# On the boundedness of multilinear Fourier multipliers on Hardy spaces

Jin Bong Lee  J.B. Lee, Research Institute of Mathematics, Seoul National
University,Seoul 08826, Republic of Korea<EMAIL_ADDRESS>and  Bae Jun Park
B. Park, Department of Mathematics, Sungkyunkwan University, Suwon 16419,
Republic of Korea<EMAIL_ADDRESS>
###### Abstract.

In this paper, we study multilinear Fourier multiplier operators on Hardy
spaces. In particular, we prove that the multilinear Fourier multiplier
operator of Hörmander type is bounded from $H^{p_{1}}\times\cdots\times
H^{p_{m}}$ to $H^{p}$ for $0<p_{1},\dots,p_{m}\leq 1$ with $1/p_{1}+\cdots
1/p_{m}=1/p$, under suitable cancellation conditions. As a result, we extend
the trilinear estimates in [17] to general multilinear ones and improve the
boundedness result in [18] in limiting situations.

###### Key words and phrases:

Fourier muitipliers, Multilinear operators, Hardy spaces

###### 2020 Mathematics Subject Classification:

Primary 42B15, 42B25, 42B30, 47H60

J.B. Lee is supported by NRF grant 2021R1C1C2008252. B. Park is supported in
part by NRF grant 2019R1F1A1044075 and was supported in part by a KIAS
Individual Grant MG070001 at the Korea Institute for Advanced Study

##  1\. Introduction

Let $m$ be a positive integer greater than or equal to $2$ and $\sigma$ be a
bounded function on $(\mathbb{R}^{n})^{m}$. The $m$-linear Fourier multiplier
operator $T_{\sigma}$ associated with $\sigma$ is defined by

(1.1)
$T_{\sigma}\big{(}f_{1},\dots,f_{m}\big{)}(x)=\int_{(\mathbb{R}^{n})^{m}}\sigma(\xi_{1},\dots,\xi_{m})\widehat{f_{1}}(\xi_{1})\cdots\widehat{f_{m}}(\xi_{m})e^{2\pi
i\langle x,\xi_{1}+\dots+\xi_{m}\rangle}d\xi_{1}\cdots d\xi_{m}$

for Schwartz functions $f_{1},\dots,f_{m}$ on $\mathbb{R}^{n}$, where
$\widehat{f}(\xi):=\int_{\mathbb{R}^{n}}f(x)e^{-2\pi i\langle x,\xi\rangle}dx$
is the Fourier transform of $f$. The $L^{p_{1}}\times\cdots\times L^{p_{m}}\to
L^{p}$ boundedness of $T_{\sigma}$, $1/p=1/p_{1}+\dots+1/p_{m}$, is one of
principal questions in multilinear harmonic analysis and there have been
several satisfactory results even though multilinear operators can not be
directly attacked by tools for linear operators. Coifman and Meyer [6] proved
that if $N$ is a sufficiently large number and $\sigma$ satisfies

(1.2)
$\big{|}\partial_{\xi_{1}}^{\alpha_{m}}\cdots\partial_{\xi_{m}}^{\alpha_{m}}\sigma(\xi_{1},\dots,\xi_{m})\big{|}\lesssim_{\alpha_{1},\dots,\alpha_{m}}\big{|}(\xi_{1},\dots,\xi_{m})\big{|}^{-(|\alpha_{1}|+\dots+|\alpha_{m}|)},\quad(\xi_{1},\dots,\xi_{m})\not=\vec{\boldsymbol{0}}$

for all multi-indices $\alpha_{1},\dots,\alpha_{m}$ satisfying
$|\alpha_{1}|+\cdots+|\alpha_{m}|\leq N$, then $T_{\sigma}$ is bounded from
$L^{p_{1}}(\mathbb{R}^{n})\times\cdots\times L^{p_{m}}(\mathbb{R}^{n})$ into
$L^{p}(\mathbb{R}^{n})$ for $1<p_{1},\dots,p_{m}<\infty$ and $1\leq p<\infty$
with $1/p=1/p_{1}+\dots+1/p_{m}$. This result was refined by Tomita [24] who
obtained a multilinear version of Hörmander’s multiplier theorem. Let $\Psi$
be a Schwartz function, defined on $(\mathbb{R}^{n})^{m}$, having the
properties that $0\leq\widehat{\Psi}\leq 1$,
$\mathrm{supp}(\widehat{\Psi})\subset\big{\\{}\vec{\boldsymbol{\xi}}\in(\mathbb{R}^{n})^{m}:2^{-1}\leq|\vec{\boldsymbol{\xi}}|\leq
2\big{\\}}$, and
$\sum_{j\in\mathbb{N}}\widehat{\Psi_{j}}(\vec{\boldsymbol{\xi}})=1$ for
$\vec{\boldsymbol{\xi}}\not=\vec{0}$ where
$\Psi_{j}:=2^{jmn}\Psi(2^{j}\vec{\cdot}\;)$. That is,
$\\{\Psi_{j}\\}_{j\in\mathbb{Z}}$ is a family of Littlewood-Paley functions on
$(\mathbb{R}^{n})^{m}$. For $s\geq 0$, we define the Sobolev space
$L_{s}^{2}((\mathbb{R}^{n})^{m})$ in terms of the finiteness of the norm

(1.3)
$\|F\|_{L_{s}^{2}((\mathbb{R}^{n})^{m})}:=\Big{(}\int_{(\mathbb{R}^{n})^{m}}\big{(}1+4\pi^{2}|\vec{\boldsymbol{\xi}}|^{2}\big{)}^{s}\big{|}\widehat{F}(\vec{\boldsymbol{\xi}})\big{|}^{2}\;d\vec{\boldsymbol{\xi}}\Big{)}^{1/2}$

and then we will use the notation

$\mathcal{L}_{s}^{2}[\sigma]:=\sup_{j\in\mathbb{Z}}\big{\|}\sigma(2^{j}\vec{\cdot}\;)\widehat{\Psi}\big{\|}_{L^{2}_{s}((\mathbb{R}^{n})^{m})}$

throughout this work.

######  Theorem A.

[24] Let $1<p,p_{1},\cdots,p_{m}<\infty$ with $1/p={1}{p_{1}}+\dots+1/p_{m}$.
Suppose that

$s>\frac{mn}{2}.$

Then we have

$\big{\|}T_{\sigma}(f_{1},\dots,f_{m})\big{\|}_{L^{p}(\mathbb{R}^{n})}\lesssim\mathcal{L}_{s}^{2}[\sigma]\prod_{\mu=1}^{m}\|f_{\mu}\|_{L^{p_{\mu}}(\mathbb{R}^{n})}$

for Schwartz functions $f_{1},\dots,f_{m}$ on $\mathbb{R}^{n}$.

The above result was generalized to the full range of indices
$0<p_{1},\dots,p_{m}\leq\infty$, in some recent papers, by using a product-
type Sobolev space condition, instead of $\mathcal{L}_{s}^{2}[\sigma]<\infty$.
For this one, the space $L^{p_{\mu}}(\mathbb{R}^{n})$, $\mu=1,\dots,m$, would
be replaced by the Hardy space $H^{p_{\mu}}(\mathbb{R}^{n})$.

To recall the definition of Hardy spaces, let $\phi$ be a smooth function
defined on $\mathbb{R}^{n}$ with
$\mathrm{supp}(\phi)\subset\\{x\in\mathbb{R}^{n}:|x|\leq 1\\}$. Then the Hardy
space $H^{p}(\mathbb{R}^{n})$ for $0<p\leq\infty$ contains all tempered
distributions $f$ on $\mathbb{R}^{n}$ such that

(1.4)
$\|f\|_{H^{p}(\mathbb{R}^{n})}:=\Big{\|}\sup_{l\in\mathbb{Z}}\big{|}\phi_{l}\ast
f\big{|}\Big{\|}_{L^{p}(\mathbb{R}^{n})}<\infty$

where $\phi_{l}:=2^{ln}\phi(2^{l}\cdot)$ for $l\in\mathbb{Z}$. The space
provides an extension to $0<p\leq 1$ in the scale of classical $L^{p}$ spaces
for $1<p\leq\infty$, which is more natural and useful in many respects than
the corresponding $L^{p}$ extension. Indeed,
$L^{p}(\mathbb{R}^{n})=H^{p}(\mathbb{R}^{n})$ for $1<p\leq\infty$ and several
essential operators, such as singular integrals of Calderón-Zygmund type, that
are well-behaved on $L^{p}(\mathbb{R}^{n})$ only for $1<p\leq\infty$ are also
well-behaved on $H^{p}(\mathbb{R}^{n})$ for $0<p\leq 1$. Moreover, especially
when $0<p\leq 1$, every $f\in H^{p}(\mathbb{R}^{n})$ can be written as

(1.5) $f=\sum_{k}\lambda_{k}a_{k}\qquad\text{ in the sense of tempered
distributions}$

where $\lambda_{k}$ are coefficients satisfying
$\big{(}\sum_{k\in\mathbb{N}}|\lambda_{k}|^{p}\big{)}^{1/p}\lesssim\|f\|_{H^{p}(\mathbb{R}^{n})}$
and $a_{k}$ are $H^{p}$-atoms, which means that there exist cubes $Q_{k}$ such
that $\mathrm{supp}(a_{k})\subset Q_{k}$,
$\|a_{k}\|_{L^{\infty}(\mathbb{R}^{n})}\leq|Q_{k}|^{-1/p}$, and

(1.6) $\int_{\mathbb{R}^{n}}x^{\alpha}a_{k}(x)dx=0\qquad\text{ for all multi-
indices }~{}|\alpha|\leq L$

for $L>n/p-n$. Here, $L$ may be actually an arbitrarily large number greater
than $n/p-n$. Now let $\mathscr{S}(\mathbb{R}^{n})$ denote the Schwartz space
on $\mathbb{R}^{n}$ and for a given positive integer $L$ let
$\mathscr{S}^{L}(\mathbb{R}^{n})$ be its subspace consisting of $f$ satisfying

$\int_{\mathbb{R}^{n}}x^{\alpha}f(x)dx=0\quad\text{ for all multi-
indices}~{}\alpha~{}\text{with }~{}|\alpha|\leq L.$

Then as mentioned in [13, Proposition 2.1], for all $0<p<\infty$

(1.7) $\mathscr{S}^{L}(\mathbb{R}^{n})~{}\text{ is dense in
}~{}H^{p}(\mathbb{R}^{n})\quad\text{ if }~{}L>\frac{n}{p}-n.$

We remark that $\mathscr{S}(\mathbb{R}^{n})$ is also dense in
$H^{p}(\mathbb{R}^{n})=L^{p}(\mathbb{R}^{n})$ for $1<p<\infty$, but not for
$0<p\leq 1$. See [22, Chapter III, §5.2] for more details. Moreover, as
discussed in [22, Chapter III, §5.4], if $f\in L^{1}(\mathbb{R}^{n})\cap
H^{p}(\mathbb{R}^{n})$ for $0<p\leq 1$, then we must necessarily have

(1.8) $\int_{\mathbb{R}^{n}}x^{\alpha}f(x)dx=0\quad\text{ for all multi-
indices }~{}|\alpha|\leq\frac{n}{p}-n.$

We refer to [2, 3, 8, 22, 26] for more details.

To study multilinear multiplier operators on Hardy spaces, we now assume that
the operator $T_{\sigma}$, defined in (1.1), initially acts on functions
$f_{\mu}\in\mathscr{S}^{L}(\mathbb{R}^{n})$, $\mu=1,\dots,m$, in view of
(1.7). Here, we set $L$ to be a sufficiently large number greater than
$n/p-n$. That is, $L>n/p-n\geq n/p_{\mu}-n$ for all $\mu=1,\dots,m$. Then we
have the following result.

######  Theorem B.

[11, 12, 15, 19] Let $0<p<\infty$ and $0<p_{1},\dots,p_{m}\leq\infty$ with
${1}/{p}=1/p_{1}+\dots+1/p_{m}$. Suppose that $L$ is sufficiently large and

(1.9) $s_{1},\dots,s_{m}>\frac{n}{2},\qquad\sum_{\mu\in
J}\Big{(}s_{\mu}-\frac{n}{p_{\mu}}\Big{)}>-\frac{n}{2}$

for any nonempty subset $J$ of $\\{1,\cdots,m\\}$. Then we have

(1.10)
$\displaystyle\big{\|}T_{\sigma}(f_{1},\dots,f_{m})\big{\|}_{L^{p}(\mathbb{R}^{n})}\lesssim\sup_{j\in\mathbb{Z}}\big{\|}\sigma(2^{j}\vec{\cdot}\;)\widehat{\Psi}\big{\|}_{L^{2}_{(s_{1},\dots,s_{m})}((\mathbb{R}^{n})^{m})}\prod_{\mu=1}^{m}\|f_{\mu}\|_{H^{p_{\mu}}(\mathbb{R}^{n})}$

for $f_{1},\dots,f_{m}\in\mathscr{S}^{L}(\mathbb{R}^{n})$.

Here, the space $L_{(s_{1},\dots,s_{m})}^{2}((\mathbb{R}^{n})^{m})$ indicates
the product type Sobolev space on $(\mathbb{R}^{n})^{m}$, in which the norm is
defined by replacing the term $(1+4\pi^{2}|\vec{\boldsymbol{\xi}}|)^{s}$ in
(1.3) by $\prod_{\mu=1}^{m}\big{(}1+4\pi^{2}|\xi_{\mu}|^{2}\big{)}^{s_{\mu}}$
for $\vec{\boldsymbol{\xi}}:=(\xi_{1},\dots,\xi_{m})\in(\mathbb{R}^{n})^{m}$.
We remark that the condition (1.9) is sharp in the sense that if the condition
does not hold, then there exists $\sigma$ such that the estimate (1.10) fails
for the corresponding operators $T_{\sigma}$. Refer to [21] for more details.

In the paper of Lee, Heo, Hong, Park, Yang, and the authors [18], there has
been some recent progress concerning multilinear multiplier theory associated
with the original Hörmander condition, which provides a generalization of
Theorem A.

######  Theorem C.

[18] Let $0<p<\infty$ and $0<p_{1},\dots,p_{m}\leq\infty$ with
${1}/{p}=1/p_{1}+\dots+1/p_{m}$. Suppose that $L$ is sufficiently large and

(1.11) $s>\frac{mn}{2},\qquad\frac{1}{p}-\frac{1}{2}<\frac{s}{n}+\sum_{\mu\in
J}\Big{(}\frac{1}{p_{\mu}}-\frac{1}{2}\Big{)}$

for any subset $J$ of $\\{1,\dots,m\\}$, which may be empty, where we adopt
the convention that the sum over the empty set is zero. Then we have

(1.12)
$\big{\|}T_{\sigma}(f_{1},\cdots,f_{m})\big{\|}_{L^{p}(\mathbb{R}^{n})}\lesssim\mathcal{L}_{s}^{2}[\sigma]\,\prod_{\mu=1}^{m}\|f_{\mu}\|_{H^{p_{\mu}}(\mathbb{R}^{n})}$

for $f_{1},\cdots,f_{m}\in\mathscr{S}^{L}(\mathbb{R}^{n})$.

The optimality of the condition (1.11) is already studied by Grafakos, He, and
Honzík [9] who proved that if (1.12) holds, then we must necessarily have
$s\geq mn/2$ and $1/p-1/2\leq s/n+\sum_{\mu\in J}(1/p_{\mu}-1/2)$ for all
subsets $J$ of $\\{1,\dots,m\\}$.

The estimate (1.12) has been recently improved in the trilinear case by
replacing the target space $L^{p}(\mathbb{R}^{n})$ by the Hardy space
$H^{p}(\mathbb{R}^{n})$ for $0<p\leq 1$, under the additional cancellation
condition

(1.13)
$\int_{\mathbb{R}^{n}}x^{\alpha}T_{\sigma}\big{(}f_{1},f_{2},f_{3}\big{)}(x)dx=0\qquad\text{
for multi-indices }~{}|\alpha|\leq\frac{n}{p}-n$

for all $f_{1},f_{2},f_{3}\in\mathscr{S}^{L}(\mathbb{R}^{n})$.

######  Theorem D.

[17] Let $0<p_{1},p_{2},p_{3}<\infty$ and $0<p\leq 1$ with
$1/p=1/p_{1}+1/p_{2}+1/p_{3}$. Suppose that $L$ is sufficiently large and

$s>\frac{3n}{2}\quad\text{ and
}\quad\frac{1}{p}-\frac{1}{2}<\frac{s}{n}+\sum_{\mu\in
J}\Big{(}\frac{1}{p_{\mu}}-\frac{1}{2}\Big{)}$

where $J$ is an arbitrary subset of $\\{1,2,3\\}$, which may be empty, where
we adopt the convention that the sum over the empty set is zero. Let $\sigma$
be a function on $(\mathbb{R}^{n})^{3}$ satisfying (1.13). Then we have

$\big{\|}T_{\sigma}(f_{1},f_{2},f_{3})\big{\|}_{H^{p}(\mathbb{R}^{n})}\lesssim\mathcal{L}_{s}^{2}[\sigma]\|f_{1}\|_{H^{p_{1}}(\mathbb{R}^{n})}\|f_{2}\|_{H^{p_{2}}(\mathbb{R}^{n})}\|f_{3}\|_{H^{p_{3}}(\mathbb{R}^{n})}$

for $f_{1},f_{2},f_{3}\in\mathscr{S}^{L}(\mathbb{R}^{n})$.

######  Remark 1.

The analogous result is still true in the bilinear setting, but the arguments
used there could not be applied to general $m$-linear operators for $m\geq 4$.

######  Remark 2.

The initial function space $\mathscr{S}^{L}(\mathbb{R}^{n})$ in Theorem D is
actually
$\mathscr{S}_{0}(\mathbb{R}^{n}):=\bigcap_{L\in\mathbb{N}}\mathscr{S}^{L}(\mathbb{R}^{n})$
in the original version [17, Theorem 1], but it is easy to see that everything
still works with $\mathscr{S}^{L}(\mathbb{R}^{n})$ as long as $L>n/p-n$. In
the present work, we employ $\mathscr{S}^{L}(\mathbb{R}^{n})$ to avoid
unnecessary situations.

######  Remark 3.

The condition (1.11) is equivalent to

$s>\frac{mn}{2},\qquad\frac{1}{p}-\frac{1}{2}<\frac{s}{n}+\sum_{\mu=1}^{m}\min\Big{\\{}0,\Big{(}\frac{1}{p_{\mu}}-\frac{1}{2}\Big{)}\Big{\\}}.$

Note that $J$ would be empty when $p_{1},\dots p_{m}\leq 2$.

The purpose of this paper is to establish multilinear extensions of Theorem D
in some limiting cases. Precisely, we obtain $H^{p_{1}}\times\cdots\times
H^{p_{m}}\to H^{p}$ estimates for general $m$-linear operators when
$0<p_{1},\dots,p_{m}\leq 1$. As mentioned in [17], the condition (1.13) is
very natural and necessary in the boundedness into the Hardy space $H^{p}$ for
$0<p\leq 1$, in view of (1.8). For a multilinear extension of the above
boundedness result, we now consider a bounded function $\sigma$ on
$(\mathbb{R}^{n})^{m}$ having the property

(1.14)
$\int_{\mathbb{R}^{n}}x^{\alpha}T_{\sigma}\big{(}f_{1},\dots,f_{m}\big{)}(x)dx=0\qquad\text{
for multi-indices }~{}|\alpha|\leq\frac{n}{p}-n$

where $f_{1},\dots,f_{m}\in\mathscr{S}^{L}(\mathbb{R}^{n})$ for $L>n/p-n$.
Then our main result is

######  Theorem 1.

Let $0<p_{1},\cdots,p_{m}\leq 1$ and $0<p\leq 1/m$ with $1/p=1/p_{1}+\cdots
1/p_{m}$. Suppose that $L$ is sufficiently large and

(1.15) $s>\frac{n}{p}-\frac{n}{2}.$

We also assume that the operator $T_{\sigma}$ satisfies the vanishing moment
condition (1.14). Then we have

$\big{\|}T_{\sigma}(f_{1},\cdots,f_{m})\big{\|}_{H^{p}(\mathbb{R}^{n})}\lesssim\mathcal{L}_{s}^{2}[\sigma]\prod_{\mu=1}^{m}\|f_{\mu}\|_{H^{p_{\mu}}(\mathbb{R}^{n})}$

for $f_{1},\cdots,f_{m}\in\mathscr{S}^{L}(\mathbb{R}^{n})$.

We note that the condition (1.15) clearly implies (1.11) when
$0<p_{1},\dots,p_{m}\leq 1$.

The study of boundedness into the Hardy space $H^{p}(\mathbb{R}^{n})$,
$0<p\leq 1$, for multilinear operators with vanishing integral, such as
(1.14), was initiated by Coifman, Lions, Meyer and Semmes [5] who proved that
certain bilinear operators are bounded from $L^{q}\times L^{q^{\prime}}$ into
$H^{1}$ for $1<q<\infty$ with $1/q+1/q^{\prime}=1$. Later, the result was
generalized by Dobyinski [7] for Coifman-Meyer multiplier operators and by
Coifman and Grafakos [4] for finite sums of products of Calderón-Zygmund
operators. These works, for $H^{1}$-boundedness, are related to $compensated$
$compactness$ which arises in theory of partial differential equations to
assure convergence of certain nonlinear quantities. For more details, we refer
to [1, 5, 20, 23] and references therein. In the works of Grafakos, Nakamura,
Nguyen and Sawano [13, 14], the $H^{p_{1}}\times\cdots\times H^{p_{m}}\to
H^{p}$ boundedness was established for multilinear multiplier operators
$T_{\sigma}$ with some cancellation condition, provided that $\sigma$
satisfies the multilinear Mihlin’s condition (1.2) for multi-indices
$\alpha_{1},\dots,\alpha_{m}$ satisfying $|\alpha_{1}|+\dots+|\alpha_{m}|\leq
N$ for sufficiently large $N$. However, the optimal regularity conditions were
not pursued at all as it requires sufficiently large $N$. From this point of
view, Theorem D improves the results of Grafakos, Nakamura, Nguyen, and Sawano
for trilinear operators, and Theorem 1 does for general multilinear operators
although the input function spaces are limited.

The proof of Theorem 1 heavily relies on the atomic decomposition (1.5), but
it is not guaranteed that we can interchange infinite sums of atoms and the
action of the operator $T_{\sigma}$ as in

$T_{\sigma}\big{(}f_{1},\dots,f_{m}\big{)}=\sum_{k_{1},\dots,k_{m}\in\mathbb{N}}\lambda_{1,k_{1}}\cdots\lambda_{m,k_{m}}T_{\sigma}\big{(}a_{1,k_{1}},\dots,a_{m,k_{m}}\big{)}$

for functions $f_{\mu}\in H^{p_{\mu}}(\mathbb{R}^{n})$ with atomic
representation
$f_{\mu}=\sum_{k_{\mu}\in\mathbb{N}}{\lambda_{\mu,k_{\mu}}a_{\mu,k_{\mu}}}$,
$1\leq\mu\leq m$. To settle this issue, we will additionally assume that the
multiplier $\sigma$ has compact support in $(\mathbb{R}^{n})^{m}$, which
allows us to write, with the aid of Theorem C,

$\big{|}\phi_{l}\ast
T_{\sigma}\big{(}f_{1},\dots,f_{m}\big{)}(x)\big{|}\leq\sum_{k_{1},\dots,k_{m}\in\mathbb{Z}}\Big{(}\prod_{\mu=1}^{m}|\lambda_{\mu,k_{\mu}}|\Big{)}\big{|}\phi_{l}\ast
T_{\sigma}\big{(}a_{1,k_{1}},\dots,a_{m,k_{m}}\big{)}(x)\big{|}.$

See Section 3 below for the details. Then we apply the method of Grafakos and
Kalton [10] and pointwise estimates of the form

$\big{|}\phi_{l}\ast
T_{\sigma}\big{(}a_{1,k_{1}},\dots,a_{m,k_{m}}\big{)}(x)\big{|}\lesssim\mathcal{L}_{s}^{2}[\sigma]b_{1}(x)\cdots
b_{m}(x)\quad\text{ uniformly in }~{}l\in\mathbb{Z}$

where $\|b_{\mu}\|_{L^{p_{\mu}}(\mathbb{R}^{n})}\lesssim 1$ for
$\mu=1,\dots,m$. Such arguments are stated in Lemma 4.1 below, which is
actually the key part of this paper. Since the above estimate separates the
left-hand side to $m$ distinct functions of $x$, we may now apply Hölder’s
inequality with exponents $1/p=1/p_{1}+\dots+1/p_{m}$. For general multipliers
$\sigma$ ( without the compact support condition ), we apply a suitable
approximation argument, which will be presented in (3.6).

### Notation

Let $\mathbb{N}$ and $\mathbb{Z}$ denote the sets of the natural numbers and
integers, respectively, and we define $\mathbb{N}_{0}:=\mathbb{N}\cup\\{0\\}$.
We use the symbol $A\lesssim B$ to denote $A\leq CB$ for some independent
constant $C$, and $A\sim B$ if $A\lesssim B$ and $B\lesssim A$ hold
simultaneously. For a given cube $Q$ in $\mathbb{R}^{n}$, let $c_{Q}$,
$\ell(Q)$ denote its center and side length, and $\chi_{Q}$ be the
characteristic function of $Q$. We denote by $Q^{*}$, $Q^{**}$, $Q^{***}$ the
concentric dilates of a cube $Q$ with $\ell(Q^{*})=10\sqrt{n}\ell(Q)$,
$\ell(Q^{**})=\big{(}10\sqrt{n}\big{)}^{2}\ell(Q)$, and
$\ell(Q^{***})=\big{(}10\sqrt{n}\big{)}^{3}\ell(Q)$. We use the notation
$\langle y\rangle:=\big{(}1+4\pi^{2}|y|^{2}\big{)}^{1/2}$ for
$y\in\mathbb{R}^{M}$, $M\in\mathbb{N}$.

##  2\. Preliminaries

For a locally integrable function $f$ on $\mathbb{R}^{n}$, let

$\mathcal{M}f(x):=\sup_{Q:x\in Q}\frac{1}{|Q|}\int_{Q}|f(y)|dy$

be the Hardy-Littlewood maximal function of $f$ where the supremum ranges over
all cubes in $\mathbb{R}^{n}$ containing $x$. Then it turns out that the
maximal operator $\mathcal{M}$ is bounded in $L^{p}(\mathbb{R}^{n})$ for all
$1<p\leq\infty$.

We now review some lemmas that will be repeatedly needed in the proof of the
main result. The first lemma is very useful in estimating the $L^{p}$ norm of
the sum of functions having a compact support for $0<p\leq 1$.

######  Lemma 2.1.

[10, Lemma 2.1] Let $0<p\leq 1$ and $J$ be a finite or countable index set.
Suppose that $\\{Q_{\mu}\\}_{\mu\in J}$ is a sequence of cubes in
$\mathbb{R}^{n}$, and $\\{F_{\mu}\\}_{\mu\in J}$ is a family of nonnegative
integrable functions with $\mathrm{supp}(F_{\mu})\subset Q_{\mu}$. Then we
have

$\Big{\|}\sum_{\mu\in
J}\chi_{Q_{\mu}}F_{\mu}\;\Big{\|}_{L^{p}(\mathbb{R}^{n})}\lesssim\bigg{\|}\sum_{\mu\in
J}\Big{(}\frac{1}{|Q_{\mu}|}\int_{Q_{\mu}}F_{\mu}(y)dy\Big{)}\chi_{Q_{\mu}}\bigg{\|}_{L^{p}(\mathbb{R}^{n})}$

where the constant in the inequality depends only on $p$.

We now introduce a variant of Bernstein’s inequality, which is contained in
[11, 19], modulo a few minor modifications.

######  Lemma 2.2.

[11, 19] Let $1\leq p\leq q\leq\infty$ and $s\geq 0$. Let $K$ be a function,
defined on $(\mathbb{R}^{n})^{m}$, whose Fourier transform is supported in a
ball of a constant radius. Then for $\mu=1,\dots,m$ and any multi-indices
$\alpha\in(\mathbb{N}_{0})^{n}$, there exists a constant $C_{p,q,{\alpha}}$
such that

$\displaystyle\big{\|}\big{\langle}(x_{1},\dots,x_{m})\big{\rangle}^{s}\partial_{{\mu}}^{{\alpha}}K(x_{1},\dots,x_{m})\big{\|}_{L^{q}(x_{\mu})}\leq
C_{p,q,{\alpha}}\big{\|}\big{\langle}(x_{1},\dots,x_{m})\big{\rangle}^{s}K(x_{1},\cdots,x_{m})\big{\|}_{L^{p}(x_{\mu})}$

for all $x_{j}\in\mathbb{R}^{n},j\not=\mu$.

The last one will play a significant role in the proof of Theorem 1, in
conjunction with the vanishing moment condition of $T_{\sigma}$ in (1.14).

######  Lemma 2.3.

[17, Lemma 4.1] Let $N\in\mathbb{N}$ and $a\in\mathbb{R}^{n}$. Suppose that a
function $F$, defined on $\mathbb{R}^{n}$, satisfies

$\int_{\mathbb{R}^{n}}{x^{\alpha}F(x)}dx=0\quad\text{for all multi-
indices}~{}\alpha~{}\text{ with }~{}|\alpha|\leq N.$

Then for any $0\leq\epsilon\leq 1$, there exists a constant $C_{\epsilon}>0$
such that

$\big{\|}\phi_{l}\ast F\big{\|}_{L^{\infty}(\mathbb{R}^{n})}\leq
C_{\epsilon}2^{l(N+n+\epsilon)}\int_{\mathbb{R}^{n}}{|y-a|^{N+\epsilon}|F(y)|}dy.$

##  3\. Proof of Theorem 1 : Reduction

We first show that the multiplier $\sigma$ can be assumed to be compactly
supported in Theorem 1, using an approximation argument. Suppose that

(3.1) $\mathcal{L}_{s}^{2}[\sigma]<\infty\quad\text{ for }~{}s>n/p-n/2.$

Let $\varphi$ be a Schwartz function on $\mathbb{R}^{n}$ such that
$\mathrm{supp}(\widehat{\varphi})\subset\\{\xi\in\mathbb{R}^{n}:|\xi|\leq
2\\}$ and $\widehat{\varphi}(\xi)=1$ for $|\xi|\leq 1$. For $R>0$, let
$\varphi_{R}:=R^{n}\varphi(R\,\cdot\,)$ and

$\widehat{\Phi_{R}}(\xi_{1},\dots,\xi_{m}):=\widehat{\varphi_{R}}(\xi_{1})\cdots\widehat{\varphi_{R}}(\xi_{m})\quad\text{
for }~{}(\xi_{1},\dots,\xi_{m})\in(\mathbb{R}^{n})^{m}.$

Now we define

$\sigma_{R}(\vec{\boldsymbol{\xi}}):=\sigma(\vec{\boldsymbol{\xi}})\widehat{\Phi_{R}}(\vec{\boldsymbol{\xi}})$

so that

(3.2) $T_{\sigma_{R}}(f_{1},\dots,f_{m})=T_{\sigma}\big{(}\varphi_{R}\ast
f_{1},\dots,\varphi_{R}\ast f_{m}\big{)}.$

It follows from Theorem C with (3.1) that

$T_{\sigma_{R}}(f_{1},\dots,f_{m})\to T_{\sigma}(f_{1},\dots,f_{m})~{}\text{
in the sense of tempered distribution}$

as $R\to\infty$, because $\varphi_{R}$ is an approximation of identity (as
$R\to\infty$). This clearly deduces

$\lim_{R\to\infty}\phi_{l}\ast
T_{\sigma_{R}}\big{(}f_{1},\dots,f_{m}\big{)}(x)=\phi_{l}\ast
T_{\sigma}\big{(}f_{1},\dots,f_{m}\big{)}(x)\quad\text{ for
}x\in\mathbb{R}^{n},$

and thus we have

(3.3)
$\displaystyle\big{\|}T_{\sigma}(f_{1},\dots,f_{m})\big{\|}_{H^{p}(\mathbb{R}^{n})}$
$\displaystyle=\Big{\|}\sup_{l\in\mathbb{Z}}\big{|}\phi_{l}\ast
T_{\sigma}(f_{1},\dots,f_{m})\big{|}\Big{\|}_{L^{p}(\mathbb{R}^{n})}$
$\displaystyle\leq\Big{\|}\liminf_{R\to\infty}\sup_{l\in\mathbb{Z}}\big{|}\phi_{l}\ast
T_{\sigma_{R}}(f_{1},\dots,f_{m})\big{|}\Big{\|}_{L^{p}(\mathbb{R}^{n})}$
$\displaystyle\leq\liminf_{R\to\infty}\Big{\|}\sup_{l\in\mathbb{Z}}\big{|}\phi_{l}\ast
T_{\sigma_{R}}(f_{1},\dots,f_{m})\big{|}\Big{\|}_{L^{p}(\mathbb{R}^{n})}$
$\displaystyle=\liminf_{R\to\infty}\big{\|}T_{\sigma_{R}}(f_{1},\dots,f_{m})\big{\|}_{H^{p}(\mathbb{R}^{n})}$

where Fatou’s lemma is applied. Moreover, using the compact support conditions
of both $\widehat{\Psi}$ and $\widehat{\Phi_{R}}$, it is easy to see that

$\displaystyle\mathcal{L}_{s}^{2}[\sigma_{R}]$
$\displaystyle\leq\sup_{j:2R<2^{j}\leq
4R\sqrt{m}}\big{\|}\sigma(2^{j}\vec{\cdot}\;)\widehat{\Psi}(\vec{\cdot}\;)\widehat{\Phi_{R}}(2^{j}\vec{\cdot}\;)\big{\|}_{L^{2}_{s}((\mathbb{R}^{n})^{m})}+\sup_{j:2^{j}\leq
2R}\big{\|}\sigma(2^{j}\vec{\cdot}\;)\widehat{\Psi}\big{\|}_{L_{s}^{2}((\mathbb{R}^{n})^{m})}.$

The second term is clearly less than $\mathcal{L}_{s}^{2}[\sigma]$ and the
Kato-Ponce inequality ( see [16] ) yields that for $2^{j}\sim R$

$\big{\|}\sigma(2^{j}\vec{\cdot}\;)\widehat{\Psi}(\vec{\cdot}\;)\widehat{\Phi}(2^{j}R^{-1}\vec{\cdot}\;)\big{\|}_{L^{2}_{s}((\mathbb{R}^{n})^{m})}\lesssim\big{\|}\sigma(2^{j}\vec{\cdot}\;)\widehat{\Psi}\big{\|}_{L^{2}_{s}((\mathbb{R}^{n})^{m})}+\big{\|}\sigma(2^{j}\vec{\cdot}\;)\widehat{\Psi}\big{\|}_{L^{2}((\mathbb{R}^{n})^{m})}\lesssim\mathcal{L}_{s}^{2}[\sigma],$

which finally proves

(3.4)
$\mathcal{L}_{s}^{2}[\sigma_{R}]\lesssim\mathcal{L}_{s}^{2}[\sigma]\quad\text{
uniformly in }R>0.$

In view of (3.3) and (3.4), we reduce the proof of Theorem 1 to the estimate

(3.5)
$\big{\|}T_{\sigma_{R}}(f_{1},\dots,f_{m})\big{\|}_{H^{p}(\mathbb{R}^{n})}\lesssim\mathcal{L}_{s}^{2}[\sigma_{R}]\prod_{\mu=1}^{m}\|f_{\mu}\|_{H^{p_{\mu}}(\mathbb{R}^{n})}\quad\text{
uniformly in }~{}R>0$

as this can be improved to any multipliers $\sigma$ in general:

(3.6)
$\displaystyle\big{\|}T_{\sigma}(f_{1},\dots,f_{m})\big{\|}_{H^{p}(\mathbb{R}^{n})}$
$\displaystyle\leq\liminf_{R\to\infty}\big{\|}T_{\sigma_{R}}(f_{1},\dots,f_{m})\big{\|}_{H^{p}(\mathbb{R}^{n})}$
$\displaystyle\lesssim\liminf_{R\to\infty}\mathcal{L}_{s}^{2}[\sigma_{R}]\prod_{\mu=1}^{m}\|f_{\mu}\|_{H^{p_{\mu}}(\mathbb{R}^{n})}$
$\displaystyle\lesssim\mathcal{L}_{s}^{2}[\sigma]\prod_{\mu=1}^{m}\|f_{\mu}\|_{H^{p_{\mu}}(\mathbb{R}^{n})}.$

In order to establish (3.5), we apply the atomic decomposition for Hardy
spaces. Each $f_{\mu}\in H^{p_{\mu}}(\mathbb{R}^{n})$ can be written as

$f_{\mu}=\sum_{k_{\mu}\in\mathbb{N}}\lambda_{\mu,k_{\mu}}a_{\mu,k_{\mu}}$

where $\\{a_{\mu,k_{\mu}}\\}_{k_{\mu}\in\mathbb{N}}$ is a sequence of
$H^{p_{\mu}}$-atoms so that $\mathrm{supp}(a_{\mu,k_{\mu}})\subset
Q_{\mu,k_{\mu}}$,
$\|a_{\mu,k_{\mu}}\|_{L^{\infty}(\mathbb{R}^{n})}\leq|Q_{\mu,k_{\mu}}|^{-1/p_{\mu}}$
for some cube $Q_{\mu,k_{\mu}}$,
$\int_{Q_{\mu,k_{\mu}}}x^{\gamma}a_{\mu,k_{\mu}}(x)dx=0$ for all multi-indices
$|\gamma|\leq L$, and
$\big{(}\sum_{k_{\mu}\in\mathbb{N}}|\lambda_{\mu,k_{\mu}}|^{p_{\mu}}\big{)}^{1/p_{\mu}}\lesssim\|f_{\mu}\|_{H^{p_{\mu}}(\mathbb{R}^{n})}$
where $L$ is a sufficiently large number greater than $n/p-n$. Then we claim
that

(3.7) $\big{|}\phi_{l}\ast
T_{\sigma_{R}}\big{(}f_{1},\dots,f_{m}\big{)}(x)\big{|}\leq\sum_{k_{1},\dots,k_{m}\in\mathbb{N}}\Big{(}\prod_{\mu=1}^{m}|\lambda_{\mu,k_{\mu}}|\Big{)}\big{|}\phi_{l}\ast
T_{\sigma_{R}}\big{(}a_{1,k_{1}},\dots,a_{m,k_{m}}\big{)}(x)\big{|}.$

For this one, we define

$f_{\mu}^{N}:=\sum_{k_{\mu}=1}^{N}\lambda_{\mu,k_{\mu}}a_{\mu,k_{\mu}},\qquad
N\in\mathbb{N}$

so that

(3.8)
$\big{\|}f_{\mu}-f_{\mu}^{N}\big{\|}_{H^{p_{\mu}}(\mathbb{R}^{n})}\lesssim\Big{(}\sum_{k_{\mu}=N+1}^{\infty}\big{|}\lambda_{\mu,k_{\mu}}\big{|}^{p_{\mu}}\Big{)}^{1/p_{\mu}}\to
0\quad\text{ as }~{}N\to\infty.$

Then the left-hand side of (3.7) is less than

$\displaystyle\big{|}\phi_{l}\ast
T_{\sigma_{R}}\big{(}f_{1}-f_{1}^{N},f_{2},\dots,f_{m}\big{)}(x)\big{|}$
$\displaystyle~{}+\big{|}\phi_{l}\ast
T_{\sigma_{R}}\big{(}f_{1}^{N},f_{2}-f_{2}^{N},f_{3},\dots,f_{m}\big{)}(x)\big{|}$
$\displaystyle\qquad\qquad\vdots$ $\displaystyle\qquad+\big{|}\phi_{l}\ast
T_{\sigma_{R}}\big{(}f_{1}^{N},\dots,f_{m-1}^{N},f_{m}-f_{m}^{N}\big{)}(x)\big{|}$
$\displaystyle\qquad\quad+\big{|}\phi_{l}\ast
T_{\sigma_{R}}\big{(}f_{1}^{N},\dots,f_{m}^{N}\big{)}(x)\big{|}.$

The last term is clearly dominated by

$\displaystyle\sum_{k_{1}=1}^{N}\cdots\sum_{k_{m}=1}^{N}\Big{(}\prod_{\mu=1}^{m}|\lambda_{\mu,k_{\mu}}|\Big{)}\big{|}\phi_{l}\ast
T_{\sigma_{R}}\big{(}a_{1,k_{1}},\dots,a_{m,k_{m}}\big{)}(x)\big{|}$
$\displaystyle\leq\sum_{k_{1},\dots,k_{m}\in\mathbb{N}}\Big{(}\prod_{\mu=1}^{m}|\lambda_{\mu,k_{\mu}}|\Big{)}\big{|}\phi_{l}\ast
T_{\sigma_{R}}\big{(}a_{1,k_{1}},\dots,a_{m,k_{m}}\big{)}(x)\big{|}$

by the multi-linearity of the operator $\phi_{l}\ast T_{\sigma_{R}}$.
Therefore, it suffices to show that the remaining terms approach $0$ as
$N\to\infty$. We will be concerned only with the first term as similar
arguments can be applied to the others. We note that the Fourier transform of
$T_{\sigma_{R}}(f_{1}-f_{1}^{N},f_{2},\dots,f_{m})$ is supported on the ball
$\\{\xi\in\mathbb{R}^{n}:|\xi|\leq 2\sqrt{m}R\\}$ due to the support of
$\widehat{\varphi_{R}}$. This shows that

$\phi_{l}\ast
T_{\sigma_{R}}\big{(}f_{1}-f_{1}^{N},f_{2},\dots,f_{m}\big{)}=\phi_{l}\ast\Upsilon_{R}\ast
T_{\sigma_{R}}\big{(}f_{1}-f_{1}^{N},f_{2},\dots,f_{m}\big{)}$

for some Schwartz function $\Upsilon_{R}$ on $\mathbb{R}^{n}$ whose Fourier
transform is equal to $1$ on the ball $\\{\xi\in\mathbb{R}^{n}:|\xi|\leq
2\sqrt{m}R\\}$ and is supported on the larger ball
$\\{\xi\in\mathbb{R}^{n}:|\xi|\leq 4\sqrt{m}R\\}$. Now, using Nikolskii’s
inequality (sometimes called Bernstein’s inequality) in [25, Proposition
1.3.2], we have

$\displaystyle\big{|}\phi_{l}\ast
T_{\sigma_{R}}\big{(}f_{1}-f_{1}^{N},f_{2},\dots,f_{m}\big{)}(x)\big{|}$
$\displaystyle=\Big{|}\int_{\mathbb{R}^{n}}\phi_{l}\ast\Upsilon_{R}(x-y)T_{\sigma_{R}}\big{(}f_{1}-f_{1}^{N},f_{2},\dots,f_{m}\big{)}(y)dy\Big{|}$
$\displaystyle\lesssim
R^{n/p-n}\Big{(}\int_{\mathbb{R}^{n}}\big{|}\phi_{l}\ast\Upsilon_{R}(x-y)\big{|}^{p}\big{|}T_{\sigma_{R}}\big{(}f_{1}-f_{1}^{N},f_{2},\dots,f_{m}\big{)}(y)\big{|}^{p}dy\Big{)}^{1/p}$
$\displaystyle\lesssim_{R,l,p}\big{\|}T_{\sigma_{R}}(f_{1}-f_{1}^{N},f_{2},\dots,f_{m})\big{\|}_{L^{p}(\mathbb{R}^{n})}$
$\displaystyle\lesssim\mathcal{L}_{s}^{2}[\sigma_{R}]\big{\|}f_{1}-f_{1}^{N}\big{\|}_{H^{p_{1}}(\mathbb{R}^{n})}\prod_{\mu=2}^{m}\|f_{\mu}\|_{H^{p_{\mu}}(\mathbb{R}^{n})}\to
0$

as $N\to\infty$, thanks to (3.8). Here, the last inequality follows from
Theorem C. This completes the proof of (3.7).

We also remark that the vanishing moment condition (1.14), together with
(3.2), guarantees

(3.9)
$\int_{\mathbb{R}^{n}}x^{\alpha}T_{\sigma_{R}}\big{(}a_{1,k_{1}},\dots,a_{m,k_{m}}\big{)}(x)dx=0\quad\text{
for multi-indices }~{}|\alpha|\leq\frac{n}{p}-n$

as each $\varphi_{R}\ast a_{\mu,k_{\mu}}$ belongs to
$\mathscr{S}^{L}(\mathbb{R}^{n})$.

In conclusion, matters reduce to the following proposition, in view of (1.4),
(3.7), and (3.9).

######  Proposition 3.1.

Let $0<p_{1},\dots,p_{m}\leq 1$ and $0<p\leq 1/m$ with
$1/p=1/p_{1}+\dots+1/p_{m}$. Suppose that $L$ is sufficiently large and

$s>\frac{n}{p}-\frac{n}{2}.$

We also assume that the multilinear multiplier operator $T_{\sigma}$ satisfies
the vanishing moment condition

$\int_{\mathbb{R}^{n}}x^{\alpha}T_{\sigma}\big{(}a_{1,k_{1}},\dots,a_{m,k_{m}}\big{)}(x)dx=0\quad\text{
for multi-indices }~{}|\alpha|\leq\frac{n}{p}-n$

for functions $f_{\mu}\in\mathscr{S}^{L}(\mathbb{R}^{n})$ having the atomic
representation
$f_{\mu}=\sum_{k_{\mu}\in\mathbb{N}}\lambda_{\mu,k_{\mu}}a_{\mu,k_{\mu}}$,
$\mu=1,\dots,m$, where each $a_{\mu,k_{\mu}}$ is an
$H^{p_{\mu}}(\mathbb{R}^{n})$ atom with the property (1.6) upto the degree
$L$, and $\\{\lambda_{\mu,k_{\mu}}\\}_{k_{\mu}\in\mathbb{N}}$ is a sequence of
coefficients satisfying
$\big{(}\sum_{k_{m}\in\mathbb{N}}|\lambda_{\mu,k_{\mu}}|^{p_{\mu}}\big{)}^{1/p{\mu}}\lesssim\|f_{\mu}\|_{H^{p_{\mu}}(\mathbb{R}^{n})}$.
Then we have

$\displaystyle\bigg{\|}\sup_{l\in\mathbb{Z}}\sum_{k_{1},\dots,k_{m}\in\mathbb{N}}\Big{(}\prod_{\mu=1}^{m}|\lambda_{\mu,k_{\mu}}|\Big{)}\big{|}\phi_{l}\ast
T_{\sigma}(a_{1,k_{1}},\dots,a_{m,k_{m}})\big{|}\bigg{\|}_{L^{p}(\mathbb{R}^{n})}\lesssim\mathcal{L}_{s}^{2}[\sigma]\prod_{\mu=1}^{m}\|f_{\mu}\|_{H^{p_{\mu}}(\mathbb{R}^{n})}.$

The proof of the proposition will be given in the next section.

##  4\. Proof of Theorem 1 : Proof of Proposition 3.1

Suppose that each $H^{p_{\mu}}$ atom $a_{\mu,k_{\mu}}$ is associated with a
cube $Q_{\mu,k_{\mu}}$ so that $\mathrm{supp}(a_{\mu,k_{\mu}})\subset
Q_{\mu,k_{\mu}}$,
$\|a_{\mu,k_{\mu}}\|_{L^{\infty}(\mathbb{R}^{n})}\leq|Q_{\mu,k_{\mu}}|^{-1/p_{\mu}}$,
and $\int_{Q_{\mu,k_{\mu}}}x^{\gamma}a_{\mu,k_{\mu}}(x)dx=0$ for multi-indices
$|\gamma|\leq L$. Then we write

$\sum_{k_{1},\dots,k_{m}\in\mathbb{N}}\Big{(}\prod_{\mu=1}^{m}|\lambda_{\mu,k_{\mu}}|\Big{)}\big{|}\phi_{l}\ast
T_{\sigma}\big{(}a_{1,k_{1}},\dots,a_{m,k_{m}}\big{)}(x)\big{|}\leq\mathcal{G}_{\mathrm{in}}^{l}(x)+\mathcal{G}_{\mathrm{out}}^{l}(x)$

where

$\mathcal{G}_{\mathrm{in}}^{l}(x):=\sum_{k_{1},\dots,k_{m}\in\mathbb{N}}\Big{(}\prod_{\mu=1}^{m}|\lambda_{\mu,k_{\mu}}|\Big{)}\big{|}\phi_{l}\ast
T_{\sigma}\big{(}a_{1,k_{1}},\dots,a_{m,k_{m}}\big{)}(x)\big{|}\chi_{Q_{1,k_{1}}^{**}\cap\cdots\cap
Q_{m,k_{m}}^{**}}(x),$
$\mathcal{G}_{\mathrm{out}}^{l}(x):=\sum_{k_{1},\dots,k_{m}\in\mathbb{N}}\Big{(}\prod_{\mu=1}^{m}|\lambda_{\mu,k_{\mu}}|\Big{)}\big{|}\phi_{l}\ast
T_{\sigma}\big{(}a_{1,k_{1}},\dots,a_{m,k_{m}}\big{)}(x)\big{|}\chi_{(Q_{1,k_{1}}^{**}\cap\cdots\cap
Q_{m,k_{m}}^{**})^{c}}(x).$

Now it suffices to show that

$\Big{\|}\sup_{l\in\mathbb{Z}}\mathcal{G}_{\mathrm{in}/\mathrm{out}}^{l}\Big{\|}_{L^{p}(\mathbb{R}^{n})}\lesssim\mathcal{L}_{s}^{2}[\sigma]\prod_{\mu=1}^{m}\|f_{\mu}\|_{H^{p_{\mu}}(\mathbb{R}^{n})}.$

The term associated with $\mathcal{G}_{\mathrm{in}}^{l}$ can be easily handled
by the argument of Grafakos and Kalton [10]. Suppose that
$Q_{1,k_{1}}^{**}\cap\cdots\cap Q_{m,k_{m}}^{**}\not=\emptyset$ as we are done
if the intersection is empty. We may also assume that the side length of the
cube $Q_{1,k_{1}}$ is the smallest among those of the cubes
$Q_{1,k_{1}},\dots,Q_{m,k_{m}}$. Then as done in [10], we choose a cube
$R_{\vec{\boldsymbol{k}}}$, $\vec{\boldsymbol{k}}:=(k_{1},\dots,k_{m})$, such
that

$Q_{1,k_{1}}^{**}\cap\cdots\cap Q_{m,k_{m}}^{**}\subset
R_{\vec{\boldsymbol{k}}}\subset Q_{1,k_{1}}^{***}\cap\cdots\cap
Q_{m,k_{m}}^{***}$

and $\ell(R_{\vec{\boldsymbol{k}}})\gtrsim\ell(Q_{1,k_{1}})$. By applying
Lemma 2.1, we have

$\displaystyle\Big{\|}\sup_{l\in\mathbb{Z}}\mathcal{G}_{\mathrm{in}}^{l}\Big{\|}_{L^{p}(\mathbb{R}^{n})}$
$\displaystyle\lesssim\bigg{\|}\sum_{k_{1},\cdots,k_{m}\in\mathbb{N}}\Big{(}\frac{1}{|R_{\vec{\boldsymbol{k}}}|}\int_{R_{\vec{\boldsymbol{k}}}}\big{|}\mathcal{M}T_{\sigma}\big{(}a_{1,k_{1}},\cdots,a_{m,k_{m}}\big{)}(y)\big{|}dy\Big{)}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\times\Big{(}\prod_{\mu=1}^{m}|\lambda_{\mu,k_{\mu}}|\Big{)}\chi_{R_{\vec{\boldsymbol{k}}}}\bigg{\|}_{L^{p}(\mathbb{R}^{n})}$

and the integral over $R_{\vec{\boldsymbol{k}}}$ is less than

$\displaystyle|R_{\vec{\boldsymbol{k}}}|^{1/2}\big{\|}\mathcal{M}T_{\sigma}\big{(}a_{1,k_{1}},\cdots,a_{m,k_{m}}\big{)}\big{\|}_{L^{2}(\mathbb{R}^{n})}$
$\displaystyle\lesssim|R_{\vec{\boldsymbol{k}}}|^{1/2}\mathcal{L}_{s}^{2}[\sigma]\|a_{1,k_{1}}\|_{L^{2}(\mathbb{R}^{n})}\prod_{\mu=2}^{m}\|a_{\mu,k_{\mu}}\|_{L^{\infty}(\mathbb{R}^{n})}$
$\displaystyle\leq|R_{\vec{\boldsymbol{k}}}|^{1/2}\mathcal{L}_{s}^{2}[\sigma]|Q_{1,k_{1}}|^{1/2}\prod_{\mu=1}^{m}|Q_{\mu,k_{\mu}}|^{-1/p_{\mu}}\lesssim|R_{\vec{\boldsymbol{k}}}|\mathcal{L}_{s}^{2}[\sigma]\prod_{\mu=1}^{m}|Q_{\mu,k_{\mu}}|^{-1/p_{\mu}}$

where we applied the maximal inequality for $\mathcal{M}$ and the $L^{2}\times
L^{\infty}\times\cdots\times L^{\infty}$ boundedness of $T_{\sigma}$, which
holds due to Theorem C, in the first inequality. This, together with Hölder’s
inequality, finally concludes that

$\displaystyle\Big{\|}\sup_{l\in\mathbb{Z}}\mathcal{G}_{\mathrm{in}}^{l}\Big{\|}_{L^{p}(\mathbb{R}^{n})}$
$\displaystyle\lesssim\mathcal{L}_{s}^{2}[\sigma]\bigg{\|}\prod_{\mu=1}^{m}\Big{(}\sum_{k_{\mu}\in\mathbb{N}}|\lambda_{\mu,k_{\mu}}||Q_{\mu,k_{\mu}}|^{-1/p_{\mu}}\chi_{Q_{\mu,k_{\mu}}^{***}}\Big{)}\bigg{\|}_{L^{p}(\mathbb{R}^{n})}$
$\displaystyle\lesssim\mathcal{L}_{s}^{2}[\sigma]\prod_{\mu=1}^{m}\Big{\|}\sum_{k_{\mu}\in\mathbb{N}}|\lambda_{\mu,k_{\mu}}||Q_{\mu,k_{\mu}}|^{-1/p_{\mu}}\chi_{Q_{\mu,k_{\mu}}^{***}}\Big{\|}_{L^{p_{\mu}}(\mathbb{R}^{n})}$
$\displaystyle\lesssim\mathcal{L}_{s}^{2}[\sigma]\prod_{\mu=1}^{m}\Big{(}\sum_{k_{\mu}\in\mathbb{N}}\big{|}\lambda_{\mu,k_{\mu}}\big{|}^{p_{\mu}}\Big{)}^{1/p_{\mu}}\lesssim\mathcal{L}_{s}^{2}[\sigma]\prod_{\mu=1}^{m}\|f_{\mu}\|_{H^{p_{\mu}}(\mathbb{R}^{n})}$

as desired.

To estimate the remaining term associated with
$\mathcal{G}_{\mathrm{out}}^{l}$, we claim the following lemma.

######  Lemma 4.1.

Let $0<p_{1},\dots,p_{m}\leq 1$ and $0<p\leq 1/m$ with
$1/p_{1}+\dots+1/p_{m}=1/p$. Suppose that $L$ is a sufficiently large number
and let $a_{\mu}$, $1\leq\mu\leq m$, be $H^{p_{\mu}}$-atoms such that

$\mathrm{supp}(a_{\mu})\subset
Q_{\mu},\quad\|a_{\mu}\|_{L^{\infty}(\mathbb{R}^{n})}\leq|Q_{\mu}|^{-1/p_{\mu}},\quad\int_{Q_{\mu}}x^{\alpha}a_{\mu}(x)dx=0$

for all $|\alpha|\leq L$. For each $\mu=1,\dots,m$, let

$B_{\mu,l}:=\big{\\{}x\in\mathbb{R}^{n}:|x-c_{Q_{\mu}}|\leq
100n^{2}2^{-l}\big{\\}}$

where $c_{Q_{\mu}}$ is the center of the cube $Q_{\mu}$. Let $J_{0}$ be any
nonempty subset of $J_{m}:=\\{1,\dots,m\\}$ and let $I_{0}$ be any nonempty
subset of $J_{0}$. Suppose that $s>n/p-n/2$ and $\sigma$ satisfies the
vanishing moment condition

(4.1)
$\int_{\mathbb{R}^{n}}x^{\alpha}T_{\sigma}\big{(}a_{1},\dots,a_{m}\big{)}(x)dx=0\quad\text{
for multi-indices }~{}|\alpha|\leq\frac{n}{p}-n.$

Then the following two statements are valid:

  1. (1)

There exist nonnegative functions
$\Gamma_{1}^{J_{0}},\dots,\Gamma_{m}^{J_{0}}$ such that

$\|\Gamma_{\mu}^{J_{0}}\|_{L^{p_{\mu}}(\mathbb{R}^{n})}\lesssim 1\quad\text{
for all }~{}1\leq\mu\leq m,$

and

(4.2) $\big{|}\phi_{l}\ast
T_{\sigma}\big{(}a_{1},\dots,a_{m}\big{)}(x)\big{|}\lesssim\mathcal{L}_{s}^{2}[\sigma]\Gamma_{1}^{J_{0}}(x)\cdots\Gamma_{m}^{J_{0}}(x)\quad\text{
uniformly in }~{}l\in\mathbb{Z}$

for all $x\in\Big{[}\big{(}\bigcap_{\mu\in J_{m}\setminus
J_{0}}Q_{\mu}^{**}\big{)}\setminus\big{(}\bigcup_{\mu\in
J_{0}}Q_{\mu}^{**}\big{)}\Big{]}\cap\big{(}\bigcap_{\mu\in
J_{0}}B_{\mu,l}\big{)}$.

  2. (2)

There exist nonnegative functions
$\Lambda_{1}^{J_{0}},\dots,\Lambda_{m}^{J_{0}}$ such that

$\|\Lambda_{\mu}^{J_{0}}\|_{L^{p_{\mu}}(\mathbb{R}^{n})}\lesssim 1\quad\text{
for all }~{}1\leq\mu\leq m,$

and

(4.3) $\big{|}\phi_{l}\ast
T_{\sigma}\big{(}a_{1},\dots,a_{m}\big{)}(x)\big{|}\lesssim\mathcal{L}_{s}^{2}[\sigma]\Lambda_{1}^{J_{0}}(x)\cdots\Lambda_{m}^{J_{0}}(x)\quad\text{
uniformly in }~{}l\in\mathbb{Z}$

for all $x\in\Big{[}\big{(}\bigcap_{\mu\in J_{m}\setminus
J_{0}}Q_{\mu}^{**}\big{)}\setminus\big{(}\bigcup_{\mu\in
J_{0}}Q_{\mu}^{**}\big{)}\Big{]}\cap\Big{[}\big{(}\bigcap_{\mu\in
J_{0}\setminus I_{0}}B_{\mu,l}\big{)}\setminus\big{(}\bigcup_{\mu\in
I_{0}}B_{\mu,l}\big{)}\Big{]}$.

Here, we adopt the convention that an intersection over an empty set is
considered $\mathbb{R}^{n}$.

The proof of the lemma will be provided in the next section. Taking the lemma
for granted and using the fact that

$\displaystyle\Big{(}\bigcap_{\mu=1}^{m}Q_{\mu,k_{\mu}}^{**}\Big{)}^{c}$
$\displaystyle=\bigcup_{J_{0}\subset
J_{m},J_{0}\not=\emptyset}\Big{[}\Big{(}\bigcap_{\mu\in J_{m}\setminus
J_{0}}Q_{\mu}^{**}\Big{)}\cap\Big{(}\bigcap_{\mu\in
J_{0}}\big{(}Q_{\mu}^{**}\big{)}^{c}\Big{)}\Big{]}$
$\displaystyle=\bigcup_{J_{0}\subset
J_{m},J_{0}\not=\emptyset}\Big{[}\Big{(}\bigcap_{\mu\in J_{m}\setminus
J_{0}}Q_{\mu}^{**}\Big{)}\setminus\Big{(}\bigcup_{\mu\in
J_{0}}Q_{\mu}^{**}\Big{)}\Big{]},$

we have

$\big{|}\phi_{l}\ast
T_{\sigma}\big{(}a_{1,k_{1}},\dots,a_{m,k_{m}}\big{)}(x)\big{|}\chi_{(\bigcap_{\mu=1}^{m}Q_{\mu,k_{\mu}}^{**})^{c}}(x)\lesssim\mathcal{L}_{s}^{2}[\sigma]\prod_{\mu=1}^{m}\Big{[}\sum_{J_{0}\subset
J_{m},J_{0}\not=\emptyset}\big{(}\Gamma_{\mu}^{J_{0}}(x)+\Lambda_{\mu}^{J_{0}}(x)\big{)}\Big{]}$

and this implies that

$\displaystyle\Big{\|}\sup_{l\in\mathbb{Z}}\mathcal{G}_{\mathrm{out}}^{l}\bigg{\|}_{L^{p}(\mathbb{R}^{n})}$
$\displaystyle\lesssim\mathcal{L}_{s}^{2}[\sigma]\bigg{\|}\prod_{\mu=1}^{m}\Big{(}\sum_{k_{\mu}\in\mathbb{N}}|\lambda_{\mu,k_{\mu}}|\sum_{J_{0}\subset
J_{m},J_{0}\not=\emptyset}\big{(}\Gamma_{\mu}^{J_{0}}+\Lambda_{\mu}^{J_{0}}\big{)}\Big{)}\bigg{\|}_{L^{p}(\mathbb{R}^{n})}$
$\displaystyle\leq\mathcal{L}_{s}^{2}[\sigma]\prod_{\mu=1}^{m}\bigg{\|}\sum_{k_{\mu}\in\mathbb{N}}|\lambda_{\mu,k_{\mu}}|\sum_{J_{0}\subset
J_{m},J_{0}\not=\emptyset}\big{(}\Gamma_{\mu}^{J_{0}}+\Lambda_{\mu}^{J_{0}}\big{)}\bigg{\|}_{L^{p_{\mu}}(\mathbb{R}^{n})}$

by using Hölder’s inequality. In addition, it follows that for each
$\mu=1,\dots,m$,

$\displaystyle\bigg{\|}\sum_{k_{\mu}\in\mathbb{N}}|\lambda_{\mu,k_{\mu}}|\sum_{J_{0}\subset
J_{m},J_{0}\not=\emptyset}\big{(}\Gamma_{\mu}^{J_{0}}+\Lambda_{\mu}^{J_{0}}\big{)}\bigg{\|}_{L^{p_{\mu}}(\mathbb{R}^{n})}$
$\displaystyle\leq\bigg{(}\sum_{k_{\mu}\in\mathbb{N}}|\lambda_{\mu,k_{\mu}}|^{p_{\mu}}\sum_{J_{0}\subset
J_{m},J_{0}\not=\emptyset}\Big{(}\big{\|}\Gamma_{\mu}^{J_{0}}\big{\|}_{L^{p_{\mu}}(\mathbb{R}^{n})}^{p_{\mu}}+\big{\|}\Lambda_{\mu}^{J_{0}}\big{\|}_{L^{p_{\mu}}(\mathbb{R}^{n})}^{p_{\mu}}\Big{)}\bigg{)}^{1/p_{\mu}}$
$\displaystyle\lesssim\Big{(}\sum_{k_{\mu}\in\mathbb{N}}|\lambda_{\mu,k_{\mu}}|^{p_{\mu}}\Big{)}^{1/p_{\mu}}\lesssim\|f_{\mu}\|_{H^{p_{\mu}}(\mathbb{R}^{n})}$

and this deduces

$\Big{\|}\sup_{l\in\mathbb{Z}}\mathcal{G}_{\mathrm{out}}^{l}\bigg{\|}_{L^{p}(\mathbb{R}^{n})}\lesssim\mathcal{L}_{s}^{2}[\sigma]\prod_{\mu=1}^{m}\|f_{\mu}\|_{H^{p_{\mu}}(\mathbb{R}^{n})}.$

This completes the proof of Proposition 3.1.

##  5\. Proof of Lemma 4.1

Without loss of generality, we may assume that $J_{0}=\\{1,\dots,\nu\\}$ for
some $1\leq\nu\leq m$. Let $[r]$ denote the greatest integer less than or
equal to $r\in\mathbb{R}$. Since $s>n/p-n/2$, there exists $0<\delta<1$ such
that

$n/p-n<[n/p-n]+\delta<s-n/2,$

and then we choose $0<\epsilon<1/(m+1)$ satisfying

(5.1) $n/p-n+(m+1)\epsilon<[n/p-n]+\delta<s-n/2,$

which implies that

(5.2) $[n/p-n+(m+1)\epsilon]\leq[n/p-n]\leq n/p-n$

because $0<\delta<1$. We may also assume that

(5.3)
$\ell(Q_{1})=\min_{1\leq{\mathrm{i}}\leq\nu}\\{\ell(Q_{{\mathrm{i}}})\\}.$

By using the Littlewood-Paley decomposition with
$\\{\Psi_{j}\\}_{j\in\mathbb{Z}}$, we can write

(5.4)
$T_{\sigma}(a_{1},\dots,a_{m})=\sum_{j\in\mathbb{Z}}T_{j}(a_{1},\dots,a_{m})$

where

$K_{j}(\vec{\boldsymbol{y}}):=\big{(}\sigma(2^{j}\vec{\cdot}\;)\widehat{\Psi}\big{)}^{\vee}(\vec{\boldsymbol{y}})$

and

$T_{j}\big{(}a_{1},\dots,a_{m}\big{)}(x):=2^{jmn}\int_{(\mathbb{R}^{n})^{m}}K_{j}\big{(}2^{j}(x-y_{1}),\dots,2^{j}(x-y_{m})\big{)}f_{1}(y_{1})\cdots
f_{m}(y_{m})\;d\vec{\boldsymbol{y}}.$

###  5.1. The first assertion

Suppose that

(5.5) $x\in\Big{(}\bigcap_{\mu=1}^{\nu}(Q_{\mu}^{**})^{c}\cap
B_{\mu,l}\Big{)}\mathbin{\scalebox{1.5}{$\cap$}}\Big{(}\bigcap_{\mu\in
J_{m}\setminus J_{0}}Q_{\mu}^{**}\Big{)}.$

We will prove (4.2) by setting

$\Gamma_{\mu}^{J_{0}}(x):=\begin{cases}\frac{\ell(Q_{\mu})^{\epsilon}}{|x-c_{Q_{\mu}}|^{n/p_{\mu}+\epsilon}}\chi_{(Q_{\mu}^{**})^{c}}(x)&\mu\in
J_{0}\\\ \ell(Q_{\mu})^{-n/p_{\mu}}\chi_{Q_{\mu}^{**}}(x)&\mu\in
J_{m}\setminus J_{0}\end{cases}.$

Then it is clear that
$\big{\|}\Gamma_{\mu}^{J_{0}}\big{\|}_{L^{p_{\mu}}(\mathbb{R}^{n})}\lesssim 1$
for all $\mu\in J_{m}$. In order to establish (4.2), let

$N_{\nu}:=[n/p_{1}+\dots+n/p_{\nu}+\nu\epsilon-n]$

and

$\epsilon_{\nu}:=n/p_{1}+\dots+n/p_{\nu}+\nu\epsilon-n-N_{\nu}$

so that

(5.6)
$N_{\nu}+n+\epsilon_{\nu}=\sum_{\mu=1}^{\nu}\big{(}n/p_{\mu}+\epsilon\big{)}.$

We note that (5.2) yields

(5.7) $N_{\nu}\leq N_{m}\leq n/p-n,$

and for any $1\leq\mu\leq\nu$,

$2^{l}\lesssim\frac{1}{|x-c_{Q_{\mu}}|}$

because of (5.5). By using the vanishing moment condition (4.1), together with
(5.7) and Lemma 2.3, we write

$\displaystyle\big{|}\phi_{l}\ast
T_{\sigma}\big{(}a_{1},\dots,a_{m}\big{)}(x)\big{|}$ $\displaystyle\lesssim
2^{l(N_{\nu}+n+\epsilon_{\nu})}\int_{\mathbb{R}^{n}}|y-c_{Q_{1}}|^{N_{\nu}+\epsilon_{\nu}}\big{|}T_{\sigma}\big{(}a_{1},\dots,a_{m}\big{)}(y)\big{|}\;dy$
$\displaystyle\lesssim\Big{(}\prod_{\mu=1}^{\nu}\frac{1}{|x-c_{Q_{\mu}}|^{n/p_{\mu}+\epsilon}}\Big{)}\int_{\mathbb{R}^{n}}|y-c_{Q_{1}}|^{N_{\nu}+\epsilon_{\nu}}\big{|}T_{\sigma}\big{(}a_{1},\dots,a_{m}\big{)}(y)\big{|}\;dy$
$\displaystyle\leq\mathcal{I}_{1}(x)+\mathcal{I}_{2}(x)$

where

$\mathcal{I}_{1}(x):=\Big{(}\prod_{\mu=1}^{\nu}\frac{\chi_{(Q_{\mu}^{**})^{c}}(x)}{|x-c_{Q_{\mu}}|^{n/p_{\mu}+\epsilon}}\Big{)}\Big{(}\prod_{\mu\in
J_{m}\setminus
J_{0}}\chi_{Q_{\mu}^{**}}(x)\Big{)}\int_{Q_{1}^{*}}|y-c_{Q_{1}}|^{N_{\nu}+\epsilon_{\nu}}\big{|}T_{\sigma}\big{(}a_{1},\dots,a_{m}\big{)}(y)\big{|}\;dy$

and

$\mathcal{I}_{2}(x):=\Big{(}\prod_{\mu=1}^{\nu}\frac{\chi_{(Q_{\mu}^{**})^{c}}(x)}{|x-c_{Q_{\mu}}|^{n/p_{\mu}+\epsilon}}\Big{)}\Big{(}\prod_{\mu\in
J_{m}\setminus
J_{0}}\chi_{Q_{\mu}^{**}}(x)\Big{)}\int_{(Q_{1}^{*})^{c}}|y-c_{Q_{1}}|^{N_{\nu}+\epsilon_{\nu}}\big{|}T_{\sigma}\big{(}a_{1},\dots,a_{m}\big{)}(y)\big{|}\;dy.$

Now it suffices to show that

(5.8)
$\mathcal{I}_{{\mathrm{i}}}(x)\lesssim\mathcal{L}_{s}^{2}[\sigma]\Gamma^{J_{0}}_{1}(x)\dots\Gamma^{J_{0}}_{m}(x),\qquad{\mathrm{i}}=1,2.$

To estimate $\mathcal{I}_{1}$, we see that

$\displaystyle\int_{Q_{1}^{*}}|y-c_{Q_{1}}|^{N_{\nu}+\epsilon_{\nu}}\big{|}T_{\sigma}\big{(}a_{1},\dots,a_{m}\big{)}(y)\big{|}\;dy\lesssim\ell(Q_{1})^{N_{\nu}+\epsilon_{\nu}}\big{\|}T_{\sigma}(a_{1},\dots,a_{m})\big{\|}_{L^{1}(\mathbb{R}^{n})}$
$\displaystyle\lesssim\mathcal{L}_{s}^{2}[\sigma]\ell(Q_{1})^{N_{\nu}+\epsilon_{\nu}}\|a_{1}\|_{H^{1}(\mathbb{R}^{n})}\prod_{\mu=2}^{m}\|a_{\mu}\|_{L^{\infty}(\mathbb{R}^{n})}\leq\mathcal{L}_{s}^{2}[\sigma]\ell(Q_{1})^{N_{\nu}+n+\epsilon_{\nu}}\prod_{\mu=1}^{m}\ell(Q_{\mu})^{-n/p_{\mu}}$
$\displaystyle\lesssim\mathcal{L}_{s}^{2}[\sigma]\Big{(}\prod_{\mu=1}^{\nu}\ell(Q_{\mu})^{\epsilon}\Big{)}\Big{(}\prod_{\mu\in
J_{m}\setminus J_{0}}\ell(Q_{\mu})^{-n/p_{\mu}}\Big{)}$

where we applied the $H^{1}\times L^{\infty}\times\cdots\times L^{\infty}\to
L^{1}$ boundedness of $T_{\sigma}$ in Theorem C and the conditions (5.6),
(5.3). Here, we notice that our assumption $s>\frac{n}{p}-\frac{n}{2}\geq
mn-\frac{n}{2}$ implies $s>\frac{mn}{2}$ whenever $m\geq 1$, and
$|Q_{1}|^{1/p_{1}-1}a_{1}$ is an $H^{1}$-atom for which
$\|a_{1}\|_{H^{1}(\mathbb{R}^{n})}\lesssim|Q_{1}|^{1-1/p_{1}}$. This proves
(5.8) for ${\mathrm{i}}=1$.

For the other term $\mathcal{I}_{2}$, we decompose $T_{\sigma}$ as in (5.4)
and then investigate the behavior of the operators $T_{j}$. We actually claim
that for $y\in(Q_{1}^{*})^{c}$

(5.9) $\displaystyle\big{|}T_{j}\big{(}a_{1},\dots,a_{m}\big{)}(y)\big{|}$
$\displaystyle\lesssim 2^{j\nu
n}\Big{(}\prod_{\mu=1}^{\nu}\ell(Q_{\mu})^{-n/p_{\mu}+n}\Big{)}\Big{(}\prod_{\mu\in
J_{m}\setminus J_{0}}\ell(Q_{\mu})^{-n/p_{\mu}}\Big{)}$
$\displaystyle\qquad\qquad\times\frac{1}{\langle
2^{j}(y-c_{Q_{1}})\rangle^{N_{\nu}+\epsilon_{\nu}+n/2+\epsilon}}\min\big{\\{}h_{j,s}^{Q_{1},0}(y),h_{j,s}^{Q_{1},L}(y)\big{\\}}$

where

(5.10) $\displaystyle h_{j,s}^{Q_{1},0}(y)$
$\displaystyle:=\int_{Q_{1}}|Q_{1}|^{-1}\Big{\|}\big{\langle}2^{j}(y-z_{1}),z_{2},\dots,z_{m}\big{\rangle}^{s}\big{|}K_{j}\big{(}2^{j}(y-z_{1}),z_{2},\dots,z_{m}\big{)}\big{|}\Big{\|}_{L^{2}(z_{2},\dots,z_{m})}dz_{1}$

and

(5.11) $\displaystyle h_{j,s}^{Q_{1},L}(y)$
$\displaystyle:=\big{(}2^{j}\ell(Q_{1})\big{)}^{L}\sum_{|\alpha|=L}\int_{0}^{1}\bigg{(}\int_{Q_{1}}|Q_{1}|^{-1}$
$\displaystyle\qquad\times\Big{\|}\big{\langle}2^{j}y_{c_{Q_{1}},z_{1}}^{\theta},z_{2},\dots,z_{m}\big{\rangle}^{s}\big{|}K_{j}\big{(}2^{j}y_{c_{Q_{1}},z_{1}}^{\theta},z_{2},\dots,z_{m}\big{)}\big{|}\Big{\|}_{L^{2}(z_{2},\dots,z_{m})}dz_{1}\bigg{)}d\theta$

by setting
$y_{c_{Q_{1}},z_{1}}^{\theta}:=y-c_{Q_{1}}-\theta(z_{1}-c_{Q_{1}})$. Here, it
is easy to see by Minkowski’s inequality that

(5.12) $\big{\|}h_{j,s}^{Q_{1},0}\big{\|}_{L^{2}(\mathbb{R}^{n})}\lesssim
2^{-jn/2}\mathcal{L}_{s}^{2}[\sigma],\qquad\big{\|}h_{j,s}^{Q_{1},L}\big{\|}_{L^{2}(\mathbb{R}^{n})}\lesssim
2^{-jn/2}\big{(}2^{j}\ell(Q_{1})\big{)}^{L}\mathcal{L}_{s}^{2}[\sigma].$

Once the claim (5.9) is established, it follows that

$\displaystyle\int_{(Q_{1}^{*})^{c}}|y-c_{Q_{1}}|^{N_{\nu}+\epsilon_{\nu}}\big{|}T_{j}\big{(}a_{1},\dots,a_{m}\big{)}(y)\big{|}dy$
$\displaystyle\lesssim 2^{-j(N_{\nu}+\epsilon_{\nu}-\nu
n)}\Big{(}\prod_{\mu=1}^{\nu}\ell(Q_{\mu})^{-n/p_{\mu}+n}\Big{)}\Big{(}\prod_{\mu\in
J_{m}\setminus J_{0}}\ell(Q_{\mu})^{-n/p_{\mu}}\Big{)}$
$\displaystyle\qquad\qquad\qquad\times\int_{(Q_{1}^{*})^{c}}\frac{1}{\langle
2^{j}(y-c_{Q_{1}})\rangle^{n/2+\epsilon}}\min\big{\\{}h_{j,\nu}^{Q_{1},0}(y),h_{j,\nu}^{Q_{1},L}(y)\big{\\}}dy$

and the integral in the last displayed expression is, via the Cauchy-Schwarz
inequality, less than

$\displaystyle\bigg{\|}\frac{1}{\langle 2^{j}(\cdot-
c_{Q_{1}})\rangle^{n/2+\epsilon}}\bigg{\|}_{L^{2}((Q_{1}^{*})^{c})}\min\Big{\\{}\big{\|}h_{j,\nu}^{Q_{1},0}\big{\|}_{L^{2}(\mathbb{R}^{n})},\big{\|}h_{j,\nu}^{Q_{1},L}\big{\|}_{L^{2}(\mathbb{R}^{n})}\Big{\\}}$
$\displaystyle\lesssim\mathcal{L}_{s}^{2}[\sigma]\ell(Q_{1})^{-\epsilon}2^{-jn-j\epsilon}\min\big{\\{}1,\big{(}2^{j}\ell(Q_{1})\big{)}^{L}\big{\\}}$

by applying (5.12). Therefore, in view of (5.4), we have

$\displaystyle\int_{(Q_{1}^{*})^{c}}|y-c_{Q_{1}}|^{N_{\nu}+\epsilon_{\nu}}\big{|}T_{\sigma}\big{(}a_{1},\dots,a_{m}\big{)}(y)\big{|}dy$
$\displaystyle\lesssim\mathcal{L}_{s}^{2}[\sigma]\Big{(}\prod_{\mu=1}^{\nu}\ell(Q_{\mu})^{-n/p_{\mu}+n}\Big{)}\Big{(}\prod_{\mu\in
J_{m}\setminus J_{0}}\ell(Q_{\mu})^{-n/p_{\mu}}\Big{)}$
$\displaystyle\qquad\qquad\qquad\times\ell(Q_{1})^{-\epsilon}\sum_{j\in\mathbb{Z}}2^{-j\big{(}\epsilon+\sum_{\mu=1}^{\nu}(n/p_{\mu}-n+\epsilon)\big{)}}\min\Big{\\{}1,\big{(}2^{j}\ell(Q_{1})\big{)}^{L}\Big{\\}}$
$\displaystyle\lesssim\mathcal{L}_{s}^{2}[\sigma]\Big{(}\prod_{\mu=1}^{\nu}\ell(Q_{\mu})^{\epsilon}\Big{)}\Big{(}\prod_{\mu\in
J_{m}\setminus J_{0}}\ell(Q_{\mu})^{-n/p_{\mu}}\Big{)}$

due to (5.6), (5.3), and the fact that

$L>n/p-n\geq\epsilon+\sum_{\mu=1}^{\nu}(n/p_{\mu}-n+\epsilon)\quad\text{ for
all }\nu=1,\dots,m.$

This finally proves (5.8) for ${\mathrm{i}}=2$.

Therefore, it remains to show the estimate (5.9). For notational convenience,
let

$M_{\nu}^{\epsilon}:=N_{\nu}+\epsilon_{\nu}+n/2+\epsilon.$

Moreover, for any $r>0$ and multi-indices $\alpha\in(\mathbb{N}_{0})^{n}$, we
simply write

(5.13) $\displaystyle K_{j}^{r}(z_{1},\dots,z_{m})$ $\displaystyle:=\langle
z_{1},\dots,z_{m}\rangle^{r}K_{j}(z_{1},\dots,z_{m}),$
$\displaystyle\big{(}\partial_{1}^{\alpha}K_{j}\big{)}^{r}(z_{1},\dots,z_{m})$
$\displaystyle:=\langle
z_{1},\dots,z_{m}\rangle^{r}\partial_{1}^{\alpha}K_{j}(z_{1},\dots,z_{m}).$

We now observe that for $y\in(Q_{1}^{*})^{c}$ and $z_{1}\in Q_{1}$,

(5.14) $|y-c_{Q_{1}}|\lesssim|y-z_{1}|$

and this implies that

$\displaystyle\langle
2^{j}(y-c_{Q_{1}})\rangle^{M_{\nu}^{\epsilon}}\big{|}T_{j}\big{(}a_{1},\dots,a_{m}\big{)}(y)\big{|}$
$\displaystyle\lesssim
2^{jmn}\Big{(}\prod_{\mu=1}^{m}\ell(Q_{\mu})^{-n/p_{\mu}}\Big{)}\int_{Q_{1}\times\cdots\times
Q_{m}}\big{|}K_{j}^{M_{\nu}^{\epsilon}}\big{(}2^{j}(y-z_{1}),\dots,2^{j}(y-z_{m})\big{)}\big{|}d\vec{\boldsymbol{z}}$
$\displaystyle\leq 2^{j\nu
n}\Big{(}\prod_{\mu=1}^{m}\ell(Q_{\mu})^{-n/p_{\mu}}\Big{)}$
$\displaystyle\qquad\quad\times\int_{Q_{1}\times\cdots\times
Q_{\nu}\times(\mathbb{R}^{n})^{m-\nu}}\big{|}K_{j}^{M_{\nu}^{\epsilon}}\big{(}2^{j}(y-z_{1}),\dots,2^{j}(y-z_{\nu}),z_{\nu+1},\dots,z_{m}\big{)}\big{|}d\vec{\boldsymbol{z}}$

with the obvious interpretations when $\nu=m$. If $\nu=1$, then the integral
in the preceding expression is bounded, using the Cauchy-Schwarz inequality,
by

$\displaystyle\ell(Q_{1})^{n}\int_{Q_{1}}|Q_{1}|^{-1}\Big{\|}K_{j}^{M_{1}^{\epsilon}+\sum_{\mu\in
J_{m}\setminus
J_{0}}(n/p_{\mu}+\epsilon)}\big{(}2^{j}(y-z_{1}),z_{2},\dots,z_{m}\big{)}\Big{\|}_{L^{2}(z_{2},\dots,z_{m})}dz_{1}$
$\displaystyle\lesssim\ell(Q_{1})^{n}h_{j,s}^{Q_{1},0}(y)$

where the inequality follows from the fact that
$s>n/p-n/2+(m+1)\epsilon=M_{1}^{\epsilon}+\sum_{\mu\in J_{m}\setminus
J_{0}}(n/p_{\mu}+\epsilon)$, which is due to (5.1). When $\nu\geq 2$, we
estimate the integral by

$\displaystyle\Big{(}\prod_{\mu=1}^{\nu}\ell(Q_{\mu})^{n}\Big{)}\int_{Q_{1}}|Q_{1}|^{-1}\Big{(}\int_{(\mathbb{R}^{n})^{m-\nu}}\Big{\|}K_{j}^{M_{\nu}^{\epsilon}}\big{(}2^{j}(y-z_{1}),z_{2},\dots,z_{m}\big{)}\Big{\|}_{L^{\infty}(z_{2},\dots,z_{\nu})}dz_{\nu+1}\cdots
dz_{m}\Big{)}dz_{1}$

and then, in view of Lemma 2.2, we may replace the $L^{\infty}$ norm inside
the integral by $L^{2}$ norm. Now by applying the Cauchy-Schwarz inequality,
the last expression is controlled by

$\displaystyle\Big{(}\prod_{\mu=1}^{\nu}\ell(Q_{\mu})^{n}\Big{)}\int_{Q_{1}}|Q_{1}|^{-1}\Big{\|}K_{j}^{M_{\nu}^{\epsilon}+\sum_{\mu\in
J_{m}\setminus
J_{0}}(n/p_{\mu}+\epsilon)}\big{(}2^{j}(y-z_{1}),z_{2},\dots,z_{m}\big{)}\Big{\|}_{L^{2}(z_{2},\dots,z_{m})}\;dz_{1}$
$\displaystyle\lesssim\Big{(}\prod_{\mu=1}^{\nu}\ell(Q_{\mu})^{n}\Big{)}h_{j,s}^{Q_{1},0}(y)$

as $s>M_{\nu}^{\epsilon}+\sum_{\mu\in J_{m}\setminus
J_{0}}(n/p_{\mu}+\epsilon)$. In both cases ( $\nu=1$ and $\nu\geq 2$ ), we
have

(5.15) $\displaystyle\big{|}T_{j}\big{(}a_{1},\dots,a_{m}\big{)}(y)\big{|}$
$\displaystyle\lesssim 2^{j\nu
n}\Big{(}\prod_{\mu=1}^{\nu}\ell(Q_{\mu})^{-n/p_{\mu}+n}\Big{)}\Big{(}\prod_{\mu\in
J_{m}\setminus J_{0}}\ell(Q_{\mu})^{-n/p_{\mu}}\Big{)}$
$\displaystyle\qquad\qquad\qquad\qquad\times\frac{1}{\langle
2^{j}(y-c_{Q_{1}})\rangle^{N_{\nu}+\epsilon_{\nu}+n/2+\epsilon}}h_{j,\nu}^{Q_{1},0}(y).$

Moreover, it follows from the vanishing moment condition of $a_{1}$ that

$\displaystyle\big{|}T_{j}\big{(}a_{1},\dots,a_{m}\big{)}(y)\big{|}$ (5.16)
$\displaystyle\lesssim\big{(}2^{j}\ell(Q)\big{)}^{L}2^{jmn}\Big{(}\prod_{\mu=1}^{m}\ell(Q_{\mu})^{-n/p_{\mu}}\Big{)}\sum_{|\alpha|=L}\int_{0}^{1}\Big{[}\int_{Q_{1}\times\cdots\times
Q_{m}}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\times\big{|}\partial_{1}^{\alpha}K_{j}\big{(}2^{j}y_{c_{Q_{1}},z_{1}}^{\theta},2^{j}(y-z_{2}),\dots,2^{j}(y-z_{m})\big{)}\big{|}d\vec{\boldsymbol{z}}\Big{]}d\theta$

where we recall that $L$ is the number of vanishing moment of $a_{1}$, which
is greater than $n/p-n$. Similar to (5.14), we have

$|y-c_{Q_{1}}|\lesssim|y_{c_{Q_{1}},z_{1}}^{\theta}|$

for $y\in(Q_{1}^{*})^{c}$, $z_{1}\in Q_{1}$, and $0<\theta<1$. This deduces
that

$\displaystyle\langle
2^{j}(y-c_{Q_{1}})\rangle^{M_{\nu}^{\epsilon}}\int_{Q_{1}\times\cdots\times
Q_{m}}\big{|}\partial_{1}^{\alpha}K_{j}\big{(}2^{j}y_{c_{Q_{1}},z_{1}}^{\theta},2^{j}(y-z_{2}),\dots,2^{j}(y-z_{m})\big{)}\big{|}d\vec{\boldsymbol{z}}$
$\displaystyle\lesssim 2^{-j(m-\nu)n}\int_{Q_{1}\times\cdots\times
Q_{\nu}\times(\mathbb{R}^{n})^{m-\nu}}\big{|}\big{(}\partial_{1}^{\alpha}K_{j}\big{)}^{M_{\nu}^{\epsilon}}\big{(}2^{j}y_{c_{Q_{1}},z_{1}}^{\theta},2^{j}(y-z_{2}),\dots,2^{j}(y-z_{\nu}),z_{\nu+1},\dots,z_{m}\big{)}\big{|}d\vec{\boldsymbol{z}}$

with the obvious interpretation when $\nu=m$. Now using the arguments that led
to (5.15), we can bound the preceding expression by

$\displaystyle
2^{-j(m-\nu)n}\Big{(}\prod_{\mu=1}^{\nu}\ell(Q_{\mu})^{n}\Big{)}\int_{Q_{1}}|Q_{1}|^{-1}\Big{\|}K_{j}^{M_{\nu}^{\epsilon}+\sum_{\mu\in
J_{m}\setminus
J_{0}}(n/p_{\mu}+\epsilon)}\big{(}2^{j}y_{c_{Q_{1}},z_{1}}^{\theta},z_{2},\dots,z_{m}\big{)}\Big{\|}_{L^{2}(z_{2},\dots,z_{m})}dz_{1}$
$\displaystyle\lesssim
2^{-j(m-\nu)n}\Big{(}\prod_{\mu=1}^{\nu}\ell(Q_{\mu})^{n}\Big{)}\int_{Q_{1}}|Q_{1}|^{-1}\Big{\|}K_{j}^{s}\big{(}2^{j}y_{c_{Q_{1}},z_{1}}^{\theta},z_{2},\dots,z_{m}\big{)}\Big{\|}_{L^{2}(z_{2},\dots,z_{m})}dz_{1}$

as $s>M_{\nu}^{\epsilon}+\sum_{\mu\in J_{m}\setminus
J_{0}}(n/p_{\mu}+\epsilon)$. Combining all together, we finally obtain

$\displaystyle\big{|}T_{j}\big{(}a_{1},\dots,a_{m}\big{)}(y)\big{|}$
$\displaystyle\lesssim 2^{j\nu
n}\Big{(}\prod_{\mu=1}^{\nu}\ell(Q_{\mu})^{-n/p_{\mu}+n}\Big{)}\Big{(}\prod_{\mu\in
J_{m}\setminus J_{0}}\ell(Q_{\mu})^{-n/p_{\mu}}\Big{)}$
$\displaystyle\qquad\qquad\qquad\qquad\times\frac{1}{\langle
2^{j}(y-c_{Q_{1}})\rangle^{N_{\nu}+\epsilon_{\nu}+n/2+\epsilon}}h_{j,s}^{Q_{1},L}(y).$

This, together with the estimate (5.15), completes the proof of (5.9).

###  5.2. The second assertion

Suppose that

$x\in\Big{(}\bigcap_{\mu\in J_{0}\setminus I_{0}}(Q_{\mu}^{**})^{c}\cap
B_{\mu,l}\Big{)}\mathbin{\scalebox{1.5}{$\cap$}}\Big{(}\bigcap_{\mu\in
I_{0}}(Q_{\mu}^{**})^{c}\cap(B_{\mu,l})^{c}\Big{)}\mathbin{\scalebox{1.5}{$\cap$}}\Big{(}\bigcap_{\mu\in
J_{m}\setminus J_{0}}Q_{\mu}^{**}\;\Big{)}.$

Since $I_{0}\not=\emptyset$, we can choose $\nu_{0}\in I_{0}$ where we set
$\nu_{0}=1$ if $I_{0}$ contains $1$. We first note that for any $\mu\in
J_{0}\setminus I_{0}$,

$|x-c_{Q_{\mu}}|\leq|x-c_{Q_{\nu_{0}}}|$

since $x\in B_{\mu,l}\cap B_{\nu_{0},l}^{c}$. Moreover, if $\mu\in I_{0}$,
$x\in(Q_{\mu}^{**})^{c}\cap(B_{\mu,l})^{c}$, and $|x-y|\leq 2^{-l}$, then we
have

$|x-c_{Q_{\mu}}|\lesssim|y-c_{Q_{\mu}}|.$

Therefore, it follows that

$\displaystyle\big{\langle}2^{j}(x-c_{Q_{1}})\big{\rangle}^{n/p_{1}-n/2+\epsilon}\Big{(}\prod_{\mu=2}^{\nu}\big{\langle}2^{j}(x-c_{Q_{\mu}})\big{\rangle}^{n/p_{\mu}+\epsilon}\Big{)}\big{|}\phi_{l}\ast
T_{j}\big{(}a_{1},\dots,a_{m}\big{)}(x)\big{|}$ (5.17) $\displaystyle\lesssim
2^{ln}\int_{|x-y|\leq
2^{-l}}\big{\langle}2^{j}(y-c_{Q_{\nu_{0}}})\big{\rangle}^{-n/2+\sum_{\mu\in
J_{0}\setminus I_{0}}(n/p_{\mu}+\epsilon)}$
$\displaystyle\qquad\qquad\qquad\times\Big{(}\prod_{\mu\in
I_{0}}\big{\langle}2^{j}(y-c_{Q_{\mu}})\big{\rangle}^{n/p_{\mu}+\epsilon}\Big{)}\big{|}T_{j}\big{(}a_{1},\dots,a_{m}\big{)}(y)\big{|}\chi_{\cap_{\mu\in
I_{0}}(Q_{\mu}^{*})^{c}}(y)dy$

as the conditions $x\in(Q_{\mu}^{**})^{c}\cap(B_{\mu,l})^{c}$ and $|x-y|\leq
2^{l}$ imply $y\in(Q_{\mu}^{*})^{c}$. In order to proceed further we claim
that the following lemma holds.

######  Lemma 5.1.

For $y\in\bigcap_{\mu\in I_{0}}(Q_{\mu}^{*})^{c}$,

$\displaystyle\big{\langle}2^{j}(y-c_{Q_{\nu_{0}}})\big{\rangle}^{-n/2+\sum_{\mu\in
J_{0}\setminus I_{0}}(n/p_{\mu}+\epsilon)}\Big{(}\prod_{\mu\in
I_{0}}\big{\langle}2^{j}(y-c_{Q_{\mu}})\big{\rangle}^{n/p_{\mu}+\epsilon}\Big{)}\big{|}T_{j}\big{(}a_{1},\dots,a_{m}\big{)}(y)\big{|}$
(5.18) $\displaystyle\lesssim 2^{j\nu
n}\Big{(}\prod_{\mu=1}^{m}\ell(Q_{\mu})^{-n/p_{\mu}}\Big{)}\Big{(}\prod_{\mu=1}^{\nu}\ell(Q_{\mu})^{n}\Big{)}\min\big{\\{}h_{j,s}^{Q_{1},0}(y),h_{j,s}^{Q_{1},L}(y)\big{\\}}$

where $h_{j,s}^{Q_{1},0}$ and $h_{j,s}^{Q_{1},L}$ are defined as in (5.10) and
(5.11).

The proof of (5.1) will be given in the last part of this section. For now we
take (5.1) for granted. Then the left-hand side of (5.2) is bounded by a
constant multiple of

$2^{j\nu
n}\Big{(}\prod_{\mu=1}^{m}\ell(Q_{\mu})^{-n/p_{\mu}}\Big{)}\Big{(}\prod_{\mu=1}^{\nu}\ell(Q_{\mu})^{n}\Big{)}\min\big{\\{}\mathcal{M}h_{j,s}^{Q_{1},0}(x),\mathcal{M}h_{j,s}^{Q_{1},L}(x)\big{\\}},$

which, together with (5.4), implies that

$\displaystyle\big{|}\phi_{l}\ast
T_{\sigma}\big{(}a_{1},\dots,a_{m}\big{)}(x)\big{|}$
$\displaystyle\lesssim\sum_{j\in\mathbb{Z}}2^{j\nu n}\Big{(}\prod_{\mu\in
J_{m}\setminus
J_{0}}\ell(Q_{\mu})^{-n/p_{\mu}}\chi_{Q_{\mu}^{**}}(x)\Big{)}\Big{(}\prod_{\mu=2}^{\nu}\frac{\ell(Q_{\mu})^{-(n/p_{\mu}-n)}}{\langle{2^{j}(x-c_{Q_{\mu}})}\rangle^{n/p_{\mu}+\epsilon}}\chi_{(Q_{\mu}^{**})^{c}}(x)\Big{)}$
$\displaystyle\qquad\qquad\times\frac{\ell(Q_{1})^{-(n/p_{1}-n)}}{\langle
2^{j}(x-c_{Q_{1}})\rangle^{n/p_{1}-n/2+\epsilon}}\chi_{(Q_{1}^{**})^{c}}(x)\min\big{\\{}\mathcal{M}h_{j,s}^{Q_{1},0}(x),\mathcal{M}h_{j,s}^{Q_{1},L}(x)\big{\\}}$
$\displaystyle\leq\Big{(}\prod_{\mu\in J_{m}\setminus
J_{0}}\ell(Q_{\mu})^{-n/p_{\mu}}\chi_{Q_{\mu}^{**}}(x)\Big{)}\Big{(}\prod_{\mu=2}^{\nu}\frac{1}{|{x-c_{Q_{\mu}}}|^{n/p_{\mu}+\epsilon}}\chi_{(Q_{\mu}^{**})^{c}}(x)\Big{)}\ell(Q_{1})^{-\sum_{\mu=1}^{\nu}(n/p_{\mu}-n)}$
$\displaystyle\qquad\qquad\times\frac{\chi_{(Q_{1}^{**})^{c}}(x)}{|x-c_{Q_{1}}|^{n/p_{1}-n/2+\epsilon}}\sum_{j\in\mathbb{Z}}2^{j\nu
n}2^{jn/2}2^{-j(\sum_{\mu=1}^{\nu}(n/p_{\mu}+\epsilon))}\min\big{\\{}\mathcal{M}h_{j,s}^{Q_{1},0}(x),\mathcal{M}h_{j,s}^{Q_{1},L}(x)\big{\\}}$
$\displaystyle\leq\Big{(}\prod_{\mu\in J_{m}\setminus
J_{0}}\ell(Q_{\mu})^{-n/p_{\mu}}\chi_{Q_{\mu}^{**}}(x)\Big{)}\Big{(}\prod_{\mu=2}^{\nu}\frac{\ell(Q_{\mu})^{\epsilon}}{|{x-c_{Q_{\mu}})}|^{n/p_{\mu}+\epsilon}}\chi_{(Q_{\mu}^{**})^{c}}(x)\Big{)}\ell(Q_{1})^{-\sum_{\mu=1}^{\nu}(n/p_{\mu}-n+\epsilon)}$
$\displaystyle\qquad\times\frac{\ell(Q_{1})^{\epsilon}}{|x-c_{Q_{1}}|^{n/p_{1}-n/2+\epsilon}}\chi_{(Q_{1}^{**})^{c}}(x)\sum_{j\in\mathbb{Z}}2^{jn/2}2^{-j(\sum_{\mu=1}^{\nu}(n/p_{\mu}-n+\epsilon))}\min\big{\\{}\mathcal{M}h_{j,s}^{Q_{1},0}(x),\mathcal{M}h_{j,s}^{Q_{1},L}(x)\big{\\}}.$

Here, we used the assumption that $\ell(Q_{1})\leq\ell(Q_{\mu})$,
$\mu=2,\dots,\nu$, in the last two inequalities. Finally, the inequality (4.3)
follows from taking

$\displaystyle\Lambda_{1}^{J_{0}}(x)$
$\displaystyle:=\big{(}\mathcal{L}_{s}^{2}[\sigma]\big{)}^{-1}\ell(Q_{1})^{-\sum_{\mu=1}^{\nu}(n/p_{\mu}-n+\epsilon)}\frac{\ell(Q_{1})^{\epsilon}}{|x-c_{Q_{1}}|^{n/p_{1}-n/2+\epsilon}}\chi_{(Q_{1}^{**})^{c}}(x)$
$\displaystyle\qquad\times\sum_{j\in\mathbb{Z}}2^{jn/2}2^{-j(\sum_{\mu=1}^{\nu}(n/p_{\mu}-n+\epsilon))}\min\big{\\{}\mathcal{M}h_{j,s}^{Q_{1},0}(x),\mathcal{M}h_{j,s}^{Q_{1},L}(x)\big{\\}}$
$\displaystyle\Lambda_{\mu}^{J_{0}}(x)$
$\displaystyle:=\begin{cases}\frac{\ell(Q_{\mu})^{\epsilon}}{|x-c_{Q_{\mu}}|^{n/p_{\mu}+\epsilon}}\chi_{(Q_{\mu}^{**})^{c}}(x),\qquad&\mu\in
J_{0}\setminus\\{1\\}\\\
\ell(Q_{\mu})^{-n/p_{\mu}}\chi_{Q_{\mu}^{**}}(x),\qquad&\mu\in J_{m}\setminus
J_{0}\end{cases}.$

Moreover, using Hölder’s inequality with $(1/p_{1}-1/2)+1/2=1/p_{1}$,

$\displaystyle\big{\|}\Lambda_{1}^{J_{0}}\big{\|}_{L^{p_{1}}(\mathbb{R}^{n})}$
$\displaystyle\leq\big{(}\mathcal{L}_{s}^{2}[\sigma]\big{)}^{-1}\ell(Q_{1})^{-\sum_{\mu=1}^{\nu}(n/p_{\mu}-n+\epsilon)}\bigg{\|}\frac{\ell(Q_{1})^{\epsilon}}{|\cdot-
c_{Q_{1}}|^{n/p_{1}-n/2+\epsilon}}\bigg{\|}_{L^{\frac{1}{1/p_{1}-1/2}}((Q_{1}^{**})^{c})}$
$\displaystyle\qquad\times\Big{\|}\sum_{j\in\mathbb{Z}}2^{jn/2}2^{-j(\sum_{\mu=1}^{\nu}(n/p_{\mu}-n+\epsilon))}\min\big{\\{}\mathcal{M}h_{j,s}^{Q_{1},0},\mathcal{M}h_{j,s}^{Q_{1},L}\big{\\}}\Big{\|}_{L^{2}(\mathbb{R}^{n})}.$

The $L^{\frac{1}{1/p_{1}-1/2}}$ norm is controlled by a constant and the
$L^{2}$ norm is less than

$\displaystyle\sum_{j\in\mathbb{Z}}2^{jn/2}2^{-j(\sum_{\mu=1}^{\nu}(n/p_{\mu}-n+\epsilon))}\min\Big{\\{}\big{\|}\mathcal{M}h_{j,s}^{Q_{1},0}\big{\|}_{L^{2}(\mathbb{R}^{n})},\big{\|}\mathcal{M}h_{j,s}^{Q_{1},L}\big{\|}_{L^{2}(\mathbb{R}^{n})}\Big{\\}}$
$\displaystyle\lesssim\mathcal{L}_{s}^{2}[\sigma]\sum_{j\in\mathbb{Z}}2^{-j(\sum_{\mu=1}^{\nu}(n/p_{\mu}-n+\epsilon))}\min\Big{\\{}1,\big{(}2^{j}\ell(Q_{1})\big{)}^{L}\Big{\\}}$
$\displaystyle\lesssim\mathcal{L}_{s}^{2}[\sigma]\ell(Q_{1})^{\sum_{\mu=1}^{\nu}(n/p_{\mu}-n+\epsilon)}$

for $L>n/p-n\geq n/p-mn+m\epsilon$, where we applied the $L^{2}$ boundedness
of $\mathcal{M}$ and the estimates (5.12) in the first inequality. This
completes the proof of

$\big{\|}\Lambda_{1}^{J_{0}}\big{\|}_{L^{p_{1}}(\mathbb{R}^{n})}\lesssim 1.$

It is also clear that

$\big{\|}\Lambda_{\mu}^{J_{0}}\big{\|}_{L^{p_{1}}(\mathbb{R}^{n})}\lesssim
1,\qquad\mu=2,\dots,m.$

###  5.3. Proof of Lemma 5.1

Now let us prove the claim (5.1). Suppose that $y\in\bigcap_{\mu\in
I_{0}}(Q_{\mu}^{*})^{c}$. For simplicity, we use the notations

$s_{\mu}:=\begin{cases}n/p_{\nu_{0}}-n/2+\epsilon+\sum_{\eta\in J_{0}\setminus
I_{0}}(n/p_{\eta}+\epsilon),\qquad&\mu=\nu_{0}\\\
n/p_{\mu}+\epsilon,\qquad&\mu\not=\nu_{0},~{}\mu\in I_{0}\end{cases}$

and

$S_{I_{0}}:=\sum_{\mu\in I_{0}}s_{\mu}=\Big{(}\sum_{\mu\in
J_{0}}\frac{n}{p_{\mu}}\Big{)}-\frac{n}{2}+\nu\epsilon.$

Then the left-hand side of (5.1) can be written as

(5.19) $\Big{(}\prod_{\mu\in
I_{0}}{\big{\langle}2^{j}(y-c_{Q_{\mu}})\big{\rangle}^{s_{\mu}}}\Big{)}\big{|}T_{j}\big{(}a_{1},\dots,a_{m}\big{)}(y)\big{|}.$

We first observe that for $y\in(Q_{\mu}^{*})^{c}$ and $z_{\mu}\in Q_{\mu}$,

(5.20) $|y-c_{Q_{\mu}}|\lesssim|y-z_{\mu}|$

and this implies that (5.19) is controlled by a constant times

$\displaystyle
2^{jmn}\Big{(}\prod_{\mu=1}^{m}\ell(Q_{\mu})^{-n/p_{\mu}}\Big{)}\int_{Q_{1}\times\cdots\times
Q_{m}}\big{|}K_{j}^{S_{I_{0}}}\big{(}2^{j}(y-z_{1}),\dots,2^{j}(y-z_{m})\big{)}\big{|}d\vec{\boldsymbol{z}}.$

where $K_{j}^{r}$ is defined as in (5.13). Now we apply a change of variables
to bound the above expression by

$\displaystyle 2^{j\nu
n}\Big{(}\prod_{\mu=1}^{m}\ell(Q_{\mu})^{-n/p_{\mu}}\Big{)}\int_{Q_{1}\times\cdots\times
Q_{\nu}\times(\mathbb{R}^{n})^{m-\nu}}\big{|}K_{j}^{S_{I_{0}}}\big{(}2^{j}(y-z_{1}),\dots,2^{j}(y-z_{\nu}),z_{\nu+1},\dots,z_{m}\big{)}\big{|}d\vec{\boldsymbol{z}}$
(5.21) $\displaystyle\leq 2^{j\nu
n}\Big{(}\prod_{\mu=1}^{m}\ell(Q_{\mu})^{-n/p_{\mu}}\Big{)}\Big{(}\prod_{\mu=1}^{\nu}\ell(Q_{\mu})^{n}\Big{)}\int_{Q_{1}}|Q_{1}|^{-1}$
$\displaystyle\qquad\qquad\qquad\times\Big{(}\int_{(\mathbb{R}^{n})^{m-\nu}}\big{\|}K_{j}^{S_{I_{0}}}\big{(}2^{j}(y-z_{1}),z_{2},\dots,z_{m}\big{)}\big{\|}_{L^{\infty}(z_{2},\dots,z_{\nu})}dz_{\nu+1}\cdots
dz_{m}\Big{)}\;dz_{1}$

with the obvious interpretations when $\nu=1$ or $\nu=m$ as in the proof of
(5.15). In view of Lemma 2.2, we may replace the $L^{\infty}$ norm inside the
integral by $L^{2}$ norm and then apply the Cauchy-Schwarz inequality to
dominate the above expression by

$\displaystyle 2^{j\nu
n}\Big{(}\prod_{\mu=1}^{m}\ell(Q_{\mu})^{-n/p_{\mu}}\Big{)}\Big{(}\prod_{\mu=1}^{\nu}\ell(Q_{\mu})^{n}\Big{)}\int_{Q_{1}}|Q_{1}|^{-1}$
$\displaystyle\qquad\qquad\times\Big{\|}K_{j}^{S_{I_{0}}+\sum_{\mu\in
J_{m}\setminus
J_{0}}(n/p_{\mu}+\epsilon)}\big{(}2^{j}(y-z_{1}),z_{2},\dots,z_{m}\big{)}\Big{\|}_{L^{2}(z_{2},\dots,z_{m})}dz_{1}$
$\displaystyle\leq 2^{j\nu
n}\Big{(}\prod_{\mu=1}^{m}\ell(Q_{\mu})^{-n/p_{\mu}}\Big{)}\Big{(}\prod_{\mu=1}^{\nu}\ell(Q_{\nu})^{n}\Big{)}h_{j,s}^{Q_{1},0}(y)$

as

(5.22) $s>n/p-n/2+m\epsilon=S_{I_{0}}+\sum_{\mu\in J_{m}\setminus
J_{0}}(n/p_{\mu}+\epsilon).$

Moreover, using (5.1) and (5.20), the term (5.19) is no more than a constant
multiple of

(5.23)
$\displaystyle\big{(}2^{j}\ell(Q_{1})\big{)}^{L}2^{jmn}\Big{(}\prod_{\mu=1}^{m}\ell(Q_{\mu})^{-n/p_{\mu}}\Big{)}\sum_{|\alpha|=L}\int_{0}^{1}\Big{[}\int_{Q_{1}\times\cdots\times
Q_{m}}$
$\displaystyle\qquad\qquad\qquad\qquad\times\big{|}\big{(}\partial_{1}^{\alpha}K_{j}\big{)}^{S_{I_{0}}}\big{(}2^{j}y_{c_{Q_{1}},z_{1}}^{\theta},2^{j}(y-z_{2}),\dots,2^{j}(y-z_{m})\big{)}\big{|}d\vec{\boldsymbol{z}}\Big{]}d\theta$

where $\big{(}\partial_{1}^{\alpha}K_{j}\big{)}^{r}$ is defined as in (5.13).
In the above estimate, if $I_{0}$ contains $1$, then we also applied the fact
that for $y\in(Q_{1}^{*})^{c}$, $z_{1}\in Q_{1}$, and $0<\theta<1$,

$|y-c_{Q_{1}}|\lesssim\big{|}y_{c_{Q_{1}},z_{1}}^{\theta}\big{|}.$

Then, similar to (5.3), we can bound (5.23) by

$\displaystyle 2^{j\nu
n}\Big{(}\prod_{\mu=1}^{m}\ell(Q_{\mu})^{-n/p_{\mu}}\Big{)}\Big{(}\prod_{\mu=1}^{\nu}\ell(Q_{\mu})^{n}\Big{)}\big{(}2^{j}\ell(Q_{1})\big{)}^{L}\sum_{|\alpha|=L}\int_{0}^{1}\Big{[}\int_{Q_{1}}|Q_{1}|^{-1}$
$\displaystyle\qquad\qquad\times\Big{(}\int_{(\mathbb{R}^{n})^{m-\nu}}\big{\|}\big{(}\partial_{1}^{\alpha}K_{j}\big{)}^{S_{I_{0}}}\big{(}2^{j}y_{c_{Q_{1}},z_{1}}^{\theta},z_{2},\dots,z_{m}\big{)}\big{\|}_{L^{\infty}(z_{2},\dots,z_{\nu})}dz_{\nu+1}\cdots
dz_{m}\Big{)}dz_{1}\Big{]}\;d\theta$

with the usual modification when $\nu=1$ or $\nu=m$. Clearly, the $L^{\infty}$
norm inside the integral can be replaced, thanks to Lemma 2.2, by

$\big{\|}K_{j}^{S_{I_{0}}}\big{(}2^{j}y_{c_{Q_{1}},z_{1}}^{\theta},z_{2},\dots,z_{m}\big{)}\big{\|}_{L^{2}(z_{2},\dots,z_{\nu})}$

and thus we finally arrive at the inequality, via the Cauchy-Schwarz
inequality,

$\displaystyle\eqref{mainestterm}$ $\displaystyle\lesssim 2^{j\nu
n}\Big{(}\prod_{\mu=1}^{m}\ell(Q_{\mu})^{-n/p_{\mu}}\Big{)}\Big{(}\prod_{\mu=1}^{\nu}\ell(Q_{\mu})^{n}\Big{)}h_{j,s}^{Q_{1},L}(y).$

with the aid of (5.22). This ends the proof of Lemma 5.1.

## Acknowledgement

We would like to thank the anonymous referees for the careful reading and
their helpful comments.

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