B_k^l)^c}(x)\; \phi_l\ast\big( \mathcal{U}_{3}^{\eta,\mathrm{in}/\mathrm{out}}(x,\cdot)\big)(x) \Big|\bigg\Vert_{L^p(x)}, \quad \eta=1,2. \end{align*} $$

Now, Equation (7.5) for $\nu =3$ follows from the lemma below.

Lemma 7.3. Let $0<p_1\le 1$ and $2<p_2,p_3<\infty $ , and let $\mathcal {U}_3^{\eta ,\mathrm {in}/\mathrm {out}}$ be defined as in Equation (7.10). Suppose that $\Vert f_1\Vert _{H^{p_1}({{\mathbb R}^n})}=\Vert f_2\Vert _{H^{p_2}({{\mathbb R}^n})}=\Vert f_3\Vert _{H^{p_3}({{\mathbb R}^n})}=1$ and $\mathcal {L}_s^2[\sigma ]=1$ for $s>n/p_1+n/2$ . Then there exist nonnegative functions $u_1^{\mathrm {in}}$ , $u_1^{\mathrm {out}}$ , $u_2$ , and $u_3$ on ${{\mathbb R}^n}$ such that

$$ \begin{align*}\big\Vert u_1^{\mathrm{in}/\mathrm{out}}\big\Vert_{L^{p_1}({{\mathbb R}^n})}\lesssim 1, \qquad \Vert u_{{\mathrm{i}}}\Vert_{L^{p_{{\mathrm{i}}}}({{\mathbb R}^n})}\lesssim 1 \quad \text{ for }~ {\mathrm{i}}=2,3,\end{align*} $$

and for each $\eta =1,2$

$$ \begin{align*} \sup_{l\in{\mathbb{Z}}}\; \Big| \sum_{k=0}^{\infty}\lambda _k\chi_{(Q_k^{***})^c}(x)\chi_{(B_k^l)^c}(x) \phi_l\ast\big( \mathcal{U}_{3}^{\eta,\mathrm{in}/\mathrm{out}}(x,\cdot)\big)(x)\Big| \lesssim u_1^{\mathrm{in}/\mathrm{out}}(x) u_2(x) u_3(x). \end{align*} $$

The proof of the lemma will be provided in Section 9.

7.1.4 Proof of Equation (7.5) for $\nu =4$

In this case, we divide $\mathcal {U}_4$ into eight types depending on whether x belongs to each of $B_P^l$ and $B_R^l$ and whether ${\Lambda }_ja_k$ is supported in $Q_k^*$ . Indeed, let

(7.11) $$ \begin{align} \Xi_{\eta}(P,R,l):=\begin{cases} P^c\cap R^c\cap (B_P^l)^c\cap (B_R^l)^c, & \eta=1\\ P^c\cap R^c\cap (B_P^l)^c\cap B_R^l, & \eta=2\\ P^c\cap R^c\cap B_P^l\cap (B_R^l)^c, & \eta=3\\ P^c\cap R^c\cap B_P^l\cap B_R^l, & \eta=4, \end{cases} \end{align} $$

and we define

(7.12) $$ \begin{align} \begin{aligned} \mathcal{U}_4^{\eta,\mathrm{in}}(x,y)&:=\sum_{j\in{\mathbb{Z}}} \sum_{P\in\mathcal{D}_j}\sum_{R\in\mathcal{D}_j} b_P^2 b_R^3 T_{\sigma_j}\big(\chi_{Q_k^*}{\Lambda}_ja_k,\psi^P,\theta^R\big)(y) \chi_{\Xi_{\eta}}(x),\\ \mathcal{U}_4^{\eta,\mathrm{out}}(x,y)&:=\sum_{j\in{\mathbb{Z}}} \sum_{P\in\mathcal{D}_j}\sum_{R\in\mathcal{D}_j} b_P^2 b_R^3 T_{\sigma_j}\big(\chi_{(Q_k^*)^c}{\Lambda}_ja_k,\psi^P,\theta^R\big)(y) \chi_{\Xi_{\eta}}(x) \end{aligned} \end{align} $$

for $\eta =1,2,3,4$ .

Then we use the following lemma to obtain the desired result.

Lemma 7.4. Let $0<p_1\le 1$ and $2<p_2,p_3<\infty $ , and let $\mathcal {U}_4^{\eta ,\mathrm {in}/\mathrm {out}}$ be defined as in Equation (7.12). Suppose that $\Vert f_1\Vert _{H^{p_1}({{\mathbb R}^n})}=\Vert f_2\Vert _{H^{p_2}({{\mathbb R}^n})}=\Vert f_3\Vert _{H^{p_3}({{\mathbb R}^n})}=1$ and $\mathcal {L}_s^2[\sigma ]=1$ for $s>n/p_1+n/2$ . Then there exist nonnegative functions $u_1^{\mathrm {in}}$ , $u_1^{\mathrm {out}}$ , $u_2$ , and $u_3$ on ${{\mathbb R}^n}$ such that

(7.13) $$ \begin{align} \big\Vert u_1^{\mathrm{in}/\mathrm{out}}\big\Vert_{L^{p_1}({{\mathbb R}^n})}\lesssim 1, \qquad \Vert u_{{\mathrm{i}}}\Vert_{L^{p_{{\mathrm{i}}}}({{\mathbb R}^n})}\lesssim 1 \quad \text{ for }~ {\mathrm{i}}=2,3, \end{align} $$

and for each $\eta =1,2,3,4,$

(7.14) $$ \begin{align} \sup_{l\in{\mathbb{Z}}}\; \Big| \sum_{k=0}^{\infty}\lambda _k\chi_{(Q_k^{***})^c}(x)\chi_{(B_k^l)^c}(x) \phi_l\ast\big( \mathcal{U}_{4}^{\eta,\mathrm{in}/\mathrm{out}}(x,\cdot)\big)(x)\Big| \lesssim u_1^{\mathrm{in}/\mathrm{out}}(x) u_2(x) u_3(x). \end{align} $$

We will prove the lemma in Section 9.

7.2 Estimate for $\mathcal {J}_2$

Let $x\in (Q_k^{***})^c\cap B_{k,l}$ . For $\nu =1,2,3,4$ , let $\Omega _{\nu }(P,R)$ be defined as in Equation (7.3). Then as in the proof of the estimate for $\mathcal {J}_1$ , we consider the four cases: $x \in \Omega _{1}(P,R)$ , $x \in \Omega _{2}(P,R)$ , $x \in \Omega _{3}(P,R)$ and $x \in \Omega _{4}(P,R)$ . That is, for each $\nu =1,2,3,4$ , let $\mathcal {U}_{\nu }$ be defined as in Equation (7.4) and

$$ \begin{align*}\mathcal{J}_2^{\nu}:=\bigg\Vert \sup_{l\in{\mathbb{Z}}} \Big| \sum_{k=0}^{\infty} \lambda _k \chi_{(Q_k^{***})^c}(x)\chi_{B_k^l}(x) \phi_l\ast \big(\mathcal{U}_{\nu}(x,\cdot) \big)(x)\Big| \bigg\Vert_{L^p(x)}.\end{align*} $$

Then it suffices to show that for each $\nu =1,2,3,4$ ,

(7.15) $$ \begin{align} \mathcal{J}_2^{\nu}\lesssim 1. \end{align} $$

7.2.1 Proof of Equation (7.15) for $\nu =1$

In this case, the proof can be simply reduced to the following lemma, which will be proved in Section 9.

Lemma 7.5. Let $0<p_1\le 1$ and $2<p_2,p_3<\infty $ , and let $\mathcal {U}_1$ be defined as in Equation (7.4). Suppose that $\Vert f_1\Vert _{H^{p_1}({{\mathbb R}^n})}=\Vert f_2\Vert _{H^{p_2}({{\mathbb R}^n})}=\Vert f_3\Vert _{H^{p_3}({{\mathbb R}^n})}=1$ and $\mathcal {L}_s^2[\sigma ]=1$ for $s>n/p_1+n/2$ . Then there exist nonnegative functions $u_1$ , $u_2$ and $u_3$ on ${{\mathbb R}^n}$ such that

$$ \begin{align*}\Vert u_{{\mathrm{i}}}\Vert_{L^{p_{{\mathrm{i}}}}({{\mathbb R}^n})}\lesssim 1 \quad \text{ for }~ {\mathrm{i}}=1,2,3, \end{align*} $$

and for $x\in {{\mathbb R}^n}$

(7.16) $$ \begin{align} \sup_{l\in{\mathbb{Z}}}\; \Big| \sum_{k=0}^{\infty}\lambda _k\chi_{(Q_k^{***})^c}(x)\chi_{B_k^l}(x) \phi_l\ast\big( \mathcal{U}_{1}(x,\cdot)\big)(x)\Big| \lesssim u_1(x) u_2(x) u_3(x). \end{align} $$

Then it follows from Hölder’s inequality that

$$ \begin{align*}\mathcal{J}_2^1\lesssim \Vert u_1\Vert_{L^{p_1}({{\mathbb R}^n})}\Vert u_2\Vert_{L^{p_2}({{\mathbb R}^n})}\Vert u_3\Vert_{L^{p_3}({{\mathbb R}^n})}\lesssim 1.\end{align*} $$

7.2.2 Proof of Equation (7.15) for $\nu =2$

For $P\in \mathcal {D}$ and $l\in {\mathbb {Z}}$ , let $B_P^l:=B_{{\mathbf {x}}_P}^l$ be the ball of center ${\mathbf {x}}_P$ and radius $100n2^{-l}$ as before. We define

(7.17) $$ \begin{align} \begin{aligned} \mathcal{U}_2^{1}(x,y)&:=\sum_{j\in{\mathbb{Z}}} \sum_{P\in\mathcal{D}_j}\sum_{R\in\mathcal{D}_j} b_P^2 b_R^3 T_{\sigma_j}\big({\Lambda}_ja_k,\psi^P,\theta^R\big)(y) \chi_{P^c\cap (B_{P}^l)^c\cap R}(x),\\ \mathcal{U}_2^{2}(x,y)&:=\sum_{j\in{\mathbb{Z}}} \sum_{P\in\mathcal{D}_j}\sum_{R\in\mathcal{D}_j} b_P^2 b_R^3 T_{\sigma_j}\big({\Lambda}_ja_k,\psi^P,\theta^R\big)(y) \chi_{P^c\cap B_P^l\cap R}(x) \end{aligned} \end{align} $$

and write

$$ \begin{align*}\mathcal{J}_2^2\lesssim\mathcal{J}_2^{2,1}+\mathcal{J}_2^{2,2}\end{align*} $$

where

$$ \begin{align*}\mathcal{J}_2^{2,\eta}:=\bigg\Vert \sup_{l\in{\mathbb{Z}}} \Big| \sum_{k=0}^{\infty} \lambda _k \chi_{(Q_k^{***})^c}(x)\chi_{B_k^l}(x) \phi_l\ast \big(\mathcal{U}_{2}^{\eta}(x,\cdot) \big)(x)\Big|\; \bigg\Vert_{L^p(x)}, \quad \eta=1,2.\end{align*} $$

Then we need the following lemmas.

Lemma 7.6. Let $0<p_1\le 1$ and $2<p_2,p_3<\infty $ and let $\mathcal {U}_2^{1}$ be defined as in (7.17). Suppose that $\Vert f_1\Vert _{H^{p_1}({{\mathbb R}^n})}=\Vert f_2\Vert _{H^{p_2}({{\mathbb R}^n})}=\Vert f_3\Vert _{H^{p_3}({{\mathbb R}^n})}=1$ and $\mathcal {L}_s^2[\sigma ]=1$ for $s>n/p_1+n/2$ . Then there exist nonnegative functions $u_1$ , $u_2$ and $u_3$ on ${{\mathbb R}^n}$ such that

$$ \begin{align*} \Vert u_{{\mathrm{i}}}\Vert_{L^{p_{{\mathrm{i}}}}({{\mathbb R}^n})}\lesssim 1 \quad \text{ for }~ {\mathrm{i}}=1,2,3, \end{align*} $$

and

(7.18) $$ \begin{align} \sup_{l\in{\mathbb{Z}}}\; \Big| \sum_{k=0}^{\infty}\lambda _k\chi_{(Q_k^{***})^c}(x)\chi_{B_k^l}(x) \phi_l\ast\big( \mathcal{U}_{2}^{1}(x,\cdot)\big)(x)\Big| \lesssim u_1(x) u_2(x) u_3(x). \end{align} $$

Lemma 7.7. Let $0<p_1\le 1$ and $2<p_2,p_3<\infty $ , and let $\mathcal {U}_2^{2}$ be defined as in Equation (7.17). Suppose that $\Vert f_1\Vert _{H^{p_1}({{\mathbb R}^n})}=\Vert f_2\Vert _{H^{p_2}({{\mathbb R}^n})}=\Vert f_3\Vert _{H^{p_3}({{\mathbb R}^n})}=1$ and $\mathcal {L}_s^2[\sigma ]=1$ for $s>n/p_1+n/2$ . Then there exist nonnegative functions $u_1$ , $u_2$ and $u_3$ on ${{\mathbb R}^n}$ such that

$$ \begin{align*} \Vert u_{{\mathrm{i}}}\Vert_{L^{p_{{\mathrm{i}}}}({{\mathbb R}^n})}\lesssim 1 \quad \text{ for }~ {\mathrm{i}}=1,2,3, \end{align*} $$

and

$$ \begin{align*} \sup_{l\in{\mathbb{Z}}}\; \Big| \sum_{k=0}^{\infty}\lambda _k\chi_{(Q_k^{***})^c}(x)\chi_{B_k^l}(x) \phi_l\ast\big( \mathcal{U}_{2}^{2}(x,\cdot)\big)(x)\Big| \lesssim u_1(x) u_2(x) u_3(x). \end{align*} $$

The above lemmas will be proved in Section 9. Using Lemmas 7.6 and 7.7, we obtain

$$ \begin{align*} \mathcal{J}_2^{2,\eta}\lesssim \Vert u_1\Vert_{L^{p_1}({{\mathbb R}^n})}\Vert u_2\Vert_{L^{p_2}({{\mathbb R}^n})}\Vert u_3\Vert_{L^{p_3}({{\mathbb R}^n})}\lesssim 1,\qquad \eta=1,2, \end{align*} $$

which finishes the proof of Equation (7.15) for $\nu =2$ .

7.2.3 Proof of Equation (7.15) for $\nu =3$

We use the notation $B_R^l:=B_{{\mathbf {x}}_R}^l$ for $R\in \mathcal {D}$ and $l\in {\mathbb {Z}}$ as before and write

$$ \begin{align*}\mathcal{J}_2^3\lesssim \mathcal{J}_2^{3,1}+\mathcal{J}_2^{3,2},\end{align*} $$

where

(7.19) $$ \begin{align} \begin{aligned} \mathcal{U}_3^{1}(x,y)&:=\sum_{j\in{\mathbb{Z}}} \sum_{P\in\mathcal{D}_j}\sum_{R\in\mathcal{D}_j} b_P^2 b_R^3 T_{\sigma_j}\big({\Lambda}_ja_k,\psi^P,\theta^R\big)(y) \chi_{P\cap R^c\cap (B_R^l)^c}(x),\\ \mathcal{U}_3^{2}(x,y)&:=\sum_{j\in{\mathbb{Z}}} \sum_{P\in\mathcal{D}_j}\sum_{R\in\mathcal{D}_j} b_P^2 b_R^3 T_{\sigma_j}\big({\Lambda}_ja_k,\psi^P,\theta^R\big)(y) \chi_{P\cap R^c\cap B_R^l}(x), \end{aligned} \end{align} $$

and

$$ \begin{align*}\mathcal{J}_2^{3,\eta}:=\bigg\Vert \sup_{l\in{\mathbb{Z}}} \Big| \sum_{k=0}^{\infty} \lambda _k \chi_{(Q_k^{***})^c}(x)\chi_{B_k^l}(x) \phi_l\ast \big(\mathcal{U}_{3}^{\eta}(x,\cdot) \big)(x)\Big| \bigg\Vert_{L^p(x)}, \quad \eta=1,2.\end{align*} $$

As in the proof of the case $\nu =2$ , it suffices to prove the following two lemmas.

Lemma 7.8. Let $0<p_1\le 1$ and $2<p_2,p_3<\infty $ , and let $\mathcal {U}_3^{1}$ be defined as in Equation (7.19). Suppose that $\Vert f_1\Vert _{H^{p_1}({{\mathbb R}^n})}=\Vert f_2\Vert _{H^{p_2}({{\mathbb R}^n})}=\Vert f_3\Vert _{H^{p_3}({{\mathbb R}^n})}=1$ and $\mathcal {L}_s^2[\sigma ]=1$ for $s>n/p_1+n/2$ . Then there exist nonnegative functions $u_1$ , $u_2$ and $u_3$ on ${{\mathbb R}^n}$ such that

(7.20) $$ \begin{align} \Vert u_{{\mathrm{i}}}\Vert_{L^{p_{{\mathrm{i}}}}({{\mathbb R}^n})}\lesssim 1 \quad \text{ for }~ {\mathrm{i}}=1,2,3, \end{align} $$

and

(7.21) $$ \begin{align} \sup_{l\in{\mathbb{Z}}}\; \Big| \sum_{k=0}^{\infty}\lambda _k\chi_{(Q_k^{***})^c}(x)\chi_{B_k^l}(x) \phi_l\ast\big( \mathcal{U}_{3}^{1}(x,\cdot)\big)(x)\Big| \lesssim u_1(x) u_2(x) u_3(x). \end{align} $$

Lemma 7.9. Let $0<p_1\le 1$ and $2<p_2,p_3<\infty $ , and let $\mathcal {U}_3^{2}$ be defined as in Equation (7.19). Suppose that $\Vert f_1\Vert _{H^{p_1}({{\mathbb R}^n})}=\Vert f_2\Vert _{H^{p_2}({{\mathbb R}^n})}=\Vert f_3\Vert _{H^{p_3}({{\mathbb R}^n})}=1$ and $\mathcal {L}_s^2[\sigma ]=1$ for $s>n/p_1+n/2$ . Then there exist nonnegative functions $u_1$ , $u_2$ and $u_3$ on ${{\mathbb R}^n}$ such that

(7.22) $$ \begin{align} \Vert u_{{\mathrm{i}}}\Vert_{L^{p_{{\mathrm{i}}}}({{\mathbb R}^n})}\lesssim 1 \quad \text{ for }~ {\mathrm{i}}=1,2,3, \end{align} $$

and

(7.23) $$ \begin{align} \sup_{l\in{\mathbb{Z}}}\; \Big| \sum_{k=0}^{\infty}\lambda _k\chi_{(Q_k^{***})^c}(x)\chi_{B_k^l}(x) \phi_l\ast\big( \mathcal{U}_{3}^{2}(x,\cdot)\big)(x)\Big| \lesssim u_1(x) u_2(x) u_3(x). \end{align} $$

The proof of Lemmas 7.8 and 7.9 will be provided in Section 9.

7.2.4 Proof of Equation (7.15) for $\nu =4$

Let $B_P^l:=B_{{\mathbf {x}}^P}^l$ and $B_R^l:=B_{{\mathbf {x}}_R}^l$ for $P,R\in \mathcal {D}$ and $l\in {\mathbb {Z}}$ , and let $\Xi _{\eta }(P,R,l)$ be defined as in Equation (7.11). Now, we write

$$ \begin{align*}\mathcal{U}_4=\mathcal{U}_4^{1}+\mathcal{U}_4^{2}+\mathcal{U}_4^{3}+\mathcal{U}_4^{4},\end{align*} $$

where

(7.24) $$ \begin{align} \mathcal{U}_4^{\eta}(x,y):=\sum_{j\in{\mathbb{Z}}} \sum_{P\in\mathcal{D}_j}\sum_{R\in\mathcal{D}_j} b_P^2 b_R^3 T_{\sigma_j}\big({\Lambda}_ja_k,\psi^P,\theta^R\big)(y) \chi_{\Xi_{\eta}(P,R,l)}(x), \quad \eta=1,2,3,4. \end{align} $$

Accordingly, we define

$$ \begin{align*}\mathcal{J}_2^{4,\eta}:=\bigg\Vert \sup_{l\in{\mathbb{Z}}} \Big| \sum_{k=0}^{\infty}\lambda _k \chi_{(Q_k^{***})^c}(x)\chi_{B_k^l}(x) \phi_l\ast \big(\mathcal{U}_{4}^{\eta}(x,\cdot) \big)(x)\Big|\; \bigg\Vert_{L^p(x)}, \quad \eta=1,2,3,4.\end{align*} $$

Then we obtain the desired result from the following lemmas.

Lemma 7.10. Let $0<p_1\le 1$ and $2<p_2,p_3<\infty $ , and let $\mathcal {U}_4^{\eta }$ , $\eta =1,2,3$ , be defined as in Equation (7.24). Suppose that $\Vert f_1\Vert _{H^{p_1}({{\mathbb R}^n})}=\Vert f_2\Vert _{H^{p_2}({{\mathbb R}^n})}=\Vert f_3\Vert _{H^{p_3}({{\mathbb R}^n})}=1$ and $\mathcal {L}_s^2[\sigma ]=1$ for $s>n/p_1+n/2$ . Then there exist nonnegative functions $u_1$ , $u_2$ and $u_3$ on ${{\mathbb R}^n}$ such that

$$ \begin{align*} \Vert u_{{\mathrm{i}}}\Vert_{L^{p_{{\mathrm{i}}}}({{\mathbb R}^n})}\lesssim 1 \quad \text{ for }~ {\mathrm{i}}=1,2,3, \end{align*} $$

and for each $\eta =1,2,3$

(7.25) $$ \begin{align} \sup_{l\in{\mathbb{Z}}}\; \Big| \sum_{k=0}^{\infty}\lambda _k\chi_{(Q_k^{***})^c}(x)\chi_{B_k^l}(x) \phi_l\ast\big( \mathcal{U}_{4}^{\eta}(x,\cdot)\big)(x)\Big| \lesssim u_1(x) u_2(x) u_3(x). \end{align} $$

Lemma 7.11. Let $0<p_1\le 1$ and $2<p_2,p_3<\infty $ , and let $\mathcal {U}_4^{4}$ be defined as in Equation (7.24). Suppose that $\Vert f_1\Vert _{H^{p_1}({{\mathbb R}^n})}=\Vert f_2\Vert _{H^{p_2}({{\mathbb R}^n})}=\Vert f_3\Vert _{H^{p_3}({{\mathbb R}^n})}=1$ and $\mathcal {L}_s^2[\sigma ]=1$ for $s>n/p_1+n/2$ . Then there exist nonnegative functions $u_1$ , $u_2$ and $u_3$ on ${{\mathbb R}^n}$ such that

(7.26) $$ \begin{align} \Vert u_{{\mathrm{i}}}\Vert_{L^{p_{{\mathrm{i}}}}({{\mathbb R}^n})}\lesssim 1 \quad \text{ for }~ {\mathrm{i}}=1,2,3, \end{align} $$

and

(7.27) $$ \begin{align} \sup_{l\in{\mathbb{Z}}}\; \Big| \sum_{k=0}^{\infty}\lambda _k\chi_{(Q_k^{***})^c}(x)\chi_{B_k^l}(x) \phi_l\ast\big( \mathcal{U}_{4}^{4}(x,\cdot)\big)(x)\Big| \lesssim u_1(x) u_2(x) u_3(x). \end{align} $$

The proof of the lemmas will be given in Section 9.

8.1 Proof of Equation (8.1)

Since

$$ \begin{align*}\big| \phi_l\ast T_{\sigma}(a_k,f_2,f_3)(x)\big|\lesssim \mathcal{M}T_{\sigma}(a_k,f_2,f_3)(x),\end{align*} $$

the left-hand side of Equation (8.1) is controlled by

$$ \begin{align*} \Big(\sum_{k=1}^{\infty}|\lambda _k|^p\big\Vert \mathcal{M}T_{\sigma}(a_k,f_2,f_3) \big\Vert_{L^p(Q^{***})}^p \Big)^{1/p}. \end{align*} $$

Using Hölder’s inequality, the $L^2$ boundedness of $\mathcal {M}$ and Theorem D, we have

$$ \begin{align*} \big\Vert \mathcal{M}T_{\sigma}(a_k,f_2,f_3) \big\Vert_{L^p(Q^{***})}\lesssim |Q_k|^{1/p-1/2}\big\Vert T_{\sigma}(a_k,f_2,f_3)\big\Vert_{L^2({{\mathbb R}^n})}\lesssim |Q_k|^{1/p-1/2}\Vert a_k\Vert_{L^2({{\mathbb R}^n})}\lesssim 1 \end{align*} $$

and thus Equation (8.1) follows from $\big (\sum _{k=1}^{\infty }|\lambda _k|^p\big )^{1/p}\lesssim 1$ .

8.2 Proof of Equation (8.2)

Let $B_k^l=B({\mathbf {x}}_{Q_k},100 n2^{-l})$ as before. We now decompose the left-hand side of Equation (8.2) as the sum of

$$ \begin{align*}\mathcal{V}_1:= \bigg\| \sup_{l\in\mathbb{Z}} \Big|\sum_{k=1}^{\infty} \lambda_k \chi_{(Q_k^{***})^c}\chi_{(B_k^l)^c}\;\phi_l \ast T_{\sigma}(a_k,f_2,f_3)\Big| \bigg\|_{L^p({{\mathbb R}^n})},\end{align*} $$

$$ \begin{align*}\mathcal{V}_2:= \bigg\| \sup_{l\in\mathbb{Z}} \Big|\sum_{k=1}^{\infty} \lambda_k \chi_{(Q_k^{***})^c}\chi_{B_k^l}\;\phi_l \ast T_{\sigma}(a_k,f_2,f_3)\Big)\Big| \bigg\|_{L^p({{\mathbb R}^n})},\end{align*} $$

and thus we need to show that

$$ \begin{align*} \mathcal{V}_{1},\mathcal{V}_2\lesssim 1. \end{align*} $$

Actually, the proof of these estimates will be complete once we have verified the following lemmas.

Lemma 8.1. Let $0<p\le 1$ . Suppose that $\Vert f_1\Vert _{H^{p}({{\mathbb R}^n})}=\Vert f_2\Vert _{H^{\infty }({{\mathbb R}^n})}=\Vert f_3\Vert _{H^{\infty }({{\mathbb R}^n})}=1$ and $\mathcal {L}_s^2[\sigma ]=1$ for $s>n/p+n/2$ . Then there exist nonnegative functions $u_1$ , $u_2$ and $u_3$ on ${{\mathbb R}^n}$ such that

$$ \begin{align*} \Vert u_{1}\Vert_{L^{p}({{\mathbb R}^n})}\lesssim 1, \qquad \Vert u_{{\mathrm{i}}}\Vert_{L^{\infty}({{\mathbb R}^n})}\lesssim 1\quad \text{ for }~ {\mathrm{i}}=2,3, \end{align*} $$

and

(8.3) $$ \begin{align} \sup_{l\in\mathbb{Z}} \Big|\sum_{k=1}^{\infty} \lambda_k \chi_{(Q_k^{***})^c}(x)\chi_{(B_k^l)^c}(x)\;\phi_l \ast T_{\sigma}\big(a_k,f_2,f_3\big)(x)\Big|\lesssim u_1(x) u_2(x) u_3(x). \end{align} $$

Lemma 8.2. Let $0<p\le 1$ . Suppose that $\Vert f_1\Vert _{H^{p}({{\mathbb R}^n})}=\Vert f_2\Vert _{H^{\infty }({{\mathbb R}^n})}=\Vert f_3\Vert _{H^{\infty }({{\mathbb R}^n})}=1$ and $\mathcal {L}_s^2[\sigma ]=1$ for $s>n/p+n/2$ . Then there exist nonnegative functions $u_1$ , $u_2$ and $u_3$ on ${{\mathbb R}^n}$ such that

$$ \begin{align*} \Vert u_{1}\Vert_{L^{p}({{\mathbb R}^n})}\lesssim 1, \qquad \Vert u_{{\mathrm{i}}}\Vert_{L^{\infty}({{\mathbb R}^n})}\lesssim 1\quad \text{ for }~ {\mathrm{i}}=2,3, \end{align*} $$

and

(8.4) $$ \begin{align} \sup_{l\in\mathbb{Z}} \Big|\sum_{k=1}^{\infty} \lambda_k \chi_{(Q_k^{***})^c}(x)\chi_{B_k^l}(x)\;\phi_l \ast T_{\sigma}\big(a_k,f_2,f_3\big)(x)\Big|\lesssim u_1(x) u_2(x) u_3(x). \end{align} $$

The proof of the two lemmas will be given in Section 9.

9.1 Proof of Lemma 6.1

Let $1<r<2$ such that $s>{3n}/{r}>{3n}/{2}$ , and we claim the pointwise estimate

(9.1) $$ \begin{align} \big|T_{\sigma_j}({\Lambda}_ja_k,{\Lambda}_jf_2,{\Gamma}_jf_3)(y)\big|\lesssim \mathcal{M}_r{\Lambda}_ja_k(y)\mathcal{M}_r{\Lambda}_jf_2(y)\mathcal{M}_r{\Gamma}_jf_3(y). \end{align} $$

Indeed, choosing t so that ${3n}/{r}<3t<s$ , we apply Hölder’s inequality to bound the left-hand side of Equation (9.1) by

$$ \begin{align*} &\int_{({{\mathbb R}^n})^3} \langle 2^j\vec{\boldsymbol{z}}\rangle^{3t} \big| \sigma_j^{\vee}(\vec{\boldsymbol{z}})\big| \frac{|{\Lambda}_ja_k(y-z_1)|}{\langle 2^jz_1\rangle^t} \frac{|{\Lambda}_jf_2(y-z_2)|}{\langle 2^jz_2\rangle^t} \frac{|{\Gamma}_jf_3(y-z_3)|}{\langle 2^jz_3\rangle^t} \; d\vec{\boldsymbol{z}}\\& \quad \le \big\Vert \langle 2^j{\vec{\cdot}\;}\rangle^{3t} \sigma_j^{\vee}\big\Vert_{L^{r'}(({{\mathbb R}^n})^3)} \bigg\Vert \frac{{\Lambda}_ja_k(y-\cdot)}{\langle 2^j\cdot \rangle^t}\bigg\Vert_{L^{r}({{\mathbb R}^n})} \bigg\Vert \frac{{\Lambda}_jf_2(y-\cdot)}{\langle 2^j\cdot \rangle^t}\bigg\Vert_{L^{r}({{\mathbb R}^n})}\bigg\Vert \frac{{\Gamma}_jf_3(y-\cdot)}{\langle 2^j\cdot \rangle^t}\bigg\Vert_{L^{r}({{\mathbb R}^n})}. \end{align*} $$

We observe that

$$ \begin{align*}\big\Vert \langle 2^j{\vec{\cdot}\;}\rangle^{3t} \sigma_j^{\vee}\big\Vert_{L^{r'}(({{\mathbb R}^n})^3)}\lesssim 2^{{3jn}/{r}}\Vert \sigma(2^j{\vec{\cdot}\;}) \Vert_{L^r_{3t}(({{\mathbb R}^n})^3)}\lesssim 2^{{3jn}/{r}}\Vert \sigma(2^j{\vec{\cdot}\;})\Vert_{L^2_s(({{\mathbb R}^n})^3)}\lesssim 2^{{3jn}/{r}}\end{align*} $$

using the Hausdorff–Young inequality, Equation (5.1) and the inclusion

$$ \begin{align*}L_{s_0}^{t_0}(A)\hookrightarrow L_{s_1}^{t_1}(A)\quad \text{ for } ~s_0\ge s_1,~ t_0\ge t_1,\end{align*} $$

where A is a ball of a constant radius, whose proof is contained in [Reference Grafakos and Park19, (1.8)]. Applying Equation (2.5) to the remaining three $L^r$ norms, we finally obtain Equation (9.1).

Now, we choose $\widetilde {r}$ and q such that $2<\widetilde {r}<p_2,p_3$ and ${1}/{q}+{2}/{\widetilde {r}}=1$ . Finally, using the estimate (9.1) and Hölder’s inequality, we have

$$ \begin{align*} &\Big|\sum_{k=0}^{\infty}\lambda _k\chi_{Q_k^{***}}(x)\phi_l\ast \Big( \sum_{j\in{\mathbb{Z}}} T_{\sigma_j}({\Lambda}_ja_k,{\Lambda}_jf_2,{\Gamma}_jf_3) \Big)(x)\Big|\\&\quad \lesssim \sum_{k=0}^{\infty}\lambda _k\chi_{Q_k^{***}}(x) 2^{ln}\int_{|x-y|\le 2^{-l} } \big\Vert \big\{ \mathcal{M}_r{\Lambda}_ja_k(y)\big\}_{j\in{\mathbb{Z}}}\big\Vert_{\ell^2}\big\Vert \big\{ \mathcal{M}_r{\Lambda}_jf_2(y)\big\}_{j\in{\mathbb{Z}}}\big\Vert_{\ell^2} \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \times \big\Vert \big\{ \mathcal{M}_r{\Gamma}_jf_3(y)\big\}_{j\in{\mathbb{Z}}}\big\Vert_{\ell^{\infty}}\; dy\\&\quad \lesssim u_1(x) u_2(x)u_3(x), \end{align*} $$

where we choose

$$ \begin{align*} u_1(x)&:= \sum_{k=0}^{\infty}\lambda _k\chi_{Q_k^{***}}(x) \mathcal{M}_{q}\big( \big\Vert \big\{ \mathcal{M}_r{\Lambda}_ja_k\big\}_{j\in{\mathbb{Z}}}\big\Vert_{\ell^2}\big)(x),\\ u_2(x)&:=\mathcal{M}_{\widetilde{r}}\big( \big\Vert \big\{ \mathcal{M}_r{\Lambda}_jf_2\big\}_{j\in{\mathbb{Z}}}\big\Vert_{\ell^2}\big)(x),\\ u_3(x)&:=\mathcal{M}_{\widetilde{r}}\big( \big\Vert \big\{ \mathcal{M}_r{\Gamma}_jf_3\big\}_{j\in{\mathbb{Z}}}\big\Vert_{\ell^{\infty}}\big)(x) \end{align*} $$

and this proves (6.1). Moreover,

$$ \begin{align*} \Vert u_1\Vert_{L^{p_1}({{\mathbb R}^n})} \le \Big( \sum_{k=0}^{\infty}|\lambda _k|^{p_1}\Big\Vert \mathcal{M}_{q}\big( \big\Vert \big\{ \mathcal{M}_r{\Lambda}_ja_k\big\}_{j\in{\mathbb{Z}}}\big\Vert_{\ell^2}\big) \Big\Vert_{L^{p_1}(Q_k^{***})}^{p_1} \Big)^{{1}/{p_1}}\lesssim 1, \end{align*} $$

where the last inequality follows from Equation (5.6) and the estimate

$$ \begin{align*} &\Big\Vert \mathcal{M}_{q}\big( \big\Vert \big\{ \mathcal{M}_r{\Lambda}_ja_k\big\}_{j\in{\mathbb{Z}}}\big\Vert_{\ell^2}\big) \Big\Vert_{L^{p_1}(Q_k^{***})}\lesssim |Q_k|^{{1}/{p_1}-{1}/{r_0}}\Big\Vert \mathcal{M}_{q}\big( \big\Vert \big\{ \mathcal{M}_r{\Lambda}_ja_k\big\}_{j\in{\mathbb{Z}}}\big\Vert_{\ell^2}\big) \Big\Vert_{L^{r_0}({{\mathbb R}^n})}\\&\quad \qquad \qquad \qquad \qquad \lesssim |Q_k|^{{1}/{p_1}-{1}/{r_0}}\big\Vert \big\{ {\Lambda}_j a_k\big\}_{j\in{\mathbb{Z}}}\big\Vert_{L^{r_0}(\ell^2)}\sim |Q_k|^{{1}/{p_1}-{1}/{r_0}}\Vert a_k\Vert_{L^{r_0}({{\mathbb R}^n})}\lesssim 1 \end{align*} $$

for $q<r_0<\infty $ . Here, we applied Hölder’s inequality, the maximal inequality (2.4), the equivalence in (2.2) and properties of the $H^{p_1}$ -atom $a_k$ . It is also easy to verify

$$ \begin{align*}\Vert u_2\Vert_{L^{p_2}({{\mathbb R}^n})}\lesssim \big\Vert \big\{ {\Lambda}_jf_2\big\}_{j\in{\mathbb{Z}}}\big\Vert_{L^{p_2}(\ell^2)}\sim 1\end{align*} $$

and

$$ \begin{align*}\Vert u_3\Vert_{L^{p_3}({{\mathbb R}^n})}\lesssim \big\Vert \big\{ {\Gamma}_jf_3\big\}_{j\in{\mathbb{Z}}}\big\Vert_{L^{p_3}(\ell^{\infty})}\sim 1\end{align*} $$

using Equations (2.4), (2.1) and (2.2).

9.2 Proof of Lemma 7.1

Since

$$ \begin{align*}s>{n}/{p_1}+{n}/{2}=\big({n}/{p_1}-{n}/{2}\big)+{n}/{2}+{n}/{2},\end{align*} $$

we can choose $s_1,s_2,s_3$ such that $s_1>{n}/{p_1}-{n}/{2}$ , $s_2,s_3>{n}/{2}$ , and $s=s_1+s_2+s_3$ .

Using the estimates

$$ \begin{align*}\big\Vert \psi^P \big\Vert_{L^{\infty}({{\mathbb R}^n})}\le |P|^{-{1}/{2}}\quad \text{ and }\quad \big\Vert \theta^R \big\Vert_{L^{\infty}({{\mathbb R}^n})}\le |R|^{-{1}/{2}},\end{align*} $$

we have

(9.2) $$ \begin{align} \big|\mathcal{U}_1^{\mathrm{in}}(x,y)\big|&\lesssim\sum_{j\in{\mathbb{Z}}} \Big(\sum_{P\in\mathcal{D}_j}|b_P^2||P|^{-{1}/{2}}\chi_P(x) \Big)\Big(\sum_{R\in\mathcal{D}_j}|b_R^3| |R|^{-{1}/{2}}\chi_R(x) \Big) \nonumber \\&\quad \times \int_{({{\mathbb R}^n})^3} \big| \sigma_j^{\vee}(y-z_1,z_2,z_3)\big|\big| {\Lambda}_ja_k(z_1)\big|\chi_{Q_k^*}(z_1) \; d\vec{\boldsymbol{z}} \nonumber \\&\le g^2\big(\{b_P^2\}_{P\in\mathcal{D}} \big)(x) g^{\infty}\big( \{b_R^3\}_{R\in\mathcal{D}}\big)(x)\nonumber\\&\quad \times \bigg( \sum_{j\in{\mathbb{Z}}}\Big(\int_{({{\mathbb R}^n})^3} \big| \sigma_j^{\vee}(y-z_1,z_2,z_3)\big|\big| {\Lambda}_ja_k(z_1)\big| \chi_{Q_k^*}(z_1) \; d\vec{\boldsymbol{z}} \Big)^2 \bigg)^{{1}/{2}} . \end{align} $$

We observe that for $|x-y|\le 2^{-l}$ , $x\in (Q_k^{***})^c\cap (B_k^l)^c$ and $z_1\in Q_k^*$ ,

(9.3) $$ \begin{align} |x-{\mathbf{x}}_{Q_k}|\lesssim |y-z_1| \end{align} $$

and thus, by using Lemma 4.5,

$$ \begin{align*} &\langle 2^j(x-{\mathbf{x}}_{Q_k})\rangle^{s_1}\int_{({{\mathbb R}^n})^3} \big| \sigma_j^{\vee}(y-z_1,z_2,z_3)\big|\big| {\Lambda}_ja_k(z_1)\big| \chi_{Q_k^*}(z_1) \; d\vec{\boldsymbol{z}} \\&\quad \lesssim \ell(Q_k)^{-{n}/{p_1}}\min \big\{1,\big(2^j\ell(Q_k)\big)^M \big\} \int_{({{\mathbb R}^n})^3} \langle 2^j(y-z_1)\rangle^{s_1}\big| \sigma_j^{\vee}(y-z_1,z_2,z_3)\big|\chi_{Q_k^*}(z_1) \; d\vec{\boldsymbol{z}} \\&\quad \lesssim \ell(Q_k)^{-{n}/{p_1}}\min \big\{1,\big(2^j\ell(Q_k)\big)^M \big\} \big\Vert \langle 2^j\cdot \rangle^{-s_2}\big\Vert_{L^2({{\mathbb R}^n})}\big\Vert \langle 2^j\cdot \rangle^{-s_3}\big\Vert_{L^2({{\mathbb R}^n})} 2^{jn} I_{k,j,s}^{\mathrm{in}}(y) \\&\quad \sim \ell(Q_k)^{-{n}/{p_1}}\min \big\{1,\big(2^j\ell(Q_k)\big)^M \big\} I_{k,j,s}^{\mathrm{in}}(y) \end{align*} $$

for sufficiently large M, where

(9.4) $$ \begin{align} I_{k,j,s}^{\mathrm{in}}(y):= 2^{-jn}\int_{Q_k^*}\Big\Vert \langle 2^j(y-z_1),2^jz_2,2^jz_3\rangle^s\big| \sigma_j^{\vee}(y-z_1,z_2,z_3)\big|\Big\Vert_{L^2(z_2,z_3)} \; dz_1. \end{align} $$

This proves that

(9.5) $$ \begin{align} &\int_{({{\mathbb R}^n})^3} \big| \sigma_j^{\vee}(y-z_1,z_2,z_3)\big|\big| {\Lambda}_ja_k(z_1)\big| \chi_{Q_k^*}(z_1) \; d\vec{\boldsymbol{z}} \nonumber\\&\quad \lesssim \ell(Q_k)^{-{n}/{p_1}}\min \big\{1,\big(2^j\ell(Q_k)\big)^M \big\} \langle 2^j(x-{\mathbf{x}}_{Q_k})\rangle^{-s_1} I_{k,j,s}^{\mathrm{in}}(y), \end{align} $$

and therefore, we obtain

(9.6) $$ \begin{align} \big| \mathcal{U}_1^{\mathrm{in}}(x,y)\big|&\lesssim g^2\big(\{b_P^2\}_{P\in\mathcal{D}} \big)(x) g^{\infty}\big( \{b_R^3\}_{R\in\mathcal{D}}\big)(x) \ell(Q_k)^{-{n}/{p_1}} | x-{\mathbf{x}}_{Q_k}|^{-s_1}\\&\quad \times \bigg(\sum_{j\in{\mathbb{Z}}}\Big(2^{-s_1j} \min \big\{1,\big(2^j\ell(Q_k)\big)^{M} \big\} I_{k,j,s}^{\mathrm{in}}(y)\Big)^2 \bigg)^{1/2}\nonumber. \end{align} $$

Similar to Equation (9.2), we write

$$ \begin{align*} \big|\mathcal{U}_1^{\mathrm{out}}(x,y)\big|&\lesssim g^2\big(\{b_P^2\}_{P\in\mathcal{D}} \big)(x) g^{\infty}\big( \{b_R^3\}_{R\in\mathcal{D}}\big)(x) \\&\quad \times \bigg( \sum_{j\in{\mathbb{Z}}}\Big(\int_{({{\mathbb R}^n})^3} \big| \sigma_j^{\vee}(y-z_1,z_2,z_3)\big|\big| {\Lambda}_ja_k(z_1)\big| \chi_{(Q_k^*)^c}(z_1) d\vec{\boldsymbol{z}} \Big)^2 \bigg)^{{1}/{2}}. \end{align*} $$

Instead of Equation (9.3), we make use of the estimate

(9.7) $$ \begin{align} \langle 2^j(x-{\mathbf{x}}_{Q_k})\rangle \lesssim \langle 2^j(y-{\mathbf{x}}_{Q_k})\rangle\le \langle 2^j(y-z_1)\rangle \langle 2^j(z_1-{\mathbf{x}}_{Q_k})\rangle \end{align} $$

for $|x-y|\le 2^{-l}$ and $x\in (Q_k^{***})^c\cap (B_k^l)^c$ . Then, using the argument that led to Equation (9.5), we have

(9.8) $$ \begin{align} &\int_{({{\mathbb R}^n})^3} \big| \sigma_j^{\vee}(y-z_1,z_2,z_3)\big|\big| {\Lambda}_ja_k(z_1)\big| \chi_{(Q_k^*)^c}(z_1) \; d\vec{\boldsymbol{z}} \nonumber\\ &\quad \lesssim \ell(Q_k)^{-{n}/{p_1}}\min\big\{ 1,\big( 2^j\ell(Q_k)\big)^M\big\} \langle 2^j(x-{\mathbf{x}}_{Q_k})\rangle^{-s_1}I_{k,j,s}^{\mathrm{out}}(y), \end{align} $$

where M, $L_0$ are sufficiently large numbers and

(9.9) $$ \begin{align} I_{k,j,s}^{\mathrm{out}}(y)&:=2^{-jn}\int_{(Q_k^*)^c} \frac{(2^j\ell(Q_k))^n}{\langle 2^j(z_1-{\mathbf{x}}_{Q_k})\rangle^{L_0-s_1}} \\ &\quad \times \Big\Vert \langle 2^j(y-z_1),2^jz_2,2^jz_3\rangle^s\big| \sigma_j^{\vee}(y-z_1,z_2,z_3)\big|\Big\Vert_{L^2(z_2,z_3)} \; dz_1.\nonumber \end{align} $$

Now, we deduce

(9.10) $$ \begin{align} \big| \mathcal{U}_1^{\mathrm{out}}(x,y)\big|&\lesssim g^2\big(\{b_P^2\}_{P\in\mathcal{D}} \big)(x) g^{\infty}\big( \{b_R^3\}_{R\in\mathcal{D}}\big)(x) \ell(Q_k)^{-{n}/{p_1}}| x-{\mathbf{x}}_{Q_k}|^{-s_1}\\ &\quad \times \bigg(\sum_{j\in{\mathbb{Z}}}\Big(2^{-s_1j} \min \big\{1,\big(2^j\ell(Q_k)\big)^{M} \big\} I_{k,j,s}^{\mathrm{out}}(y)\Big)^2 \bigg)^{{1}/{2}}.\nonumber \end{align} $$

According to Equations (9.6) and (9.10), the estimate (7.7) follows from taking

$$ \begin{align*} u_1^{\mathrm{in}/\mathrm{out}}(x)&:=\sum_{k=0}^{\infty}|\lambda _k|\ell(Q_k)^{-{n}/{p_1}}\chi_{(Q_k^{***})^c}(x)| x-{\mathbf{x}}_{Q_k}|^{-s_1}\\ &\quad \times\mathcal{M}\bigg[\bigg(\sum_{j\in{\mathbb{Z}}}\Big(2^{-s_1j}\min{\{1,(2^j\ell(Q_k))^M\}}I_{k,j,s}^{\mathrm{in}/\mathrm{out}}(\cdot) \Big)^2 \bigg)^{{1}/{2}}\bigg](x),\nonumber\\ u_2(x)&:=g^2\big(\{b_P^2\}_{P\in\mathcal{D}} \big)(x) ,\nonumber\\ u_3(x)&:=g^{\infty}\big( \{b_R^3\}_{R\in\mathcal{D}}\big)(x).\nonumber \end{align*} $$

It is clear that

(9.11) $$ \begin{align} \Vert u_2\Vert_{L^{p_2}({{\mathbb R}^n})}&=\big\Vert \{b_P^2\}_{P\in\mathcal{D}}\big\Vert_{\dot{f}^{p_2,2}}\sim 1 \end{align} $$

(9.12) $$ \begin{align} \Vert u_3\Vert_{L^{p_3}({{\mathbb R}^n})}&=\big\Vert \{b_R^3\}_{R\in\mathcal{D}}\big\Vert_{\dot{f}^{p_3,\infty}}\sim 1 \end{align} $$

in view of Equations (7.1) and (7.2). To estimate $u_1^{\mathrm {in}}$ and $u_1^{\mathrm {out}}$ , we note that

(9.13) $$ \begin{align} \big\Vert I_{k,j,s_1}^{\mathrm{in}}\big\Vert_{L^2({{\mathbb R}^n})}&\le 2^{-jn}\int_{Q_k}\bigg( \int_{{{\mathbb R}^n}}\Big\Vert \langle 2^jy,2^jz_2,2^jz_3\rangle^s \big| \sigma_j^{\vee}(y,z_2,z_3)\big|\Big\Vert_{L^2(z_2,z_3)}^2 \; dy\bigg)^{{1}/{2}} \; dz_1\nonumber\\0 &=2^{-jn}\ell(Q_k)^n\big\Vert \langle 2^j{\vec{\cdot}\;}\rangle^s\sigma_j^{\vee}\big\Vert_{L^2(({{\mathbb R}^n})^3)}\le 2^{{jn}/{2}}\ell(Q_k)^n, \end{align} $$

where we applied Minkowski’s inequality and a change of variables, and similarly,

(9.14) $$ \begin{align} \big\Vert I_{k,j,s}^{\mathrm{out}}\big\Vert_{L^2({{\mathbb R}^n})}&\lesssim 2^{-jn}\int_{(Q_k^*)^c} \frac{(2^j\ell(Q_k))^n}{\langle 2^j(z_1-{\mathbf{x}}_{Q_k})\rangle^{L_0-s_1}} \; dz_1 \big\Vert \langle 2^j{\vec{\cdot}\;}\rangle^s\sigma_j^{\vee} \big\Vert_{L^2(({{\mathbb R}^n})^3)}\nonumber\\ &\lesssim 2^{{jn}/{2}}\ell(Q_k)^n\big( 2^j\ell(Q_k)\big)^{-(L_0-s_1-n)} \end{align} $$

for $L_0>s+n$ . Now, we have

$$ \begin{align*} \big\Vert u_1^{\mathrm{in}}\big\Vert_{L^{p_1}({{\mathbb R}^n})}^{p_1}&\le \sum_{k=0}^{\infty}|\lambda _k|^{p_1}\ell(Q_k)^{-n}\int_{(Q_k^{***})^c} | x-{\mathbf{x}}_{Q_k}|^{-s_1p_1}\\ &\quad \times \bigg( \mathcal{M}\bigg[\bigg(\sum_{j\in{\mathbb{Z}}}\Big(2^{-s_1j}\min{\big\{1,\big(2^j\ell(Q_k)\big)^M\big\}}I_{k,j,s_1}^{\mathrm{in}}(\cdot) \Big)^2 \bigg)^{1/2}\bigg](x) \bigg)^{p_1}\, dx \end{align*} $$

and the integral is dominated by

$$ \begin{align*} &\big\Vert |\cdot-{\mathbf{x}}_{Q_k}|^{-{s_1p_1}}\big\Vert_{L^{({2}/{p_1})'}((Q_k^{***})^c)}\\ &\quad \times \bigg\Vert \bigg( \mathcal{M}\bigg[\bigg(\sum_{j\in{\mathbb{Z}}}\Big(2^{-s_1j}\min{\big\{1,\big(2^j\ell(Q_k)\big)^M\big\}}I_{k,j,s_1}^{\mathrm{in}}(\cdot) \Big)^2 \bigg)^{1/2}\bigg] \bigg)^{p_1} \bigg\Vert_{L^{{2}/{p_1}}({{\mathbb R}^n})}. \end{align*} $$

The first term is no more than a constant times $\ell (Q_k)^{-p_1(s_1-({n}/{p_1}-{n}/{2}))}$ , and the second one is bounded by

$$ \begin{align*} &\bigg( \sum_{j\in{\mathbb{Z}}} \Big(2^{-s_1j}\min \big\{1,\big( 2^j\ell(Q_k)\big)^N \big\}\big\Vert I_{k,j,s_1}^{\mathrm{in}}\big\Vert_{L^2({{\mathbb R}^n})} \Big)^2\bigg)^{{p_1}/{2}}\\ &\quad \lesssim \ell(Q_k)^{p_1n}\bigg(\sum_{j\in{\mathbb{Z}}}\Big( 2^{-s_1j} \min \big\{1,\big( 2^j\ell(Q_k)\big)^M \big\} 2^{{jn}/{2}} \Big)^2 \bigg)^{{p_1}/{2}}\lesssim \ell(Q_k)^{s_1p_1+{p_1n}/{2}}, \end{align*} $$

due to Equation (9.13). This proves

(9.15) $$ \begin{align} \big\Vert u_1^{\mathrm{in}}\big\Vert_{L^{p_1}({{\mathbb R}^n})}\lesssim \Big( \sum_{k=0}^{\infty}|\lambda _k|^{p_1}\Big)^{{1}/{p_1}}\lesssim 1. \end{align} $$

In a similar way, together with Equation (9.14), we can also prove

(9.16) $$ \begin{align} \big\Vert u_1^{\mathrm{out}}\big\Vert_{L^{p_1}({{\mathbb R}^n})}\lesssim \Big( \sum_{k=0}^{\infty}|\lambda _k|^{p_1}\Big)^{{1}/{p_1}}\lesssim 1, \end{align} $$

choosing $M>L_0-{3n}/{2}$ .

9.3 Proof of Lemma 7.2

As in the proof of Lemma 7.1, we pick $s_1,s_2,s_3$ satisfying $s_1>{n}/{p_1}-{n}/{2}$ , $s_2,s_3>{n}/{2}$ , and $s=s_1+s_2+s_3>n/p_1+n/2$ .

We first consider the case $\eta =1$ . For $x\in P^c\cap (B_P^l)^c$ and $|x-y|\le 2^{-l}$ , we have

(9.17) $$ \begin{align} \langle 2^j(x-{\mathbf{x}}_{P})\rangle\lesssim \langle 2^j(y-{\mathbf{x}}_P)\rangle \le \langle 2^j(y-z_2)\rangle \langle 2^j(z_2-{\mathbf{x}}_P)\rangle. \end{align} $$

By using

$$ \begin{align*}\big\Vert \theta^R\big\Vert_{L^{\infty}({{\mathbb R}^n})}\le |R|^{-{1}/{2}},\end{align*} $$

we have

(9.18) $$ \begin{align} \big| \mathcal{U}_2^{1,\mathrm{in}}(x,y)\big| &\lesssim \sum_{j\in{\mathbb{Z}}}\Big( \sum_{R\in\mathcal{D}_j}|b_R^3||R|^{-{1}/{2}}\chi_R(x)\Big)\int_{({{\mathbb R}^n})^3} \big|\sigma_j^{\vee}(y-z_1,y-z_2,z_3) \big|\\&\quad \times \big| {\Lambda}_ja_k(z_1)\big|\chi_{Q_k^*}(z_1)\Big(\sum_{P\in\mathcal{D}_j} |b_P^2|\chi_{P^c}(x)\chi_{(B_P^l)^c}(x)\big| \psi^P(z_2)\big| \Big) \; d\vec{\boldsymbol{z}}.\nonumber \end{align} $$

Using Equations (9.3) and (9.17) and Lemma 4.5, the integral in the preceding expression is bounded by

$$ \begin{align*} & \ell(Q_k)^{-{n}/{p_1}}\min\big\{1,\big(2^j\ell(Q_k) \big)^M \big\}\langle2^j(x-c_{Q_k})\rangle^{-s_1}\int_{({{\mathbb R}^n})^3} \langle 2^j(y-z_1)\rangle^{s_1}\langle 2^j(y-z_2)\rangle^{s_2} \\& \qquad \times \big| \sigma_j^{\vee}(y-z_1,y-z_2,z_3)\big|\chi_{Q_k^*}(z_1)\bigg(\sum_{P\in\mathcal{D}_j}|b_P^2|\frac{\chi_{P^c}(x)}{\langle 2^j(x-{\mathbf{x}}_P)\rangle^{s_2}}\big| \widetilde{\psi^P}(z_2)\big| \bigg) \; d\vec{\boldsymbol{z}}\\&\quad \lesssim \ell(Q_k)^{-{n}/{p_1}}\min\big\{1,\big(2^j\ell(Q_k) \big)^M \big\} \langle2^j(x-{\mathbf{x}}_{Q_k})\rangle^{-s_1}I_{k,j,s}^{\mathrm{in}}(y) \\&\qquad \times 2^{{jn}/{2}}\bigg\Vert \sum_{P\in\mathcal{D}_j}|b_P^2|\frac{\chi_{P^c}(x)}{\langle 2^j(x-{\mathbf{x}}_P)\rangle^{s_2}}\big| \widetilde{\psi^P}(\cdot)\big|\bigg\Vert_{L^2({{\mathbb R}^n})} \end{align*} $$

for sufficiently large $M>0$ , where $\widetilde {\psi ^P}(z_2):=\langle 2^j(z_2-{\mathbf {x}}_P)\rangle ^{s_2}\psi ^P(z_2)$ for $P\in \mathcal {D}_j$ and $I_{k,j,s}^{\mathrm {in}}$ is defined as in Equation (9.4). Note that

(9.19) $$ \begin{align} |b_P^2|\lesssim \mathscr{B}_P^2(f_2):=\Big\langle \big| \widetilde{{\Lambda}_j}f_2\big|,\frac{2^{{jn}/{2}}}{\langle 2^j(\cdot-{\mathbf{x}}_P)\rangle^L}\Big\rangle \quad \text{ for }~L>n,s, \end{align} $$

and thus it follows from Lemma 4.2 that the $L^2$ norm in the last displayed expression is dominated by

$$ \begin{align*}2^{-{jn}/{2}}\bigg( \sum_{P\in\mathcal{D}_j} \Big( | \mathscr{B}_P^2(f_2)| |P|^{-{1}/{2}} \frac{\chi_{P^c}(x)}{\langle 2^j(x-{\mathbf{x}}_P)\rangle^{s_2}}\Big)^2\bigg)^{{1}/{2}}.\end{align*} $$

This yields that

(9.20) $$ \begin{align} \big| \mathcal{U}_2^{1,\mathrm{in}}(x,y) \big|&\lesssim \ell(Q_k)^{-{n}/{p_1}}|x-c_{Q_k}|^{-s_1}\Big( \sum_{j\in{\mathbb{Z}}} \big(2^{-js_1}\min\big\{1,\big(2^j\ell(Q_k) \big)^M \big\}I_{k,j,s}^{\mathrm{in}}(y) \big)^2\Big)^{{1}/{2}} \nonumber\\&\quad \times \bigg( \sum_{P\in\mathcal{D}} \Big( \big| \mathscr{B}_P^2(f_2)\big| |P|^{-{1}/{2}} \frac{\chi_{P^c}(x)}{\langle 2^j(x-{\mathbf{x}}_P)\rangle^{s_2}}\Big)^2 \bigg)^{{1}/{2}}g^{\infty}\big(\{b_R^3\}_{R\in\mathcal{D}}\big)(x). \end{align} $$

Similarly, using Equations (9.7) and (9.17), Lemma 4.5 and Lemma 4.2, we have

(9.21) $$ \begin{align} \big| \mathcal{U}_2^{1,\mathrm{out}}(x,y) \big|&\lesssim \sum_{j\in{\mathbb{Z}}}\Big(\sum_{R\in\mathcal{D}_j} |b_R^3| |R|^{-{1}/{2}}\chi_R(x) \Big) \int_{({{\mathbb R}^n})^3} \big|\sigma_j^{\vee}(y-z_1,y-z_2,z_3) \big|\nonumber\\&\qquad \times \big| {\Lambda}_ja_k(z_1)\big|\chi_{(Q_k^*)^c}(z_1)\Big(\sum_{P\in\mathcal{D}_j} |b_P^2|\chi_{P^c}(x)\chi_{(B_P^l)^c}(x)\big| \psi^P(z_2)\big| \Big) \; d\vec{\boldsymbol{z}} \nonumber\\&\quad \lesssim \ell(Q_k)^{-{n}/{p_1}}|x-{\mathbf{x}}_{Q_k}|^{-s_1}\Big( \sum_{j\in{\mathbb{Z}}} \big(2^{-js_1}\min\big\{1,\big(2^j\ell(Q_k) \big)^M \big\}I_{k,j,s}^{\mathrm{out}}(y) \big)^2\Big)^{{1}/{2}} \nonumber\\&\qquad \times \bigg( \sum_{P\in\mathcal{D}} \Big( \big| \mathscr{B}_P^2(f_2)\big| |P|^{-{1}/{2}} \frac{\chi_{P^c}(x)}{\langle 2^j(x-{\mathbf{x}}_P)\rangle^{s_2}}\Big)^2 \bigg)^{{1}/{2}}g^{\infty}\big(\{b_R^3\}_{R\in\mathcal{D}}\big)(x), \end{align} $$

where $I_{k,j,s}^{\mathrm {out}}$ is defined as in Equation (9.9).

When $\eta =2$ , we use the inequality

(9.22) $$ \begin{align} \langle 2^j(x-{\mathbf{x}}_{Q_k})\rangle^{-1}\le \langle 2^j (x-{\mathbf{x}}_{P})\rangle^{-1} \end{align} $$

for $x\in (B_k^l)^c \cap B_{P}^l$ . Then, similar to Equation (9.18), we have

$$ \begin{align*} \big| \mathcal{U}_2^{2,\mathrm{in}}(x,y) \big|&\lesssim \sum_{j\in{\mathbb{Z}}}\Big( \sum_{R\in\mathcal{D}_j}|b_R^3||R|^{-{1}/{2}}\chi_R(x)\Big)\int_{({{\mathbb R}^n})^3} \big|\sigma_j^{\vee}(y-z_1,y-z_2,z_3) \big|\\[3pt]&\quad \times \big| {\Lambda}_ja_k(z_1)\big|\chi_{Q_k^*}(z_1)\Big(\sum_{P\in\mathcal{D}_j} |b_P^2|\chi_{P^c}(x)\chi_{B_P^l}(x)\big| \psi^P(z_2)\big| \Big) \; d\vec{\boldsymbol{z}}\nonumber, \end{align*} $$

and the integral is dominated by a constant times

$$ \begin{align*} &\ell(Q_k)^{-{n}/{p_1}}\min \{1,\big( 2^j\ell(Q_k)\big)^M\}\langle 2^j(x-{\mathbf{x}}_{Q_k})\rangle^{-(s_1+s_2)}\int_{({{\mathbb R}^n})^3}\langle 2^j(y-z_1)\rangle^{s_1+s_2}\\[3pt]&\qquad \times \big| \sigma_j^{\vee}(y-z_1,y-z_2,z_3)\big|\chi_{Q_k^*}(z_1)\Big( \sum_{P\in\mathcal{D}_j}|b_P^2|\chi_{P^c}(x)\chi_{B_P^l}(x)\big| {\psi^P}(z_2)\big|\Big)\;d\vec{\boldsymbol{z}}\\[3pt]&\quad \le \ell(Q_k)^{-{n}/{p_1}}\min \{1,\big( 2^j\ell(Q_k)\big)^M\}\langle 2^j(x-{\mathbf{x}}_{Q_k})\rangle^{-s_1}\int_{({{\mathbb R}^n})^3}\langle 2^j(y-z_1)\rangle^{s_1+s_2}\\[3pt]&\qquad\times \big| \sigma_j^{\vee}(y-z_1,y-z_2,z_3)\big|\chi_{Q_k^*}(z_1)\Big( \sum_{P\in\mathcal{D}_j}|b_P^2|\frac{\chi_{P^c}(x)}{\langle 2^j(x-{\mathbf{x}}_P)\rangle^{s_2}}\big| {\psi^P}(z_2)\big|\Big)\;d\vec{\boldsymbol{z}}\\[3pt]&\quad \lesssim \ell(Q_k)^{-{n}/{p_1}}\min\big\{1,\big(2^j\ell(Q_k) \big)^M \big\} \langle2^j(x-{\mathbf{x}}_{Q_k})\rangle^{-s_1}I_{k,j,s}^{\mathrm{in}}(y)\\[3pt]&\qquad \times \bigg( \sum_{P\in\mathcal{D}} \Big( \big| \mathscr{B}_P^2(f_2)\big| |P|^{-{1}/{2}} \frac{\chi_{P^c}(x)}{\langle 2^j(x-{\mathbf{x}}_P)\rangle^{s_2}}\Big)^2 \bigg)^{{1}/{2}} \end{align*} $$

due to Equations (9.3) and (9.22), Lemma 4.5 and Lemma 4.2, where $I_{k,j,s}^{\mathrm {in}}$ and $\mathscr {B}_P^2(f_2)$ are defined as in Equations (9.4) and (9.19). Therefore,

(9.23) $$ \begin{align} \big| \mathcal{U}_2^{2,\mathrm{in}}(x,y)\big|&\lesssim \ell(Q_k)^{-{n}/{p_1}}|x-{\mathbf{x}}_{Q_k}|^{-s_1}\Big( \sum_{j\in{\mathbb{Z}}} \big(2^{-js_1}\min\big\{1,\big(2^j\ell(Q_k) \big)^M \big\}I_{k,j,s}^{\mathrm{in}}(y) \big)^2\Big)^{{1}/{2}}\nonumber\\&\quad \times \bigg( \sum_{P\in\mathcal{D}} \Big( \big| \mathscr{B}_P^2(f_2)\big| |P|^{-{1}/{2}} \frac{\chi_{P^c}(x)}{\langle 2^j(x-{\mathbf{x}}_P)\rangle^{s_2}}\Big)^2 \bigg)^{{1}/{2}} g^{\infty}\big(\{b_R^3\}_{R\in\mathcal{D}}\big)(x). \end{align} $$

Similarly, we can also prove that

(9.24) $$ \begin{align} \big| \mathcal{U}_2^{2,\mathrm{out}}(x,y)\big|&\lesssim \ell(Q_k)^{-{n}/{p_1}}|x-{\mathbf{x}}_{Q_k}|^{-s_1}\Big( \sum_{j\in{\mathbb{Z}}} \big(2^{-js_1}\min\big\{1,\big(2^j\ell(Q_k) \big)^M \big\}I_{k,j,s}^{\mathrm{out}}(y) \big)^2\Big)^{{1}/{2}}\nonumber\\&\quad \times \bigg( \sum_{P\in\mathcal{D}} \Big( \big| \mathscr{B}_P^2(f_2)\big| |P|^{-{1}/{2}} \frac{\chi_{P^c}(x)}{\langle 2^j(x-{\mathbf{x}}_P)\rangle^{s_2}}\Big)^2 \bigg)^{{1}/{2}} g^{\infty}\big(\{b_R^3\}_{R\in\mathcal{D}}\big)(x). \end{align} $$

Combining Equations (9.20), (9.21), (9.23) and (9.24), the estimate (7.9) holds with

$$ \begin{align*} u_1^{\mathrm{in}/\mathrm{out}}(x)&:=\sum_{k=0}^{\infty}|\lambda _k|\ell(Q_k)^{-{n}/{p_1}}\chi_{(Q_k^{***})^c}(x)| x-{\mathbf{x}}_{Q_k}|^{-s_1}\\&\quad \times\mathcal{M}\bigg[\bigg(\sum_{j\in{\mathbb{Z}}}\Big(2^{-s_1j}\min{\{1,(2^j\ell(Q_k))^M\}}I_{k,j,s}^{\mathrm{in}/\mathrm{out}}(\cdot) \Big)^2 \bigg)^{{1}/{2}}\bigg](x),\\u_2(x)&:= \bigg(\sum_{j\in{\mathbb{Z}}} \sum_{P\in\mathcal{D}_j} \Big( \big| \mathscr{B}_P^2(f_2)\big| |P|^{-{1}/{2}} \frac{\chi_{P^c}(x)}{\langle 2^j(x-{\mathbf{x}}_P)\rangle^{s_2}}\Big)^2 \bigg)^{{1}/{2}},\\u_3(x)&:=g^{\infty}\big( \{b_R^3\}_{R\in\mathcal{D}}\big)(x). \end{align*} $$

Clearly, as in Equations (9.15), (9.16) and (9.12),

$$ \begin{align*} \big\Vert u_1^{\mathrm{in}/\mathrm{out}}\big\Vert_{L^{p_1}({{\mathbb R}^n})}\lesssim 1, \qquad \Vert u_3\Vert_{L^{p_3}({{\mathbb R}^n})}\lesssim 1 \end{align*} $$

and Lemma 4.3 proves that

$$ \begin{align*} \Vert u_2\Vert_{L^{p_2}({{\mathbb R}^n})}\lesssim 1. \end{align*} $$

9.4 Proof of Lemma 7.3

The proof is almost same as that of Lemma 7.2. By letting $M>0$ be sufficiently large and exchanging the role of terms associated with $f_2$ and $f_3$ in the estimate (9.18), we may obtain

(9.25) $$ \begin{align} \big| \mathcal{U}_3^{1,\mathrm{in}}(x,y)\big|&\lesssim \sum_{j\in{\mathbb{Z}}}\Big( \sum_{P\in\mathcal{D}_j}|b_P^2||P|^{-{1}/{2}}\chi_P(x)\Big)\int_{({{\mathbb R}^n})^3} \big|\sigma_j^{\vee}(y-z_1,z_2,y-z_3) \big|\nonumber\\&\qquad \times \big| {\Lambda}_ja_k(z_1)\big|\chi_{Q_k^*}(z_1)\Big(\sum_{R\in\mathcal{D}_j} |b_R^3|\chi_{R^c}(x)\chi_{(B_R^l)^c}(x)\big| \theta^R(z_3)\big| \Big) \; d\vec{\boldsymbol{z}}\nonumber\\&\quad \lesssim \ell(Q_k)^{-{n}/{p_1}}|x-{\mathbf{x}}_{Q_k}|^{-s_1}\bigg( \sum_{j\in{\mathbb{Z}}} \Big( 2^{-js_1}\min{\big\{ 1,\big( 2^j\ell(Q_k)\big)^M\big\}} I_{k,j,s}^{\mathrm{in}}(y)\Big)^2 \bigg)^{{1}/{2}}\nonumber\\&\qquad \times g^2\big(\{b_P^2 \}_{P\in\mathcal{D}} \big)(x)\sup_{j\in{\mathbb{Z}}}\bigg( \sum_{R\in\mathcal{D}_j} \Big(\big| \mathscr{B}_R^3(f_3) \big| |R|^{-{1}/{2}}\frac{1}{\langle 2^j(x-{\mathbf{x}}_R)\rangle ^{s_3}} \Big)^2 \bigg)^{{1}/{2}}, \end{align} $$

where $I_{k,j,s}^{\mathrm {in}}$ is defined as in Equation (9.4), $\widetilde {\theta ^R}(z_3):=\langle 2^j(z_3-{\mathbf {x}}_R)\rangle ^{s_3}\theta ^R(z_3)$ for $R\in \mathcal {D}_j$ and

(9.26) $$ \begin{align} \mathscr{B}_R^3(f_3):=\Big\langle \big| \widetilde{{\Gamma}_j}f_3\big|,\frac{2^{{jn}/{2}}}{\langle 2^j(\cdot-{\mathbf{x}}_R)\rangle^L}\Big\rangle \quad \text{ for }~L>s,n. \end{align} $$

Similarly,

(9.27) $$ \begin{align} \big| \mathcal{U}_3^{1,\mathrm{out}}(x,y) \big|&\lesssim \ell(Q_k)^{-{n}/{p_1}} |x-{\mathbf{x}}_{Q_k}|^{-s_1} \bigg( \sum_{j\in{\mathbb{Z}}} \Big(2^{-js_1} \min{\big\{ 1,\big( 2^j\ell(Q_k)\big)^M\big\}} I_{k,j,s}^{\mathrm{out}}(y)\Big)^2 \bigg)^{{1}/{2}}\nonumber\\&\quad \times g^2\big(\{b_P^2 \}_{P\in\mathcal{D}} \big)(x)\sup_{j\in{\mathbb{Z}}}\bigg( \sum_{R\in\mathcal{D}_j} \Big(\big| \mathscr{B}_R^3(f_3) \big| |R|^{-{1}/{2}}\frac{1}{\langle 2^j(x-{\mathbf{x}}_R)\rangle ^{s_3}} \Big)^2 \bigg)^{{1}/{2}}, \end{align} $$

where $I_{k,j,s}^{\mathrm {out}}$ is defined as in Equation (9.9).

For the case $\eta =2$ , we use the fact that for $x\in (B_k^l)^c\cap B_R^l$ ,

$$ \begin{align*}\langle 2^j(x-{\mathbf{x}}_{Q_k})\rangle^{-1}\le \langle 2^j(x-{\mathbf{x}}_R)\rangle^{-1},\end{align*} $$

instead of Equation (9.22). Then we have

$$ \begin{align*} \big| \mathcal{U}_3^{2,\mathrm{in}}(x,y)\big| &\lesssim \ell(Q_k)^{-{n}/{p_1}}|x-{\mathbf{x}}_{Q_k}|^{-s_1}\Big( \sum_{j\in{\mathbb{Z}}} \big(2^{-js_1}\min\big\{1,\big(2^j\ell(Q_k) \big)^M \big\}I_{k,j,s}^{\mathrm{in}}(y) \big)^2\Big)^{{1}/{2}}\\&\quad \times g^{2}\big(\{b_P^2\}_{P\in\mathcal{D}}\big)(x) \sup_{j\in{\mathbb{Z}}}\bigg( \sum_{R\in\mathcal{D}_j} \Big(\big| \mathscr{B}_R^3(f_3) \big| |R|^{-{1}/{2}}\frac{1}{\langle 2^j(x-{\mathbf{x}}_R)\rangle ^{s_3}} \Big)^2 \bigg)^{{1}/{2}} \end{align*} $$

and

$$ \begin{align*} \big| \mathcal{U}_3^{2,\mathrm{out}}(x,y) \big|&\lesssim \ell(Q_k)^{-{n}/{p_1}}\bigg( \sum_{j\in{\mathbb{Z}}} \Big( \min{\big\{ 1,\big( 2^j\ell(Q_k)\big)^M\big\}} \langle 2^j(x-{\mathbf{x}}_Q)\rangle^{-s_1}I_{k,j,s}^{\mathrm{out}}(y)\Big)^2 \bigg)^{{1}/{2}}\\&\quad \times g^2\big(\{b_P^2 \}_{P\in\mathcal{D}} \big)\sup_{j\in{\mathbb{Z}}}\bigg( \sum_{R\in\mathcal{D}_j} \Big(\big| \mathscr{B}_R^3(f_3) \big| |R|^{-{1}/{2}}\frac{1}{\langle 2^j(x-{\mathbf{x}}_R)\rangle ^{s_3}} \Big)^2 \bigg)^{{1}/{2}} \end{align*} $$

which are analogous to Equations (9.25) and (9.27).

Then Lemma 7.3 follows from Equations (9.15), (9.16) and (9.11) and Lemma 4.4 by choosing

$$ \begin{align*} u_1^{\mathrm{in}/\mathrm{out}}(x)&:= \sum_{k=0}^{\infty}|\lambda _k|\ell(Q_k)^{-{n}/{p_1}}\chi_{(Q_k^{***})^c}(x)| x-{\mathbf{x}}_{Q_k}|^{-s_1}\\&\quad \times\mathcal{M}\bigg[\bigg(\sum_{j\in{\mathbb{Z}}}\Big(2^{-s_1j}\min{\{1,(2^j\ell(Q_k))^M\}}I_{k,j,s}^{\mathrm{in}/\mathrm{out}}(\cdot) \Big)^2 \bigg)^{{1}/{2}}\bigg](x) \\u_2(x)&:= g^2\big(\{b_P^2 \}_{P\in\mathcal{D}} \big)(x) \\u_3(x)&:= \sup_{j\in{\mathbb{Z}}}\bigg( \sum_{R\in\mathcal{D}_j} \Big(\big| \mathscr{B}_R^3(f_3) \big| |R|^{-{1}/{2}}\frac{1}{\langle 2^j(x-{\mathbf{x}}_R)\rangle ^{s_3}} \Big)^2 \bigg)^{{1}/{2}}. \end{align*} $$

9.5 Proof of Lemma 7.4

Let $I_{k,j,s}^{\mathrm {in}}$ , $I_{k,j,s}^{\mathrm {out}}$ , $\mathscr {B}_P^2(f_2)$ and $\mathscr {B}_R^3(f_3)$ be defined as before. Let $M>0$ be a sufficiently large number. We claim the pointwise estimates that for each $\eta =1,2,3,4$ ,

$$ \begin{align*} \big| \mathcal{U}_4^{\eta,\mathrm{in}/\mathrm{out}}(x,y)\big|&\lesssim \ell(Q_k)^{-{n}/{p_1}}|x-x_{Q_k}|^{-s_1}\bigg( \sum_{j\in{\mathbb{Z}}}\Big( 2^{-js_1}\min\big\{1,\big( 2^j\ell(Q_k)\big)^M \big\}I_{k,j,s}^{\mathrm{in}/\mathrm{out}}(y) \Big)^2\bigg)^{{1}/{2}}\\&\quad \times \bigg( \sum_{P\in\mathcal{D}} \Big( \big| \mathscr{B}_P^2(f_2)\big| |P|^{-{1}/{2}} \frac{\chi_{P_c}(x)}{\langle 2^j(x-{\mathbf{x}}_P)\rangle^{s_2}} \Big)^2\bigg)^{{1}/{2}} \\&\quad \times \sup_{j\in{\mathbb{Z}}}\bigg( \sum_{P\in\mathcal{D}_j}\Big( \big| \mathscr{B}_R^3(f_3)\big| |R|^{-{1}/{2}} \frac{1}{\langle 2^j(x-{\mathbf{x}}_R)\rangle^{s_3}} \Big)^2\bigg)^{{1}/{2}}. \end{align*} $$

The proof of the above claim is a repetition of the arguments used in the proof of Lemmas 7.2 and 7.3, so we omit the details. We now take

$$ \begin{align*} u_1^{\mathrm{in}/\mathrm{out}}(x)&:= \sum_{k=0}^{\infty}|\lambda _k|\ell(Q_k)^{-{n}/{p_1}}\chi_{(Q_k^{***})^c}(x)| x-{\mathbf{x}}_{Q_k}|^{-s_1}\\&\quad \times\mathcal{M}\bigg[\bigg(\sum_{j\in{\mathbb{Z}}}\Big(2^{-s_1j}\min{\{1,(2^j\ell(Q_k))^M\}}I_{k,j,s}^{\mathrm{in}/\mathrm{out}}(\cdot) \Big)^2 \bigg)^{{1}/{2}}\bigg](x) \\u_2(x)&:= \bigg(\sum_{j\in{\mathbb{Z}}} \sum_{P\in\mathcal{D}_j} \Big( \big| \mathscr{B}_P^2(f_2)\big| |P|^{-{1}/{2}} \frac{\chi_{P^c}(x)}{\langle 2^j(x-{\mathbf{x}}_P)\rangle^{s_2}}\Big)^2 \bigg)^{{1}/{2}} \\u_3(x)&:= \sup_{j\in{\mathbb{Z}}}\bigg( \sum_{R\in\mathcal{D}_j} \Big(\big| \mathscr{B}_R^3(f_3) \big| |R|^{-{1}/{2}}\frac{1}{\langle 2^j(x-{\mathbf{x}}_R)\rangle ^{s_3}} \Big)^2 \bigg)^{{1}/{2}}. \end{align*} $$

Then it is obvious that Equations (7.13) and (7.14) hold.

9.6 Proof of Lemma 7.5

We choose $0<\epsilon <1$ such that

(9.28) $$ \begin{align} N_{p_1}:=\big[{n}/{p_1}-n\big]\le {n}/{p_1}-n<\big[ {n}/{p_1}-n\big]+\epsilon<s-{3n}/{2}. \end{align} $$

We note that

$$ \begin{align*} 2^l\lesssim |x-{\mathbf{x}}_{Q_k}|^{-1} \quad \text{ for }~ x\in B_k^l. \end{align*} $$

By using Lemma 4.1 with the vanishing moment condition (5.3), we have

$$ \begin{align*} \big| \phi_l\ast \big( \mathcal{U}_{1}(x,\cdot)\big)(x)\big| & \lesssim \sum_{j\in{\mathbb{Z}}}\sum_{P\in\mathcal{D}_j}\sum_{R\in\mathcal{D}_j}|b_P^2| |b_R^3|\chi_{P\cap R}(x) 2^{l(n+N_{p_1}+\epsilon)}\nonumber\\&\qquad\times \int_{{{\mathbb R}^n}}|y-{\mathbf{x}}_{Q_k}|^{N_{p_1}+\epsilon}\big|T_{\sigma_j}\big({\Lambda}_ja_k,\psi^P,\theta^R\big)(y) \big| \; dy\nonumber\\&\quad \lesssim \frac{1}{|x-{\mathbf{x}}_{Q_k}|^{n+N_{p_1}+\epsilon}}\sum_{j\in{\mathbb{Z}}}\sum_{P\in\mathcal{D}_j}\sum_{R\in\mathcal{D}_j}|b_P^2| |b_R^3|\chi_{P\cap R}(x)\\&\qquad \times \Big(\mathscr{K}_{N_{p_1}+\epsilon}^{j,\mathrm{in}}(Q_k,P,R)+\mathscr{K}_{N_{p_1}+\epsilon}^{j,\mathrm{out}}(Q_k,P,R) \Big)\nonumber, \end{align*} $$

where

$$ \begin{align*}\mathscr{K}_{N_{p_1}+\epsilon}^{j,\mathrm{in}}(Q_k,P,R):=\int_{Q_k^{**}} |y-{\mathbf{x}}_{Q_k}|^{N_{p_1}+\epsilon}\big| T_{\sigma_j}\big( {\Lambda}_ja_k,\psi^P,\theta^R\big)(y)\big| \; dy,\end{align*} $$

and

$$ \begin{align*}\mathscr{K}_{N_{p_1}+\epsilon}^{j,\mathrm{out}}(Q_k,P,R):=\int_{(Q_k^{**})^c} |y-{\mathbf{x}}_{Q_k}|^{N_{p_1}+\epsilon}\big| T_{\sigma_j}\big( {\Lambda}_ja_k,\psi^P,\theta^R\big)(y)\big| \; dy.\end{align*} $$

Now, the left-hand side of Equation (7.16) is dominated by $\mathscr {J}^{in}(x)+\mathscr {J}^{out}(x)$

$$ \begin{align*} \mathscr{J}^{\mathrm{in}/\mathrm{out}}(x)&:=\sum_{k=0}^{\infty}|\lambda _k|\chi_{(Q_{k}^{***})^c}(x)\frac{1}{|x-{\mathbf{x}}_{Q_k}|^{n+N_{p_1}+\epsilon}}\\&\quad \times \sum_{j\in{\mathbb{Z}}}\sum_{P\in\mathcal{D}_j}\sum_{R\in\mathcal{D}_j}|b_P^2| |b_R^3| \chi_{P\cap R}(x)\mathscr{K}_{N_{p_1}+\epsilon}^{j,\mathrm{in}/ \mathrm{out}}(Q_k,P,R). \end{align*} $$

To estimate $\mathscr {J}^{\mathrm {in}}$ , we first see that

$$ \begin{align*} &\mathscr{K}_{N_{p_1}+\epsilon}^{j,\mathrm{in}}(Q_k,P,R)\\&\quad \lesssim \ell(Q_k)^{N_{p_1}+\epsilon}|P|^{-{1}/{2}}|R|^{-{1}/{2}}\int_{y\in Q_k^{**}}\int_{({{\mathbb R}^n})^3}\big| \sigma_j^{\vee}(y-z_1,z_2,z_3)\big| \big|{\Lambda}_ja_k(z_1) \big| \; d\vec{\boldsymbol{z}} dy\\&\quad \lesssim \ell(Q_k)^{N_{p_1}+\epsilon}|P|^{-{1}/{2}}|R|^{-{1}/{2}}\Vert {\Lambda}_ja_k\Vert_{L^{1}({{\mathbb R}^n})}, \end{align*} $$

using the Cauchy–Schwarz inequality with $s>{3n}/{2}$ , and thus

(9.29) $$ \begin{align} &\sum_{j\in{\mathbb{Z}}}\sum_{P\in\mathcal{D}_j}\sum_{R\in\mathcal{D}_j} |b_P^2| |b_R^3| \chi_{P\cap R}(x)\mathscr{K}_{N_{p_1}+\epsilon}^{j,\mathrm{in}}(Q_k,P,R)\nonumber\\&\quad \lesssim \ell(Q_k)^{N_{p_1}+\epsilon}\big\Vert \big\{ {\Lambda}_ja_k\big\}_{j\in{\mathbb{Z}}}\big\Vert_{L^1(\ell^2)}g^2\big(\big\{ b_P^2\big\}_{P\in\mathcal{D}} \big)(x)g^{\infty}\big(\big\{ b_R^3\big\}_{R\in\mathcal{D}} \big)(x)\nonumber\\&\quad \lesssim |Q_k|^{-1/p_1} \ell(Q_k)^{n+N_{p_1}+\epsilon}g^2\big(\big\{ b_P^2\big\}_{P\in\mathcal{D}} \big)(x)g^{\infty}\big(\big\{ b_R^3\big\}_{R\in\mathcal{D}} \big)(x) \end{align} $$

by using the fact that

$$ \begin{align*} \big\Vert \big\{ {\Lambda}_ja_k\big\}_{j\in{\mathbb{Z}}}\big\Vert_{L^1(\ell^2)}\sim \Vert a_k\Vert_{H^1({{\mathbb R}^n})}\lesssim \ell(Q_k)^{-{n}/{p_1}+n}. \end{align*} $$

For the other term $\mathscr {J}^{out}$ , we choose $s_1$ such that

(9.30) $$ \begin{align} N_{p_1}+{n}/{2}+\epsilon<s_1<s-n, \end{align} $$

which is possible due to Equation (9.28), and $s_2,s_3>{n}/{2}$ such that

(9.31) $$ \begin{align} s_1+n<s_1+s_2+s_3=s. \end{align} $$

We observe that, for $y\in (Q_k^{**})^c$ ,

$$ \begin{align*} &\langle 2^j(y-{\mathbf{x}}_{Q_k})\rangle^{s_1}\big| {\Lambda}_ja_k(z_1)\big|\\&\quad \lesssim \ell(Q_k)^{-{n}/{p_1}}\min{\big\{1,\big( 2^j\ell(Q_k)\big)^M \big\}}\langle 2^j(y-{\mathbf{x}}_{Q_k})\rangle^{s_1} \\&\qquad \times \Big(\chi_{Q_k^*}(z_1)+\chi_{(Q_k^*)^c}(z_1)\frac{(2^j\ell(Q_k))^n}{\langle 2^j(z_1-{\mathbf{x}}_{Q_k})\rangle^{L_0}} \Big)\\&\quad \lesssim \ell(Q_k)^{-{n}/{p_1}}\min{\big\{1,\big( 2^j\ell(Q_k)\big)^M \big\}} \langle 2^j(y-z_1)\rangle^{s_1}\\&\qquad \times \Big(\chi_{Q_k^*}(z_1)+\chi_{(Q_k^*)^c}(z_1)\frac{(2^j\ell(Q_k))^n}{\langle 2^j(z_1-{\mathbf{x}}_{Q_k})\rangle^{L_0-s_1}} \Big), \end{align*} $$

where Lemma 4.5 is applied in the first inequality. Here, M and $L_0$ are sufficiently large numbers such that $L_0-s_1>n$ and $M-L_0+3n/2>0$ . By letting

(9.32) $$ \begin{align} A_{j,Q_k}(z_1):=\chi_{Q_k^*}(z_1)+\chi_{(Q_k^*)^c}(z_1)\frac{(2^j\ell(Q_k))^n}{\langle 2^j(z_1-{\mathbf{x}}_{Q_k})\rangle^{L_0-s_1}}, \end{align} $$

we have

(9.33) $$ \begin{align} &\big| T_{\sigma_j}\big( {\Lambda}_ja_k,\psi^P,\theta^R\big)(y)\big|\chi_{(Q_k^{**})^c}(y)\nonumber\\&\quad \lesssim \ell(Q_k)^{-{n}/{p_1}}\min \big\{ 1,\big( 2^j\ell(Q_k)\big)^M\big\}\frac{1}{\langle 2^j(y-{\mathbf{x}}_{Q_k})\rangle^{s_1}}|P|^{-{1}/{2}} |R|^{-{1}/{2}}\\&\qquad \times \int_{({{\mathbb R}^n})^3} \langle 2^j(y-z_1)\rangle^{s_1}\big| \sigma_j^{\vee}(y-z_1,z_2,z_3)\big| A_{j,Q_k}(z_1)\; d\vec{\boldsymbol{z}} \nonumber \end{align} $$

and the integral is, via the Cauchy-Schwarz inequality, less than

$$ \begin{align*}2^{-jn}\int_{{{\mathbb R}^n}} |A_{j,Q_k}(z_1)| \big\Vert \langle 2^j(y-z_1,z_2,z_3)\rangle^{s}\sigma_j^{\vee}(y-z_1,z_2,z_3)\big\Vert_{L^2(z_2,z_3)} \; dz_1. \end{align*} $$

This deduces that

$$ \begin{align*} &\mathscr{K}_{N_{p_1}+\epsilon}^{j,\mathrm{out}}(Q_k,P,R)\\&\quad \lesssim \ell(Q_k)^{-n/{p_1}}\min\big\{1,\big( 2^j\ell(Q_k)\big)^M \big\} |P|^{-{1}/{2}}|R|^{-{1}/{2}}2^{-js_1}2^{-jn} \int_{{{\mathbb R}^n}} \big|A_{j,Q_k}(z_1)\big|\\&\qquad \times \Big(\int_{(Q_k^{**})^c} \frac{1}{|y-{\mathbf{x}}_{Q_k}|^{s_1-(N_{p_1}+\epsilon)}} \big\Vert \langle 2^j(y-z_1,z_2,z_3)\rangle^{s}\sigma_j^{\vee}(y-z_1,z_2,z_3)\big\Vert_{L^2(z_2,z_3)} \, dy\Big) \; dz_1\\&\quad\lesssim \ell(Q_k)^{-n/{p_1}}\min\big\{1,\big( 2^j\ell(Q_k)\big)^M \big\} |P|^{-{1}/{2}}|R|^{-{1}/{2}}2^{-js_1}2^{-jn} \Vert A_{j,Q_k}\Vert_{L^1({{\mathbb R}^n})}\\&\qquad \times \bigg\Vert \frac{1}{|\cdot-{\mathbf{x}}_{Q_k}|^{s_1-(N_{p_1}+\epsilon)}}\bigg\Vert_{L^2((Q_k^{**})^c)}\big\Vert \langle 2^j{\vec{\cdot}\;}\rangle^s\sigma_j^{\vee}\big\Vert_{L^2(({{\mathbb R}^n})^3)}\\&\quad \lesssim \ell(Q_k)^{-n/p_1+n+N_{p_1}+\epsilon} \big( 2^j\ell(Q_k)\big)^{-(s_1-{n}/{2})} \min\big\{1,\big( 2^j\ell(Q_k)\big)^{M-L_0+s_1+n} \big\} |P|^{-{1}/{2}}|R|^{-{1}/{2}} \end{align*} $$

since

$$ \begin{align*}\Vert A_{j,Q_k}\Vert_{L^1({{\mathbb R}^n})}\lesssim \ell(Q_k)^n\Big(1+(2^j\ell(Q_k))^{-(L_0-s_1-n)} \Big) \quad \text{ for }~ L_0-s_1>n.\end{align*} $$

Therefore, by using the Cauchy–Schwarz inequality

(9.34) $$ \begin{align} &\sum_{j\in{\mathbb{Z}}}\sum_{P\in\mathcal{D}_j}\sum_{R\in\mathcal{D}_j} |b_P^2| |b_R^3| \chi_{P\cap R}(x)\mathscr{K}_{N_{p_1}+\epsilon}^{j,\mathrm{out}}(Q_k,P,R)\nonumber\\&\quad \lesssim |Q_k|^{-{1}/{p_1}}\ell(Q_k)^{n+N_{p_1}+\epsilon}g^2\big(\big\{ b_P^2\big\}_{P\in\mathcal{D}} \big)(x)g^{\infty}\big(\big\{ b_R^3\big\}_{R\in\mathcal{D}} \big)(x)\nonumber\\&\qquad \times \bigg( \sum_{j\in{\mathbb{Z}}}\Big(\big( 2^j\ell(Q_k)\big)^{-(s_1-{n}/{2})} \min\big\{1,\big( 2^j\ell(Q_k)\big)^{M-L_0+s_1+n} \big\} \Big)^2\bigg)^{{1}/{2}} \nonumber\\&\quad\lesssim |Q_k|^{-{1}/{p_1}}\ell(Q_k)^{n+N_{p_1}+\epsilon}g^2\big(\big\{ b_P^2\big\}_{P\in\mathcal{D}} \big)(x)g^{\infty}\big(\big\{ b_R^3\big\}_{R\in\mathcal{D}} \big)(x), \end{align} $$

where the last inequality holds due to $s_1>{n}/{2}$ and $M-L_0+{3n}/{2}>0$ .

In conclusion, the estimate (7.16) can be derived from Equations (9.29) and (9.34), using the choices of

$$ \begin{align*} u_1(x)&:= \sum_{k=0}^{\infty}|\lambda _k| |Q_k|^{-{1}/{p_1}} \chi_{(Q^{**})^c} \frac{\ell(Q_k)^{n+N_{p_1}+\epsilon}}{|x-{\mathbf{x}}_{Q_k}|^{n+N_{p_1}+\epsilon}} , \\ u_2(x)&:= g^2\big(\big\{ b_P^2\big\}_{P\in\mathcal{D}} \big)(x), \\ u_3(x)&:= g^{\infty}\big(\big\{ b_R^3\big\}_{R\in\mathcal{D}} \big)(x). \end{align*} $$

It is obvious from Equations (7.1) and (7.2) that $\Vert u_2\Vert _{L^{p_2}({{\mathbb R}^n})}, \Vert u_3\Vert _{L^{p_3}({{\mathbb R}^n})}\lesssim 1$ . Furthermore,

(9.35) $$ \begin{align} \Vert u_1\Vert_{L^{p_1}({{\mathbb R}^n})}&\lesssim \bigg(\sum_{k=0}^{\infty}|\lambda _k|^{p_1}|Q_k|^{-1}\int_{(Q_k^{***})^c}\frac{\ell(Q_k)^{(n+N_{p_1}+\epsilon)p_1}}{|x-{\mathbf{x}}_{Q_k}|^{(n+N_{p_1}+\epsilon)p_1}} dx \bigg)^{{1}/{p_1}}\nonumber\\ &\lesssim \Big( \sum_{k=0}^{\infty}|\lambda _k|^{p_1}\Big)^{{1}/{p_1}}\lesssim 1. \end{align} $$

This completes the proof.

9.7 Proof of Lemma 7.6

Choose $s_1$ , $s_2$ , and $s_3$ such that $s_1>{n}/{p_1}-{n}/{2}$ , $s_2>{n}/{2}$ , $s_3>{n}/{2}$ and $s=s_1+s_2+s_3$ . For $x\in B_k^l\cap (B_P^l)^c$ and $|x-y|\le 2^{-l}$ , we have

(9.36) $$ \begin{align} |x-{\mathbf{x}}_{Q_k}|\le |x-{\mathbf{x}}_{P}|\lesssim |y-{\mathbf{x}}_P|. \end{align} $$

This implies

$$ \begin{align*} &\langle 2^j(x-{\mathbf{x}}_{Q_k})\rangle^{s_1}\langle 2^j(x-{\mathbf{x}}_P)\rangle^{s_2}\big| \phi_l\ast T_{\sigma_j}\big({\Lambda}_ja_k,\psi^P,\theta^R \big)(x)\big|\\&\quad\lesssim |R|^{-{1}/{2}} 2^{ln}\int_{|x-y|\le 2^{-l}} \int_{({{\mathbb R}^n})^3}\langle 2^j(y-{\mathbf{x}}_P)\rangle^{s_1+s_2}\\&\qquad \times\big| \sigma_j^{\vee}(y-z_1,y-z_2,z_3)\big| \big| {\Lambda}_ja_k(z_1)\big| \big| {\psi^P}(z_2)\big|\;d\vec{\boldsymbol{z}}\; dy\\&\quad \le |R|^{-{1}/{2}} 2^{ln}\int_{|x-y|\le 2^{-l}}\int_{({{\mathbb R}^n})^3} \langle 2^j(y-z_2)\rangle^{s_1+s_2} \\&\qquad\times \big| \sigma_j^{\vee}(y-z_1,y-z_2,z_3)\big| \big| {\Lambda}_ja_k(z_1)\big| \big|\widetilde{\psi^P}(z_2) \big|\;d\vec{\boldsymbol{z}}\; dy, \end{align*} $$

where

$$ \begin{align*}\widetilde{\psi^P}(z_2) :=\langle 2^j(z_2-{\mathbf{x}}_P)\rangle^{s_1+s_2}\psi^P(z_2).\end{align*} $$

By using the Cauchy–Schwarz inequality and Lemma 4.2, we obtain

(9.37) $$ \begin{align} &\big| \phi_l\ast \big( \mathcal{U}_2^1(x,\cdot)\big)(x)\big| \nonumber\\[3pt]&\quad \lesssim \sum_{j\in{\mathbb{Z}}}\sum_{P\in\mathcal{D}_j}\sum_{R\in\mathcal{D}_j}|b_P^2| |b_R^3|\chi_{P^c}(x)\chi_R(x)\frac{1}{\langle 2^j(x-{\mathbf{x}}_{Q_k})\rangle^{s_1}} \frac{1}{\langle 2^j(x-{\mathbf{x}}_P)\rangle^{s_2}} |R|^{-{1}/{2}}\nonumber\\[3pt]&\qquad \times 2^{ln}\int_{|x-y|\le 2^{-l}} \int_{({{\mathbb R}^n})^3}\langle 2^j(y-z_2)\rangle^{s_1+s_2}\big| \sigma_j^{\vee}(y-z_1,y-z_2,z_3) \big| \big| {\Lambda}_ja_k(z_1)\big| \; d\vec{\boldsymbol{z}} dy\nonumber\\[3pt]&\quad \lesssim g^{\infty}\big( \big\{ b_R^3\big\}_{R\in\mathcal{D}}\big)(x) \sum_{j\in{\mathbb{Z}}}\frac{1}{\langle 2^j(x-{\mathbf{x}}_{Q_k})\rangle^{s_1}} 2^{ln}\int_{|x-y|\le 2^{-l}} \int_{({{\mathbb R}^n})^3} \langle 2^j(y-z_2)\rangle^{s_1+s_2} \nonumber\\[3pt]&\qquad \times \big| \sigma_j^{\vee}(y-z_1,y-z_2,z_3)\big| \big| {\Lambda}_ja_k(z_1)\big|\Big(\sum_{P\in\mathcal{D}_j}|b_P^2|\frac{\chi_{(P)^c}(x)}{\langle 2^j(x-{\mathbf{x}}_P)\rangle^{s_2}} \big|\widetilde{\psi^P}(z_3) \big| \Big) \; d\vec{\boldsymbol{z}} dy\nonumber\\[3pt]&\quad \lesssim g^{\infty}\big( \big\{ b_R^3\big\}_{R\in\mathcal{D}}\big)(x) \frac{1}{|x-{\mathbf{x}}_{Q_k}|^{s_1}}\Big( \sum_{j\in{\mathbb{Z}}} \big(2^{-js_1}\mathcal{M}J_{k,j,s}^1(x) \big)^2\Big)^{{1}/{2}} \nonumber\\[3pt]&\qquad \times\Big( \sum_{j\in{\mathbb{Z}}}\sum_{P\in\mathcal{D}_j} \Big( \big|\mathscr{B}_P^2(f_2)\big||P|^{-{1}/{2}}\frac{\chi_{P^c}(x)}{\langle 2^j(x-{\mathbf{x}}_P)\rangle^{s_2}}\Big)^2 \Big)^{{1}/{2}}, \end{align} $$

where $\mathscr {B}_P^2(f_2)$ is defined as in Equation (9.19) for some $L>n,s_2$ , and

(9.38) $$ \begin{align} J_{k,j,s}^1(y):= 2^{-jn}\int_{{{\mathbb R}^n}} \big| {\Lambda}_ja_k(z_1)\big|\Big\Vert \langle 2^j(y-z_1,z_2,z_3)\rangle^s\sigma_j^{\vee}(y-z_1,z_2,z_3) \Big\Vert_{L^2(z_2,z_3)} dz_1. \end{align} $$

Now, we choose

$$ \begin{align*} u_1(x)&:=\sum_{k=0}^{\infty}|\lambda _k|\chi_{(Q_k^{***})^c}(x)\frac{1}{|x-{\mathbf{x}}_{Q_k}|^{s_1}}\Big( \sum_{j\in{\mathbb{Z}}}\big(2^{-js_1}\mathcal{M}J_{k,j,s}^1(x) \big)^2\Big)^{{1}/{2}},\\ u_2(x)&:=\bigg( \sum_{j\in{\mathbb{Z}}}\sum_{P\in\mathcal{D}_j} \Big( |\mathscr{B}_P^2(f_2)||P|^{-{1}/{2}}\frac{\chi_{P^c}(x)}{\langle 2^j(x-{\mathbf{x}}_P)\rangle^{s_2}}\Big)^2 \bigg)^{{1}/{2}},\\ u_3(x)&:=g^{\infty}\big( \big\{ b_R^3\big\}_{R\in\mathcal{D}}\big)(x). \end{align*} $$

Clearly, Equation (7.18) holds and $\Vert u_2\Vert _{L^{p_2}({{\mathbb R}^n})}, \Vert u_3\Vert _{L^{p_3}({{\mathbb R}^n})} \lesssim 1$ due to Lemma 4.3 and Equation (7.2). In addition,

$$ \begin{align*} \Vert u_1\Vert_{L^{p_1}({{\mathbb R}^n})}^{p_1}\lesssim \sum_{k=0}^{\infty} |\lambda _k|^{p_1}\int_{(Q_k^{***})^c} |x-{\mathbf{x}}_{Q_k}|^{-s_1p_1}\Big(\sum_{j\in{\mathbb{Z}}}\big(2^{-s_1j}\mathcal{M}J_{k,j,s}^1(x) \big)^2 \Big)^{{p_1}/{2}} dx \end{align*} $$

and the integral is controlled by

$$ \begin{align*} &\big\Vert |\cdot-{\mathbf{x}}_{Q_k}|^{-s_1p_1}\big\Vert_{L^{({2}/{p_1})'}((Q_k^{***})^c)}\bigg\Vert \Big( \sum_{j\in{\mathbb{Z}}}\big(2^{-s_1j}\mathcal{M}J_{k,j,s}^1(x) \big)^2 \Big)^{{p_1}/{2}}\bigg\Vert_{L^{{2}/{p_1}}({{\mathbb R}^n})}\\&\quad \lesssim \ell(Q_k)^{-p_1(s_1-({n}/{p_1}-{n}/{2}))} \Big( \sum_{j\in{\mathbb{Z}}}2^{-2js_1}\big\Vert J_{k,j,s}^1 \big\Vert_{L^2({{\mathbb R}^n})}^2\Big)^{{p_1}/{2}} \end{align*} $$

by using Hölder’s inequality and the $L^2$ boundedness of $\mathcal {M}$ . It follows from Minkowski’s inequality and Lemma 4.5 that

$$ \begin{align*} \big\Vert J_{k,j,s}^{1}\big\Vert_{L^2({{\mathbb R}^n})}&\lesssim 2^{-jn}\big\Vert {\Lambda}_ja_k\big\Vert_{L^1({{\mathbb R}^n})}\Big\Vert \langle 2^j{\vec{\cdot}\;}\rangle^s\big| \sigma_j^{\vee}\big|\Big\Vert_{L^2(({{\mathbb R}^n})^3)}\\ &\lesssim \ell(Q_k)^{-{n}/{p_1}+n}2^{{jn}/{2}}\min\big\{1,\big(2^j\ell(Q_k) \big)^M \big\}, \end{align*} $$

and this finally yields that

(9.39) $$ \begin{align} \Vert u_1\Vert_{L^{p_1}({{\mathbb R}^n})}\lesssim \Big( \sum_{k=0}^{\infty}|\lambda _k|^{p_1}\Big)^{{1}/{p_1}}\lesssim 1. \end{align} $$

9.8 Proof of Lemma 7.7

For $x\in B_k^l\cap B_P^l$ ,

(9.40) $$ \begin{align} 2^l\lesssim |x-{\mathbf{x}}_{Q_k}|^{-1}, |x-{\mathbf{x}}_P|^{-1}. \end{align} $$

Since $s>{n}/{p_1}+{n}/{2}$ , there exist $0<\epsilon _0,\epsilon _1<1$ such that

$$ \begin{align*}{n}/{p_1}+{n}/{p_2}<\big[ {n}/{p_1}+{n}/{p_2}\big]+\epsilon_0 \quad \text{ and }\quad \big[ {n}/{p_1}+{n}/{p_2}\big]+\epsilon_0+\epsilon_1 <s-\big({n}/{2} -{n}/{p_2}\big). \end{align*} $$

Choose $t_1$ and $t_2$ satisfying $t_1>{n}/{p_1}$ , $t_2>{n}/{p_2}$ and $t_1+t_2=\big [ {n}/{p_1}+{n}/{p_2}\big ]+\epsilon _0$ , and let $N_0:=\big [ {n}/{p_1}+{n}/{p_2}\big ]-n $ . Then Lemma 4.1, together with the vanishing moment condition (5.3), and the estimate (9.40) yield that

$$ \begin{align*} &\big| \phi_l\ast T_{\sigma_j}\big( {\Lambda}_ja_k,\psi^P,\theta^R\big)(x)\big|\\[3pt]&\quad \lesssim 2^{l(N_0+n+\epsilon_0)}\int_{{{\mathbb R}^n}}|y-{\mathbf{x}}_P |^{N_0+\epsilon_0} \big| T_{\sigma_j} \big( {\Lambda}_ja_k,\psi^P,\theta^R\big)(y)\big| \; dy\\[3pt]&\quad \lesssim |R|^{-{1}/{2}}\frac{1}{|x-{\mathbf{x}}_{Q_k}|^{t_1}}2^{-j(t_1-n)}\int_{{{\mathbb R}^n}} \int_{({{\mathbb R}^n})^3}\langle 2^j(y-z_2)\rangle^{N_0+\epsilon_0}\\[3pt]&\qquad \times \big| \sigma_j^{\vee}(y-z_1,y-z_2,z_3)\big| \big| {\Lambda}_ja_k(z_1)\big| \frac{| \widetilde{\psi^P}(z_2)|}{\langle 2^j(x-{\mathbf{x}}_P)\rangle^{t_2}} \; d\vec{\boldsymbol{z}} dy, \end{align*} $$

where

$$ \begin{align*}\widetilde{\psi^P}(z_2):= \langle z_2-{\mathbf{x}}_P\rangle^{N_0+\epsilon_0}\psi^P(z_2).\end{align*} $$

This deduces

(9.41) $$ \begin{align} &\big| \phi_l\ast \big( \mathcal{U}_2^2(x,\cdot)\big)(x)\big| \nonumber\\[3pt]&\quad \lesssim g^{\infty}\big(\big\{ b_R^3\big\}_{R\in\mathcal{D}} \big)(x)\frac{1}{|x-{\mathbf{x}}_{Q_k}|^{t_1}}\sum_{j\in{\mathbb{Z}}}2^{-j(t_1-n)}\int_{{{\mathbb R}^n}} \int_{({{\mathbb R}^n})^3} \langle 2^j(y-z_2)\rangle^{N_0+\epsilon_0} \nonumber\\[3pt]&\qquad \times \big| \sigma_j(y-z_1,y-z_2,z_3)\big| \big| {\Lambda}_ja_k(z_1)\big| \Big( \sum_{P\in\mathcal{D}_j} |b_P^2|\frac{\chi_{P^c}(x)}{\langle 2^j(x-{\mathbf{x}}_P)\rangle^{t_2}}\big| \widetilde{\psi^P}(z_2)\big| \Big) \; d\vec{\boldsymbol{z}} dy. \end{align} $$

Using Hölder’s inequality with $\frac {1}{2}+\frac {1}{(1/p_2'-1/2)^{-1}}+\frac {1}{p_2}=1$ and Lemma 4.2, we see that

$$ \begin{align*} &\int_{{{\mathbb R}^n}} \langle 2^j(y-z_2)\rangle^{N_0+\epsilon_0} \big| \sigma_j(y-z_1,y-z_2,z_3)\big| \Big( \sum_{P\in\mathcal{D}_j} |b_P^2|\frac{\chi_{P^c}(x)}{\langle 2^j(x-{\mathbf{x}}_P)\rangle^{t_2}}\big| \widetilde{\psi^P}(z_2)\big| \Big) \; dz_2\\[3pt]&\quad \le \big\Vert \langle 2^jz_2\rangle^{s-n-\epsilon_1}\sigma_j^{\vee}(y-z_1,z_2,z_3)\big\Vert_{L^2(z_2)}\big\Vert \langle 2^j\cdot \rangle^{-(s-t_1-t_2-\epsilon_1)}\big\Vert_{L^{({1}/{p_2'}-{1}/{2})^{-1}}({{\mathbb R}^n})}\\[3pt]&\qquad \times \Big\Vert \sum_{P\in\mathcal{D}_j} |b_P^2|\frac{\chi_{P^c}(x)}{\langle 2^j(x-{\mathbf{x}}_P)\rangle^{t_2}}\big| \widetilde{\psi^P}(z_2)\big| \Big\Vert_{L^{p_2}({{\mathbb R}^n})}\\[3pt]&\quad \lesssim 2^{-\frac{jn}{2}}\big\Vert \langle 2^jz_2\rangle^{s-n-\epsilon_1}\sigma_j^{\vee}(y-z_1,z_2,z_3)\big\Vert_{L^2(z_2)}\\[3pt]&\qquad \times \bigg( \sum_{P\in\mathcal{D}_j}\Big( |\mathscr{B}_P^2(f_2)||P|^{-{1}/{2}}\frac{\chi_{P^c}(x)}{\langle 2^j(x-{\mathbf{x}}_P)\rangle^{t_2}}\Big)^{p_2} \bigg)^{{1}/{p_2}} \end{align*} $$

because $s-n-\epsilon _1=s-t_1-t_2-\epsilon _1+N_0+\epsilon _0$ , $s-t_1-t_2-\epsilon _1>n(1/p_2'-1/2)$ . This shows that the integral in the right-hand side of Equation (9.41) is dominated by a constant times

$$ \begin{align*} &\Vert {\Lambda}_ja_k\Vert_{L^1({{\mathbb R}^n})} \bigg( \sum_{P\in\mathcal{D}_j}\Big( |\mathscr{B}_P^2(f_2)||P|^{-{1}/{2}}\frac{\chi_{P^c}(x)}{\langle 2^j(x-{\mathbf{x}}_P)\rangle^{t_2}}\Big)^{p_2} \bigg)^{{1}/{p_2}} \\&\qquad \times 2^{-{jn}/{2}}\int_{({{\mathbb R}^n})^2} \big\Vert \langle 2^jz_2\rangle^{s-n-\epsilon_1}\sigma_j^{\vee}(y,z_2,z_3)\big\Vert_{L^2(z_2)} \;dy\; dz_3\\&\quad \lesssim \ell(Q_k)^{-{n}/{p_1}+n}\min\big\{1,\big( 2^j\ell(Q_k)\big)^M \big\} \bigg( \sum_{P\in\mathcal{D}_j}\Big( |\mathscr{B}_P^2(f_2)||P|^{-{1}/{2}}\frac{\chi_{P^c}(x)}{\langle 2^j(x-{\mathbf{x}}_P)\rangle^{t_2}}\Big)^{p_2} \bigg)^{{1}/{p_2}}, \end{align*} $$

where $\mathscr {B}_P^2(f_2)$ is defined as in Equation (9.19) and M is sufficiently large. Consequently,

(9.42) $$ \begin{align} &\big| \phi_l\ast \big( \mathcal{U}_2^2(x,\cdot)\big)(x)\big| \nonumber\\&\quad \lesssim g^{\infty}\big(\big\{ b_R^3\big\}_{R\in\mathcal{D}} \big)(x) \;\sup_{j\in{\mathbb{Z}}} \bigg( \sum_{P\in\mathcal{D}_j}\Big( |\mathscr{B}_P^2(f_2)||P|^{-{1}/{2}}\frac{\chi_{P^c}(x)}{\langle 2^j(x-{\mathbf{x}}_P)\rangle^{t_2}}\Big)^{p_2} \bigg)^{{1}/{p_2}}\nonumber\\&\qquad \times \frac{1}{|x-{\mathbf{x}}_{Q_k}|^{t_1}}\ell(Q_k)^{-{n}/{p_1}+n}\sum_{j\in{\mathbb{Z}}} 2^{-j(t_1-n)} \min\big\{1,\big( 2^j\ell(Q_k)\big)^M \big\}\nonumber\\&\quad \lesssim |Q_k|^{-1/p_1} \frac{\ell(Q_k)^{t_1}}{|x-{\mathbf{x}}_{Q_k}|^{t_1}} \bigg( \sum_{j\in{\mathbb{Z}}} \sum_{P\in\mathcal{D}_j}\Big( |\mathscr{B}_P^2(f_2)||P|^{-{1}/{2}}\frac{\chi_{P^c}(x)}{\langle 2^j(x-{\mathbf{x}}_P)\rangle^{t_2}}\Big)^{p_2} \bigg)^{{1}/{p_2}}\\&\qquad\times g^{\infty}\big(\big\{ b_R^3\big\}_{R\in\mathcal{D}} \big)(x).\nonumber \end{align} $$

Now, we are done with

$$ \begin{align*} u_1(x)&:= \sum_{k=0}^{\infty}|\lambda _k| |Q_k|^{-{1}/{p_1}}\frac{\ell(Q_k)^{t_1}}{|x-{\mathbf{x}}_{Q_k}|^{t_1}}\chi_{(Q_k^{***})^c}(x)\\u_2(x)&:= \bigg(\sum_{j\in{\mathbb{Z}}} \sum_{P\in\mathcal{D}_j}\Big( |\mathscr{B}_P^2(f_2)||P|^{-{1}/{2}}\frac{\chi_{P^c}(x)}{\langle 2^j(x-{\mathbf{x}}_P)\rangle^{t_2}}\Big)^{p_2} \bigg)^{{1}/{p_2}}\\u_3(x)&:= g^{\infty}\big(\big\{ b_R^3\big\}_{R\in\mathcal{D}} \big)(x) \end{align*} $$

as $\Vert u_{{\mathrm {i}}}\Vert _{L^{p_{{\mathrm {i}}}}({{\mathbb R}^n})}\lesssim 1$ , ${\mathrm {i}}=1,2,3$ , follow from Lemma 4.3, Equation (7.2) and the argument that led to (9.35) with $t_1>n/p_1$ .

9.9 Proof of Lemma 7.8

Let $s_1$ , $s_2$ and $s_3$ satisfy $s_1>{n}/{p_1}-{n}/{2}$ , $s_2>{n}/{2}$ , $s_3>{n}/{2}$ and $s=s_1+s_2+s_3$ . By mimicking the argument that led to Equation (9.37) with

$$ \begin{align*}|x-{\mathbf{x}}_{Q_k}|\le |x-{\mathbf{x}}_R|\lesssim |y-{\mathbf{x}}_R|\end{align*} $$

for $x\in B_k^l\cap (B_R^l)^c$ and $|x-y|\le 2^{-l}$ , instead of Equation (9.36), we can prove

$$ \begin{align*} \big| \phi_l\ast \big(\mathcal{U}_{3}^{1}(x,\cdot)\big)(x) \big|&\lesssim \frac{1}{|x-{\mathbf{x}}_{Q_k}|^{s_1}}\Big( \sum_{j\in{\mathbb{Z}}} \big(2^{-js_1}\mathcal{M}J_{k,j,s}^1(x) \big)^2\Big)^{{1}/{2}} \; g^2\big(\big\{ b_P^2\big\}_{P\in\mathcal{D}} \big)(x)\\&\quad \times \sup_{j\in{\mathbb{Z}}}\bigg( \sum_{R\in\mathcal{D}_j} \Big( \big|\mathscr{B}_R^3(f_3)\big||R|^{-{1}/{2}}\frac{\chi_{R^c}(x)}{\langle 2^j(x-{\mathbf{x}}_R)\rangle^{s_3}}\Big)^2\bigg)^{{1}/{2}}, \end{align*} $$

where $J_{k,j,s}^1$ and $\mathscr {B}_R^3(f_3)$ are defined as in Equations (9.38) and (9.26) for some $L>n,s_3$ .

Now, let

$$ \begin{align*} u_1(x)&:= \sum_{k=0}^{\infty}|\lambda _k|\chi_{(Q_k^{***})^c}(x)\frac{1}{|x-{\mathbf{x}}_{Q_k}|^{s_1}}\Big( \sum_{j\in{\mathbb{Z}}}\big(2^{-js_1}\mathcal{M}J_{k,j,s}^1(x) \big)^2\Big)^{{1}/{2}} \\ u_2(x)&:=g^2\big( \big\{ b_P^2\big\}_{P\in\mathcal{D}}\big)(x)\\ u_3(x)&:= \sup_{j\in{\mathbb{Z}}}\bigg( \sum_{P\in\mathcal{D}_j}\Big( |\mathscr{B}_R^3(f_3)| |R|^{-{1}/{2}}\frac{\chi_{R^c}(x)}{\langle 2^j(x-{\mathbf{x}}_R)\rangle^{s_3}}\Big)^2\bigg)^{{1}/{2}}. \end{align*} $$

Then the estimate (7.21) is clear and it follows from Equations (9.39) and (7.1) and Lemma 4.4 that Equation (7.20) holds.

9.10 Proof of Lemma 7.9

Let $0<\epsilon _0,\epsilon _1<1$ satisfy

$$ \begin{align*}{n}/{p_1}+{n}/{p_3}<\big[ {n}/{p_1}+{n}/{p_3}\big]+\epsilon_0 \quad \text{ and }\quad \big[ {n}/{p_1}+{n}/{p_3}\big]+\epsilon_0+\epsilon_1<s-\big( {n}/{2}-{n}/{p_3}\big),\end{align*} $$

and select $t_1,t_3$ so that $t_1>{n}/{p_1}$ , $t_3>{n}/{p_3}$ and $t_1+t_3=\big [{n}/{p_1}+{n}/{p_3} \big ]+\epsilon _0$ . Let $N_0:= \big [ {n}/{p_1}+{n}/{p_3}\big ]-n$ and $B_R^3(f_3)$ be defined as in Equation (9.26). Then, as the counterpart of Equation (9.42), we can get

$$ \begin{align*} \big| \phi_l\ast \big(\mathcal{U}_{3}^{2}(x,\cdot)\big)(x)\big|&\lesssim |Q_k|^{-1/p_1} \frac{\ell(Q_k)^{t_1}}{|x-{\mathbf{x}}_{Q_k}|^{t_1}}g^{2}\big( \big\{b_P^2 \big\}_{P\in\mathcal{D}}\big)(x)\\ &\qquad \qquad \times \sup_{j\in{\mathbb{Z}}}\bigg( \sum_{R\in\mathcal{D}_j}\Big( |\mathscr{B}_R^3(f_3)||R|^{-{1}/{2}}\frac{\chi_{R^c}(x)}{\langle 2^j(x-{\mathbf{x}}_R)\rangle^{t_3}}\Big)^{p_3}\bigg)^{{1}/{p_3}}, \end{align*} $$

where the embedding $\ell ^2\hookrightarrow \ell ^{\infty }$ is applied. By taking

$$ \begin{align*} u_1(x)&:= \sum_{k=0}^{\infty}|\lambda _k| |Q_k|^{-{1}/{p_1}}\frac{\ell(Q_k)^{t_1}}{|x-{\mathbf{x}}_{Q_k}|^{t_1}}\chi_{(Q_k^{***})^c}(x) \\ u_2(x)&:= g^2\big( \big\{ b_P^2\big\}_{P\in\mathcal{D}}\big)(x), \\ u_3(x)&:= \sup_{j\in{\mathbb{Z}}}\bigg( \sum_{R\in\mathcal{D}_j}\Big( |\mathscr{B}_R^3(f_3)||R|^{-{1}/{2}}\frac{\chi_{R^c}(x)}{\langle 2^j(x-{\mathbf{x}}_R)\rangle^{t_3}}\Big)^{p_3}\bigg)^{{1}/{p_3}}, \end{align*} $$

we obtain the inequality (7.22) and Equation (7.23).

9.11 Proof of Lemma 7.10

The proof is almost same as that of Lemmas 7.6 and 7.8. Let $s_1$ , $s_2$ and $s_3$ be numbers such that $s_1>{n}/{p_1}-{n}/{2}$ , $s_2>{n}/{2}$ , $s_3>{n}/{2}$ and $s=s_1+s_2+s_3$ . We claim that for $\eta =1,2,3$ ,

(9.43) $$ \begin{align} \big| \phi_l\ast \big( \mathcal{U}_4^{\eta}(x,\cdot)\big)(x)\big| &\lesssim \frac{1}{|x-{\mathbf{x}}_{Q_k}|^{s_1}} \Big( \sum_{j\in{\mathbb{Z}}}\big( 2^{-js_1}\mathcal{M}J_{k,j,s}^1(x)\big)^2\Big)^{{1}/{2}}\\&\quad \times\bigg( \sum_{j\in{\mathbb{Z}}}\sum_{P\in\mathcal{D}_j} \Big(|\mathscr{B}_P^2(f_2)| |P|^{-{1}/{2}} \frac{\chi_{P^c}(x)}{\langle 2^j(x-{\mathbf{x}}_P)\rangle^{s_2}}\Big)^2 \bigg)^{{1}/{2}}\nonumber\\&\quad \times \sup_{j\in{\mathbb{Z}}}\bigg(\sum_{R\in\mathcal{D}_j} \Big( |\mathscr{B}_R^3(f_3)| |R|^{-{1}/{2}}\frac{\chi_{R^c}(x)}{\langle 2^j(x-{\mathbf{x}}_R)\rangle^{s_3}}\Big)^2\bigg)^{{1}/{2}}\nonumber, \end{align} $$

where $J_{k,j,s}^1$ , $\mathscr {B}_P^2(f_2)$ and $\mathscr {B}_R^3(f_3)$ are defined as in Equations (9.38), (9.19) and (9.26), respectively. Then we have Equation (7.25) with the choice

$$ \begin{align*} u_1(x)&:= \sum_{k=0}^{\infty}|\lambda _k|\chi_{(Q_k^{***})^c}(x)\frac{1}{|x-{\mathbf{x}}_{Q_k}|^{s_1}}\Big( \sum_{j\in{\mathbb{Z}}}\big(2^{-js_1}\mathcal{M}J_{k,j,s}^1(x) \big)^2\Big)^{{1}/{2}}, \\ u_2(x)&:= \bigg( \sum_{j\in{\mathbb{Z}}}\sum_{P\in\mathcal{D}_j} \Big(|\mathscr{B}_P^2(f_2)| |P|^{-{1}/{2}} \frac{\chi_{P^c}(x)}{\langle 2^j(x-{\mathbf{x}}_P)\rangle^{s_2}}\Big)^2 \bigg)^{{1}/{2}}, \\ u_3(x)&:= \sup_{j\in{\mathbb{Z}}}\bigg( \sum_{P\in\mathcal{D}_j}\Big( |\mathscr{B}_R^3(f_3)| |R|^{-{1}/{2}}\frac{\chi_{R^c}(x)}{\langle 2^j(x-{\mathbf{x}}_R)\rangle^{s_3}}\Big)^2\bigg)^{{1}/{2}}. \end{align*} $$

The estimates for $u_1,u_2,u_3$ follow from Equation (9.39), Lemma 4.3 and Lemma 4.4.

Now, we return to the proof of Equation (9.43). For $x\in B_k^l\cap (B_P^l)^c\cap (B_R^l)^c$ and $|x-y|\le 2^{-l}$ , we have

(9.44) $$ \begin{align} |x-{\mathbf{x}}_{Q_k}|\le |x-{\mathbf{x}}_P|\lesssim |y-{\mathbf{x}}_P| \quad \text{and} \quad |x-{\mathbf{x}}_R|\lesssim |y-{\mathbf{x}}_R|. \end{align} $$

Then we have

$$ \begin{align*} &\langle 2^j(x-{\mathbf{x}}_{Q_k})\rangle^{s_1}\langle 2^j(x-{\mathbf{x}}_P)\rangle^{s_2}\langle 2^j(x-{\mathbf{x}}_R)\rangle^{s_3}\big| \phi_l\ast T_{\sigma_j}\big( {\Lambda}_ja_k,\psi^P,\theta^R\big)(x)\big|\\&\quad\lesssim 2^{ln}\int_{|x-y|\le 2^{-l}} \int_{({{\mathbb R}^n})^3}\langle 2^j(y-z_2)\rangle^{s_1+s_2}\langle 2^j(y-z_3)\rangle^{s_3}\big|\sigma_j^{\vee}(y-z_1,y-z_2,y-z_3) \big| \\&\qquad\times |{\Lambda}_ja_k(z_1)|\big|\widetilde{\psi^P}(z_2)\big| \big|\widetilde{\theta^R}(z_3)\big| \; d\vec{\boldsymbol{z}} dy, \end{align*} $$

where

$$ \begin{align*} \widetilde{\psi^P}(z_2)&:=\langle 2^j(z_2-{\mathbf{x}}_P)\rangle^{s_1+s_2}\psi^P(z_2),\\ \widetilde{\theta^R}(z_3)&:=\langle 2^j(z_3-{\mathbf{x}}_R)\rangle^{s_3}\theta^R(z_3). \end{align*} $$

Now, using the method similar to that used in the proof of Equation (9.37), we obtain Equation (9.43) for $\eta =1$ .

For the case $\eta =2$ , we use the fact, instead of Equation (9.44), that for $x\in B_k^l\cap (B_P^l)^c\cap B_R^l$ and $|x-y|\le 2^{-l}$ ,

$$ \begin{align*}|x-{\mathbf{x}}_{Q_k}|, |x-{\mathbf{x}}_R|\le |x-{\mathbf{x}}_P|\lesssim |y-{\mathbf{x}}_P|.\end{align*} $$

This shows that

$$ \begin{align*} &\langle 2^j(x-{\mathbf{x}}_{Q_k})\rangle^{s_1}\langle 2^j(x-{\mathbf{x}}_P)\rangle^{s_2}\langle 2^j(x-{\mathbf{x}}_R)\rangle^{s_3}\big| \phi_l\ast T_{\sigma_j}\big( {\Lambda}_ja_k,\psi^P,\theta^R\big)(x)\big|\\&\quad \lesssim 2^{ln}\int_{|x-y|\le 2^{-l}} \int_{({{\mathbb R}^n})^3}\langle 2^j(y-z_2)\rangle^{s}\big|\sigma_j^{\vee}(y-z_1,y-z_2,y-z_3) \big| \\&\qquad\times |{\Lambda}_ja_k(z_1)|\big|{\widetilde{\psi^P}}(z_2)\big| \big|{\theta^R}(z_3)\big| \; d\vec{\boldsymbol{z}} dy, \end{align*} $$

where

$$ \begin{align*}\widetilde{\psi^P}(z_2):=\langle 2^j(z_2-{\mathbf{x}}_P)\rangle^{s}\psi^P(z_2),\end{align*} $$

and then Equation (9.43) for $\eta =2$ follows.

Similarly, we can prove that for $x\in B_k^l\cap B_P^l\cap (B_R^l)^c$ and $|x-y|\le 2^{-l}$ ,

$$ \begin{align*} &\langle 2^j(x-{\mathbf{x}}_{Q_k})\rangle^{s_1}\langle 2^j(x-{\mathbf{x}}_P)\rangle^{s_2}\langle 2^j(x-{\mathbf{x}}_R)\rangle^{s_3}\big| \phi_l\ast T_{\sigma_j}\big( {\Lambda}_ja_k,\psi^P,\theta^R\big)(x)\big|\\&\quad \lesssim 2^{ln}\int_{|x-y|\le 2^{-l}} \int_{({{\mathbb R}^n})^3}\langle 2^j(y-z_3)\rangle^{s}\big|\sigma_j^{\vee}(y-z_1,y-z_2,y-z_3) \big| \\&\qquad\times |{\Lambda}_ja_k(z_1)|\big|{\psi^P}(z_2)\big| \big|\widetilde{\theta^R}(z_3)\big| \; d\vec{\boldsymbol{z}} dy, \end{align*} $$

where

$$ \begin{align*}\widetilde{\theta^R}(z_3):=\langle 2^j(z_3-{\mathbf{x}}_R)\rangle^{s}\theta^R(z_3).\end{align*} $$

This proves (9.43) for $\eta =3$ .

9.12 Proof of Lemma 7.11

We first note that

(9.45) $$ \begin{align} 2^l\lesssim |x-{\mathbf{x}}_{Q_k}|^{-1}, |x-{\mathbf{x}}_P|^{-1}, |x-{\mathbf{x}}_R|^{-1} \end{align} $$

for $x\in B_k^l\cap B_P^l\cap B_R^l$ . Since ${n}/{p}<s-\big ({n}/{2}-{n}/{p_2}-{n}/{p_3} \big )$ , there exist $0<\epsilon _0,\epsilon _1<1$ such that

$$ \begin{align*}{n}/{p}<\big[ {n}/{p}\big]+\epsilon_0 \quad \text{ and }\quad \big[ {n}/{p}\big]+\epsilon_0+\epsilon_1<s-\big({n}/{2} -{n}/{p_2}-{n}/{p_3}\big).\end{align*} $$

Choose $t_1$ , $t_2$ , and $t_3$ satisfying $t_1>{n}/{p_1}$ , $t_2>{n}/{p_2}$ , $t_3>{n}/{p_3}$ , and $t_1+t_2+t_3=\big [ {n}/{p}\big ]+\epsilon _0$ and let $N_0:=\big [ {n}/{p}\big ]-n $ . Then it follows from Lemma 4.1 and the estimate (9.45) that

$$ \begin{align*} &\big| \phi_l\ast T_{\sigma_j}\big( {\Lambda}_ja_k,\psi^P,\theta^R\big)(x)\big|\\&\quad \lesssim 2^{l(N_0+n+\epsilon_0)}\int_{{{\mathbb R}^n}}|y-{\mathbf{x}}_P |^{N_0+\epsilon_0} \big| T_{\sigma_j} \big( {\Lambda}_ja_k,\psi^P,\theta^R\big)(y)\big| \; dy\\&\quad \lesssim \frac{1}{|x-{\mathbf{x}}_{Q_k}|^{t_1}}2^{-j(t_1-n)}\int_{{{\mathbb R}^n}} \int_{({{\mathbb R}^n})^3}\langle 2^j(y-z_2)\rangle^{N_0+\epsilon_0}\big| \sigma_j^{\vee}(y-z_1,y-z_2,y-z_3)\big| \\[3pt]&\qquad \times \big| {\Lambda}_ja_k(z_1)\big| \frac{| \widetilde{\psi^P}(z_2)|}{\langle 2^j(x-{\mathbf{x}}_P)\rangle^{t_2}}\frac{|\theta^R(z_3)|}{\langle 2^j(x-{\mathbf{x}}_R)\rangle^{t_3}} \; d\vec{\boldsymbol{z}} dy, \end{align*} $$

where

$$ \begin{align*}\widetilde{\psi^P}(z_2):= \langle z_2-{\mathbf{x}}_P\rangle^{N_0+\epsilon_0}\psi^P(z_2).\end{align*} $$

This deduces that

(9.46) $$ \begin{align} &\big| \phi_l\ast \big( \mathcal{U}_4^4(x,\cdot)\big)(x)\big| \nonumber\\[3pt]&\quad \lesssim \frac{1}{|x-{\mathbf{x}}_{Q_k}|^{t_1}}\sum_{j\in{\mathbb{Z}}}2^{-j(t_1-n)}\int_{{{\mathbb R}^n}} \int_{({{\mathbb R}^n})^3} \langle 2^j(y-z_2)\rangle^{N_0+\epsilon_0} \big| \sigma_j(y-z_1,y-z_2,z_3)\big| \big| {\Lambda}_ja_k(z_1)\big|\nonumber\\[3pt]&\qquad \times \bigg( \sum_{P\in\mathcal{D}_j} |b_P^2|\frac{\chi_{P^c}(x)}{\langle 2^j(x-{\mathbf{x}}_P)\rangle^{t_2}}\big| \widetilde{\psi^P}(z_2)\big| \bigg) \bigg( \sum_{R\in\mathcal{D}_j} |b_R^3|\frac{\chi_{R^c}(x)}{\langle 2^j(x-{\mathbf{x}}_R)\rangle^{t_3}}\big| {\theta^R}(z_3)\big| \bigg) \; d\vec{\boldsymbol{z}} dy. \end{align} $$

Since $s-\big [{n}/{p} \big ]+{n}/{2}-\epsilon _0-\epsilon _1>\big ({n}/{2}-{n}/{p_2} \big )+\big ({n}/{2}-{n}/{p_3}\big )$ , there exist $\mu _2$ and $\mu _3$ such that $\mu _2>{n}/{2}-{n}/{p_2}$ , $\mu _3>{n}/{2}-{n}/{p_3}$ , and $\mu _1+\mu _2=s-\big [ {n}/{p}\big ]+{n}/{2}-\epsilon _0-\epsilon _1$ . Using Hölder’s inequality with

$$ \begin{align*}\frac{1}{2}+\frac{1}{(1/p_2'-1/2)^{-1}}+\frac{1}{p_2}=\frac{1}{2}+\frac{1}{(1/p_3'-1/2)^{-1}}+\frac{1}{p_3}=1,\end{align*} $$

we have

$$ \begin{align*} &\int_{({{\mathbb R}^n})^2} \langle 2^j(y-z_2)\rangle^{N_0+\epsilon_0} \big| \sigma_j(y-z_1,y-z_2,y-z_3)\big| \\[3pt]&\qquad \times \bigg( \sum_{P\in\mathcal{D}_j} |b_P^2|\frac{\chi_{P^c}(x)}{\langle 2^j(x-{\mathbf{x}}_P)\rangle^{t_2}}\big| \widetilde{\psi^P}(z_2)\big| \bigg) \bigg( \sum_{R\in\mathcal{D}_j} |b_R^3|\frac{\chi_{R^c}(x)}{\langle 2^j(x-{\mathbf{x}}_R)\rangle^{t_3}}\big| {\theta^R}(z_3)\big| \bigg) \; dz_2 dz_3\\[3pt]&\quad \le \big\Vert \langle 2^jz_2\rangle^{N_0+\epsilon_0+\mu_2}\langle 2^jz_3\rangle^{\mu_3}\sigma_j^{\vee}(y-z_1,z_2,z_3)\big\Vert_{L^2(z_2,z_3)}\big\Vert \langle 2^j\cdot \rangle^{-\mu_2}\big\Vert_{L^{({1}/{p_2'}-{1}/{2})^{-1}}({{\mathbb R}^n})}\\[3pt]&\qquad \times \big\Vert \langle 2^j\cdot \rangle^{-\mu_3}\big\Vert_{L^{({1}/{p_3'}-{1}/{2})^{-1}}({{\mathbb R}^n})}\bigg\Vert \sum_{P\in\mathcal{D}_j} |b_P^2|\frac{\chi_{P^c}(x)}{\langle 2^j(x-{\mathbf{x}}_P)\rangle^{t_2}}\big| \widetilde{\psi^P}(z_2)\big| \bigg\Vert_{L^{p_2}(z_2)}\\[3pt]&\qquad \times \bigg\Vert \sum_{R\in\mathcal{D}_j} |b_R^3|\frac{\chi_{R^c}(x)}{\langle 2^j(x-{\mathbf{x}}_R)\rangle^{t_3}}\big| {\theta^R}(z_3)\big| \bigg\Vert_{L^{p_2}(z_3)}, \end{align*} $$

and then Lemma 4.2 yields that the preceding expression is less than a constant times

$$ \begin{align*} & 2^{-jn}\big\Vert \langle 2^j(z_2,z_3)\rangle^{N_0+\epsilon_0+\mu_1+\mu_2}\sigma_j^{\vee}(y-z_1,z_2,z_3)\big\Vert_{L^2(z_2,z_3)}\\&\quad \times \bigg( \sum_{P\in\mathcal{D}_j}\Big( |\mathscr{B}_P^2(f_2)||P|^{-{1}/{2}}\frac{\chi_{P^c}(x)}{\langle 2^j(x-{\mathbf{x}}_P)\rangle^{t_2}}\Big)^{p_2} \bigg)^{{1}/{p_2}}\\&\quad \times \bigg( \sum_{R\in\mathcal{D}_j}\Big( |\mathscr{B}_R^3(f_3)||R|^{-{1}/{2}}\frac{\chi_{R^c}(x)}{\langle 2^j(x-{\mathbf{x}}_R)\rangle^{t_3}}\Big)^{p_3} \bigg)^{{1}/{p_3}} \end{align*} $$

because $\mu _2>n(1/p_2'-1/2)$ and $\mu _3>n(1/p_3'-1/2)$ , where $\mathscr {B}_P^2(f_2)$ and $\mathscr {B}_R(f_3)$ are defined as in Equations (9.19) and (9.26).

Now, the integral in the right-hand side of Equation (9.46) is dominated by a constant times

$$ \begin{align*} &2^{-{jn}}\Vert {\Lambda}_ja_k\Vert_{L^1({{\mathbb R}^n})} \int_{{{\mathbb R}^n} } \big\Vert \langle 2^j(z_2,z_3)\rangle^{s-{n}/{2}-\epsilon_1}\sigma_j^{\vee}(y,z_2,z_3)\big\Vert_{L^2(z_2,z_3)} dy \\&\quad \times \bigg( \sum_{P\in\mathcal{D}_j}\Big( |\mathscr{B}_P^2(f_2)||P|^{-{1}/{2}}\frac{\chi_{P^c}(x)}{\langle 2^j(x-{\mathbf{x}}_P)\rangle^{t_2}}\Big)^{p_2} \bigg)^{{1}/{p_2}} \\&\quad \times\bigg( \sum_{R\in\mathcal{D}_j}\Big( |\mathscr{B}_R^3(f_3)||R|^{-{1}/{2}}\frac{\chi_{R^c}(x)}{\langle 2^j(x-{\mathbf{x}}_R)\rangle^{t_3}}\Big)^{p_3} \bigg)^{{1}/{p_3}}, \\ \end{align*} $$

and this is no more than

$$ \begin{align*} & \ell(Q_k)^{-n/p_1+n}\min\big\{1,\big( 2^j\ell(Q_k)\big)^M \big\} \\&\quad \times \bigg( \sum_{P\in\mathcal{D}_j}\Big( |\mathscr{B}_P^2(f_2)||P|^{-{1}/{2}}\frac{\chi_{P^c}(x)}{\langle 2^j(x-{\mathbf{x}}_P)\rangle^{t_2}}\Big)^{p_2} \bigg)^{{1}/{p_2}}\\&\quad \times \bigg( \sum_{R\in\mathcal{D}_j}\Big( |\mathscr{B}_R^3(f_3)||R|^{-{1}/{2}}\frac{\chi_{R^c}(x)}{\langle 2^j(x-{\mathbf{x}}_R)\rangle^{t_3}}\Big)^{p_3} \bigg)^{{1}/{p_3}}, \end{align*} $$

where $N_0+\epsilon _0+\mu _2+\mu _3=s-\frac {n}{2}-\epsilon _1$ . Hence, it follows that

$$ \begin{align*} &\big| \phi_l\ast \big( \mathcal{U}_4^4(x,\cdot)\big)(x)\big|\\ & \lesssim \frac{1}{|x-{\mathbf{x}}_{Q_k}|^{t_1}}\ell(Q_k)^{-{n}/{p_1}+n}\sum_{j\in{\mathbb{Z}}} 2^{-j(t_1-n)} \min\big\{1,\big( 2^j\ell(Q_k)\big)^M \big\}\\ &\qquad \times \sup_{j\in{\mathbb{Z}}} \bigg( \sum_{P\in\mathcal{D}_j}\Big( |\mathscr{B}_P^2(f_2)||P|^{-{1}/{2}}\frac{\chi_{P^c}(x)}{\langle 2^j(x-{\mathbf{x}}_P)\rangle^{t_2}}\Big)^{p_2} \bigg)^{{1}/{p_2}} \\ &\qquad \times \sup_{j\in{\mathbb{Z}}} \bigg( \sum_{R\in\mathcal{D}_j}\Big( |\mathscr{B}_R^3(f_3)||R|^{-{1}/{2}}\frac{\chi_{R^c}(x)}{\langle 2^j(x-{\mathbf{x}}_R)\rangle^{t_3}}\Big)^{p_3} \bigg)^{{1}/{p_3}}\\ &\lesssim |Q_k|^{-1/p_1} \frac{\ell(Q_k)^{t_1}}{|x-{\mathbf{x}}_{Q_k}|^{t_1}} \bigg( \sum_{j\in{\mathbb{Z}}} \sum_{P\in\mathcal{D}_j}\Big( |\mathscr{B}_P^2(f_2)||P|^{-{1}/{2}}\frac{\chi_{P^c}(x)}{\langle 2^j(x-{\mathbf{x}}_P)\rangle^{t_2}}\Big)^{p_2} \bigg)^{{1}/{p_2}}\\ &\qquad \qquad \qquad \qquad \qquad \times \sup_{j\in{\mathbb{Z}}} \bigg( \sum_{R\in\mathcal{D}_j}\Big( |\mathscr{B}_R^3(f_3)||R|^{-{1}/{2}}\frac{\chi_{R^c}(x)}{\langle 2^j(x-{\mathbf{x}}_R)\rangle^{t_3}}\Big)^{p_3} \bigg)^{{1}/{p_3}}. \end{align*} $$

Now, let

$$ \begin{align*} u_1(x)&:= \sum_{k=0}^{\infty}|\lambda _k| |Q_k|^{-{1}/{p_1}}\frac{\ell(Q_k)^{t_1}}{|x-{\mathbf{x}}_{Q_k}|^{t_1}}\chi_{(Q_k^{***})^c}(x), \\ u_2(x)&:= \bigg(\sum_{j\in{\mathbb{Z}}} \sum_{P\in\mathcal{D}_j}\Big( |\mathscr{B}_P^2(f_2)||P|^{-{1}/{2}}\frac{\chi_{P^c}(x)}{\langle 2^j(x-{\mathbf{x}}_P)\rangle^{t_2}}\Big)^{p_2} \bigg)^{{1}/{p_2}}, \\ u_3(x)&:= \sup_{j\in{\mathbb{Z}}} \bigg( \sum_{R\in\mathcal{D}_j}\Big( |\mathscr{B}_R^3(f_3)||R|^{-{1}/{2}}\frac{\chi_{R^c}(x)}{\langle 2^j(x-{\mathbf{x}}_R)\rangle^{t_3}}\Big)^{p_3} \bigg)^{{1}/{p_3}}. \end{align*} $$

Then it is easy to prove Equations (7.26) and (7.27).

9.13 Proof of Lemma 8.1

Using the fact that $\sum _{j\in {\mathbb {Z}}}\widehat {\Psi }(2^{-j}\vec {\boldsymbol {\xi }})=1$ for $\vec {\boldsymbol {\xi }}\not =\vec {\boldsymbol {0}}$ , we can write

(9.47) $$ \begin{align} T_{\sigma}(a_k,f_2,f_3)=\sum_{j\in{\mathbb{Z}}}T_{\widetilde{\sigma_j}}(a_k,f_2,f_3), \end{align} $$

where $\widetilde {\sigma _j}(\vec {\boldsymbol {\xi }}):=\sigma (\vec {\boldsymbol {\xi }})\widehat {\Psi }(2^{-j}\vec {\boldsymbol {\xi }})$ so that

$$ \begin{align*}\sup_{k\in{\mathbb{Z}}}\big\Vert \widetilde{\sigma_k}(2^k{\vec{\cdot}\;})\big\Vert_{L^2_s(({{\mathbb R}^n})^3)}=\mathcal{L}_s^2[\sigma]=1.\end{align*} $$

Moreover, due to the support of $\widetilde {\sigma _j}$ ,

(9.48) $$ \begin{align} T_{\widetilde{\sigma_j}}(a_k,f_2,f_3)=T_{\widetilde{\sigma_j}}({\Gamma}_{j+1}a_k,f_2,f_3). \end{align} $$

Now, the left-hand side of Equation (8.3) is less than

$$ \begin{align*}\sup_{l\in{\mathbb{Z}}} \Big|\sum_{k=1}^{\infty}\lambda _k\chi_{(Q_k^{***})^c}(x)\chi_{(B_k^l)^c}(x)\phi_l\ast \Big(\sum_{j\in{\mathbb{Z}}}T_{\widetilde{\sigma_j}}\big( {\Gamma}_{j+1}a_k,f_2,f_3\big)(x) \Big)(x) \Big|.\end{align*} $$

Let $s_1,s_2,s_3$ be numbers such that $s_1>n/p-n/2$ , $s_2,s_3>n/2$ , and $s=s_1+s_2+s_3$ . For $x\in (Q^{***})^c\cap (B_k^l)^c$ and $|x-y| \le 2^{-l}$ ,

$$ \begin{align*} |x - {\mathbf{x}}_{Q_k}| \lesssim | y - {\mathbf{x}}_{Q_k}|. \end{align*} $$

In the same argument as in the proof of Equations (9.5) and (9.8), with Equation (4.8) replaced by Equation (4.9), we can get

$$ \begin{align*} &\langle 2^j(x-{\mathbf{x}}_{Q_k})\rangle^{s_1}\big| T_{\widetilde{\sigma_j}}\big({\Gamma}_{j+1}a_k,f_2,f_3\big)(y) \big|\\ & \lesssim\Vert f_2\Vert_{L^{\infty}({{\mathbb R}^n})}\Vert f_3\Vert_{L^{\infty}({{\mathbb R}^n})} \langle2^j(x-{\mathbf{x}}_{Q_k})\rangle^{s_1} \int_{({{\mathbb R}^n})^3}\big|\widetilde{\sigma_j}^{\vee}(y-z_1,z_2,z_3)\big| |{\Gamma}_{j+1}a_k(z_1)| \; d\vec{\boldsymbol{z}} \\ &\lesssim \ell(Q_k)^{-n/p}\min{\big\{1,\big(2^j\ell(Q_k) \big)^M \big\}}I_{k,j,s}(y), \end{align*} $$

where $I_{k,j,s}^{in}$ and $I_{k,j,s}^{out}$ are defined as in Equaitons (9.4) and (9.9), respectively, and

$$ \begin{align*}I_{k,j,s}(y):=I_{k,j,s}^{in}(y)+I_{k,j,s}^{out}(y).\end{align*} $$

This yields that

$$ \begin{align*} &\Big| \phi_l\ast \Big(\sum_{j\in{\mathbb{Z}}}T_{\widetilde{\sigma_j}}\big({\Gamma}_{j+1}a_k,f_2,f_3\big) \Big)(x)\Big|\\ &\lesssim \ell(Q_k)^{-n/p}|x-{\mathbf{x}}_{Q_k}|^{-s_1}\mathcal{M}\Big( \sum_{j\in{\mathbb{Z}}}2^{-s_1 j}\min{\big\{1,\big( 2^j\ell(Q_k)\big)^M\big\}}I_{k,j,s}(\cdot) \Big)(x) \end{align*} $$

and thus Equation (8.3) follows from choosing $u_2(x)=u_3(x):=1$ and

$$ \begin{align*} u_1(x)&:=\sum_{k=1}^{\infty}|\lambda _k|\chi_{(Q_k^{***})^c}(x)\ell(Q_k)^{-n/p}|x-{\mathbf{x}}_{Q_k}|^{-s_1}\\ &\qquad \qquad \times \mathcal{M}\Big( \sum_{j\in{\mathbb{Z}}}2^{-s_1 j}\min{\big\{1,\big( 2^j\ell(Q_k)\big)^M\big\}}I_{k,j,s}(\cdot) \Big)(x). \end{align*} $$

Now, it is straightforward that $\Vert u_1\Vert _{L^p({{\mathbb R}^n})}$ is less than

$$ \begin{align*} \bigg(\sum_{k=1}^{\infty}|\lambda _k|^p\ell(Q_k)^{-n}\bigg\Vert |\cdot-{\mathbf{x}}_{Q_k}|^{-s_1} \mathcal{M}\Big( \sum_{j\in{\mathbb{Z}}}2^{-s_1 j}\min{\big\{1,\big( 2^j\ell(Q_k)\big)^M\big\}}I_{k,j,s}(\cdot) \Big) \bigg\Vert_{L^p((Q_k^{***})^c)}^p \bigg)^{1/p}, \end{align*} $$

and the $L^p$ -norm in the preceding expression is less than

$$ \begin{align*} &\big\Vert |\cdot-{\mathbf{x}}_{Q_k}|^{-{s_1}}\big\Vert_{L^{p({2}/{p})'}((Q_k^{***})^c)} \bigg\Vert \mathcal{M}\Big(\sum_{j\in{\mathbb{Z}}}2^{-s_1j}\min{\big\{1,\big(2^j\ell(Q_k)\big)^M\big\}}I_{k,j,s_1}(\cdot) \Big) \bigg\Vert_{L^{{2}}({{\mathbb R}^n})}\\ &\lesssim \ell(Q_k)^{-(s_1-(n/p-n/2))}\sum_{j\in{\mathbb{Z}}}2^{-s_1 j}\min\big\{1,\big(2^j\ell(Q_k) \big)^M\big\}\Vert I_{k,j,s}\Vert_{L^2({{\mathbb R}^n})}\\ &\lesssim \ell(Q_k)^{-(s_1-(n/p-n/2))}\ell(Q_k)^n\sum_{j\in{\mathbb{Z}}}2^{-(s_1-n/2) j} \min\big\{1,\big(2^j\ell(Q_k) \big)^M\big\}\lesssim \ell(Q_k)^{n/p}, \end{align*} $$

where Equations (9.13) and (9.14) are applied in the penultimate inequality for sufficiently large M. This concludes that

$$ \begin{align*}\Vert u_1\Vert_{L^p({{\mathbb R}^n})}\lesssim \Big( \sum_{k=1}^{\infty}|\lambda _k|^p\Big)^{1/p}\lesssim 1.\end{align*} $$

9.14 Proof of Lemma 8.2

Select $0<\epsilon <1$ such that

$$ \begin{align*}N_p:=[n/p-n]\le n/p-n<[n/p-n]+\epsilon<s-3n/2.\end{align*} $$

Then Lemma 4.1 yields that

$$ \begin{align*} \big| \phi_l \ast T_{\sigma}\big(a_k, f_2, f_3\big)(x) \big| &\lesssim 2^{l(N_p+n+\epsilon)} \int_{{{\mathbb R}^n}} |y - {\mathbf{x}}_{Q_k}|^{N_p + \epsilon} \big|T_{\sigma}(a_k, f_2, f_3)(y) \big| \;dy\\ &\lesssim \frac{1}{ |x - {\mathbf{x}}_{Q_k}|^{N_p+n+\epsilon} } \int_{\mathbb{R}^n}| y - {\mathbf{x}}_{Q_k} |^{N_p+\epsilon } \big|T_{\sigma}(a_k, f_2, f_3)(y) \big| \; dy\\ &\le \frac{1}{ |x - {\mathbf{x}}_{Q_k}|^{N_p+n+\epsilon} } \big( \mathcal{K}_{N_p+\epsilon}^{in}(a_k,f_2,f_3)+ \mathcal{K}_{N_p+\epsilon}^{in}(a_k,f_2,f_3)\big), \end{align*} $$

where we applied $2^l\lesssim |x-{\mathbf {x}}_{Q_k}|$ for $x\in B_k^l$ in the penultimate inequality and

$$ \begin{align*}\mathcal{K}_{N_p+\epsilon}^{\mathrm{in}}(a_k,f_2,f_3):=\int_{Q_k^{**}}| y - {\mathbf{x}}_{Q_k} |^{N_p+\epsilon } \big|T_{\sigma}(a_k, f_2, f_3)(y) \big| \; dy,\end{align*} $$

$$ \begin{align*}\mathcal{K}_{N_p+\epsilon}^{\mathrm{out}}(a_k,f_2,f_3):=\int_{(Q_k^{**})^c}| y - {\mathbf{x}}_{Q_k} |^{N_p+\epsilon } \big|T_{\sigma}(a_k, f_2, f_3)(y) \big| \; dy.\end{align*} $$

Now, we claim that

(9.49) $$ \begin{align} \mathcal{K}_{N_p+\epsilon}^{\mathrm{in}/\mathrm{out}}(a_k,f_2,f_3)\lesssim \ell(Q_k)^{N_p-n/p+n+\epsilon}. \end{align} $$

Once Equation (9.49) holds, we obtain

$$ \begin{align*}\big| \phi_l \ast T_{\sigma}\big(a_k, f_2, f_3\big)(x) \big| \lesssim |Q_k|^{-1/p}\frac{\ell(Q_k)^{N_p+n+\epsilon}}{|x-{\mathbf{x}}_{Q_k}|^{N_p+n+\epsilon}},\end{align*} $$

which implies (8.4) with $u_2(x)=u_3(x):=1$ and

$$ \begin{align*}u_1(x):=\sum_{k=1}^{\infty}|\lambda _k||Q_k|^{-1/p}\frac{\ell(Q_k)^{N_p+n+\epsilon}}{|x-{\mathbf{x}}_{Q_k}|^{N_p+n+\epsilon}}\chi_{(Q_k^{***})^c}(x).\end{align*} $$

Moreover,

$$ \begin{align*} \Vert u_1\Vert_{L^p({{\mathbb R}^n})}&\le \Big( \sum_{k=1}^{\infty}|\lambda _k|^p|Q_k|^{-1}\ell(Q_k)^{p(N_p+n+\epsilon)}\big\Vert |\cdot-{\mathbf{x}}_{Q_k}|^{-(N_p+n+\epsilon)}\big\Vert_{L^p((Q_k^{***})^c)}^p \Big)^{1/p}\\ &\lesssim \Big(\sum_{k=1}^{\infty}|\lambda _k|^p \Big)^{1/p}\lesssim 1 \end{align*} $$

because $N_p+n+\epsilon>n/p$ .

Therefore, it remains to show Equation (9.49). Indeed, it follows from Theorem D that

$$ \begin{align*} \mathcal{K}_{N_p+\epsilon}^{\mathrm{in}}(a_k,f_2,f_3)&\lesssim \ell(Q_k)^{N_p+\epsilon}\big\Vert T_{\sigma}(a_k,f_2,f_3)\big\Vert_{L^1({{\mathbb R}^n})}\\ &\lesssim \ell(Q_k)^{N_p+\epsilon}\Vert a_k\Vert_{L^1({{\mathbb R}^n})}\lesssim \ell(Q_k)^{N_p-n/p+n+\epsilon}. \end{align*} $$

For the other term, we use both Equations (9.47) and (9.48) to write

$$ \begin{align*}\mathcal{K}_{\hspace{1pt}N_p+\epsilon}^{\hspace{1pt}\mathrm{out}}(a_k,f_2,f_3)\lesssim \sum_{j\in{\mathbb{Z}}}2^{-j(N_p+\epsilon)}\int_{(Q_k^{**})^c}{ \langle 2^j(y-{\mathbf{x}}_{Q_k})\rangle^{N_p+\epsilon}\big| T_{\widetilde{\sigma_j}}\big({\Gamma}_{j+1}a_k,f_2,f_3\big)(y)\big|} \; dy.\end{align*} $$

Let $s_1,s_2,s_3$ be numbers satisfying

$$ \begin{align*}N_p+n/2+\epsilon<s_1<s-n,\quad s_2,s_3>n/2,\quad s_1+n<s_1+s_2+s_3=s,\end{align*} $$

similar to Equations (9.30) and (9.31). Then, using the argument in Equation (9.33), we have

$$ \begin{align*} &\big| T_{\widetilde{\sigma_j}}\big({\Gamma}_{j+1}a_k,f_2,f_3 \big)(y)\big|\chi_{(Q_k^{**})^c}(y)\lesssim \ell(Q_k)^{-n/p}\min\big\{1,\big( 2^j\ell(Q_k)\big)^M \big\}\frac{1}{\langle 2^j(y-{\mathbf{x}}_{Q_k})\rangle^{s_1}}\\ &\qquad \qquad \qquad \times 2^{-jn}\int_{{{\mathbb R}^n}} |A_{j,Q_k}(z_1)|\big\Vert \langle 2^j(y-z_1,z_2,z_3)\rangle^{s}\widetilde{\sigma_j}^{\vee}(y-z_1,z_2,z_3)\big\Vert_{L^2(z_2,z_3)} dz_1, \end{align*} $$

where $A_{j,Q_k}$ is defined as in Equation (9.32). This finally yields that

$$ \begin{align*} &\mathcal{K}_{N_p+\epsilon}^{\mathrm{out}}(a_k,f_2,f_3)\lesssim \ell(Q_k)^{-n/p}\sum_{j\in{\mathbb{Z}}}2^{-j(N_p+n+\epsilon)}\min\big\{1,\big( 2^j\ell(Q_k)\big)^M\big\} \int_{{{\mathbb R}^n}}|A_{j,Q_k}(z_1)| \\ &\qquad \times \Big(\int_{(Q_k^{**})^c} \frac{1}{\langle 2^j(y-{\mathbf{x}}_{Q_k})\rangle^{s_1-(N_p+\epsilon)}} \big\Vert \langle 2^j(y-z_1,z_2,z_3)\rangle^{s}\widetilde{\sigma_j}^{\vee}(y-z_1,z_2,z_3)\big\Vert_{L^2(z_2,z_3)} dy \Big) \; dz_1\\ &\lesssim \ell(Q_k)^{-n/p}\sum_{j\in{\mathbb{Z}}}2^{-j(s_1+n)}\min\big\{1,\big( 2^j\ell(Q_k)\big)^M\big\}\Vert A_{j,Q_k}\Vert_{L^1({{\mathbb R}^n})}\\ &\qquad \qquad \qquad \qquad \times \big\Vert \langle 2^j{\vec{\cdot}\;} \rangle^s\widetilde{\sigma_j}^{\vee}\big\Vert_{L^2(({{\mathbb R}^n})^3)} \big\Vert |\cdot-{\mathbf{x}}_{Q_k}|^{-(s_1-(N_p+\epsilon))}\big\Vert_{L^2({{\mathbb R}^n})}\\ &\lesssim \ell(Q_k)^{-n/p+n}\ell(Q_k)^{-(s_1-(N_p+n/2+\epsilon))}\sum_{j\in{\mathbb{Z}}}2^{-j(s_1-n/2)}\min \big\{1,\big( 2^j\ell(Q_k)\big)^{M-(L_0-s_1-n)} \big\}\\ &\lesssim \ell(Q_k)^{N_p-n/p+n+\epsilon} \end{align*} $$

for M and $L_0$ satisfying $M>L_0-s_1-n$ , which completes the proof of Equation (9.49).