# Improved curvature conditions on $L^{2}\times\cdots\times L^{2}\to L^{2/m}$
bounds for multilinear maximal averages

Chu-hee Cho  C. Cho, Research Institute of Mathematics, Seoul National
University, 08826 Gwanak-ro 1, Seoul, Republic of Korea<EMAIL_ADDRESS>,
Jin Bong Lee  J. B. Lee, Research Institute of Mathematics, Seoul National
University, 08826 Gwanak-ro 1, Seoul, Republic of Korea<EMAIL_ADDRESS>and
Kalachand Shuin  K. Shuin, Department of Mathematical Sciences, Seoul National
University, 08826 Gwanak-ro 1, Seoul, Republic of Korea<EMAIL_ADDRESS>
###### Abstract.

In this article, we focus on $L^{2}(\mathbb{R}^{d})\times\cdots\times
L^{2}(\mathbb{R}^{d})\rightarrow L^{2/m}(\mathbb{R}^{d})$ estimates for
multilinear maximal averages over non-degenerate hypersurfaces. Our findings
is new for $m$-linear averages with $m\geq 3$, and represent a reproof of the
recent result of T. Borges, B. Foster, and Y. Ou on the curvature conditions
of the hypersurfaces required in establishing $L^{2}(\mathbb{R}^{d})\times
L^{2}(\mathbb{R}^{d})\rightarrow L^{1}(\mathbb{R}^{d})$ estimates of bilinear
maximal functions.

###### Key words and phrases:

Multilinear maximal averages, Curvature conditions

###### 2020 Mathematics Subject Classification:

42B25, 47H60

##  1\. Introduction

Let $\sigma$ be the normalized surface measure on $\Sigma$ supported in a unit
ball $\mathbb{B}^{md}(0,1)\subset\mathbb{R}^{md}$. For
$\mathrm{F}=(f_{1},\dots,f_{m})$ with Schwartz functions
$f_{1},\dots,f_{m}\in\mathscr{S}(\mathbb{R}^{d})$, define a multilinear
averaging operator $\mathrm{A}_{\sigma}$ by

(1.1)
$\displaystyle\mathrm{A}_{\sigma}(\mathrm{F})(x,t)\coloneqq\int_{\Sigma}\prod_{i=1}^{m}f_{i}(x-ty_{i})~{}\mathrm{d}\sigma(y),$

and a maximal average by

(1.2)
$\displaystyle\mathrm{M}_{\sigma}(\mathrm{F})(x)\coloneqq\sup_{t>0}|\mathrm{A}_{\sigma}(\mathrm{F})(x,t)|.$

One classical example of multilinear maximal function is the bilinear
spherical maximal function, which is defined by

$\mathcal{M}_{full}(\mathrm{F})(x):=\sup_{t>0}\Big{|}\int_{\mathbb{S}^{2d-1}}f_{1}(x-ty)f_{2}(x-tz)~{}d\sigma(y,z)\Big{|},$

where $\sigma$ is the normalized surface measure on $\mathbb{S}^{2d-1}$,
$d\geq 1$. This operator was initially introduced by Barrionuevo, Grafakos,
He, Honzík, and Oliveira [1], and then Heo, Hong, and Yang [12] improved upon
the previous results. After, Jeong and Lee [13] have successfully demonstrated
the complete (except few border line cases) $L^{p_{1}}\times
L^{p_{2}}\rightarrow L^{p}$ boundedness of this operator using a clever idea
of slicing argument in dimensions $d\geq 2$.

Boundedness of the bilinear spherical maximal function in dimension $d=1$ was
later investigated by Christ, Zhou [7] and Dosidis, Ramos [9] independently.
Recently, Bhojak, Choudhary, Shrivastava, and the third author of this article
[2] have established end point estimates of bilinear spherical maximal
function in dimensions $d=1,2$. Lee and the third author of this article [15]
extended the slicing technique to deal with boundedness of bilinear maximal
functions defined on degenerate hypersurfaces, like
$\Sigma_{a_{1},a_{2}}:=\\{(y,z)\in\mathbb{R}^{2d}:\Phi(y,z)=|y|^{a_{1}}+|z|^{a_{2}}-1=0\\}$,
for $a_{1},a_{2}\in[1,\infty)$. Despite of the vanishing curvature conditions
of $\Sigma_{a_{1},a_{2}}$, the authors of [15] were able to apply the slicing
argument to study boundedness of the bilinear maximal function defined on
$\Sigma_{a_{1},a_{2}}$ and proved sharp $L^{p_{1}}\times L^{p_{2}}\rightarrow
L^{p}$ estimates except for border line cases. In the slicing argument of
[15], the non-vanishing gradient $(|\nabla\Phi|\neq 0)$ of the hypersurface
representing function $\Phi$ plays a crucial role.

Therefore, a natural question arises, when we only have information about the
curvature conditions of hypersurfaces but not the equation, what can we infer
about the boundedness of bilinear and multilinear maximal functions associated
with the hypersurfaces? This problem was first addressed by Grafakos, He, and
Honzík [10] and later Chen, Grafakos, He, Honzík, Slavíková [6] considered as
bilinear analogues of [17]. For $m=2$, Chen $et.$ $al.$ [6] proved the
following result.

######  Theorem 1.1 ([6], Theorem 2).

Let $\sigma$ be the surface measure of a compact and smooth surface $\Sigma$
without boundary such that $k$ of its $2d-1$ principal curvatures are non-
zero. Then $\mathrm{M}_{\sigma}$ maps $L^{2}(\mathbb{R}^{d})\times
L^{2}(\mathbb{R}^{d})\rightarrow L^{1}(\mathbb{R}^{d})$ when $k>d+2$.

Very recently, Borges, Foster, and Ou [4] have improved the curvature
condition to $k\geq d+2$. Their results are based on studies of Sobolev
smoothing estimates for various bilinear maximal operators given by Fourier
multipliers, which contain multi-scale maximal functions, and maximal
functions with fractal dilation sets.

To state our main results, we begin with mutlilinear local maximal functions.

(1.3)
$\displaystyle\mathrm{M}_{\sigma}^{loc}(\mathrm{F})(x)\coloneqq\sup_{1<t<2}\Big{|}\int\prod_{i=1}^{m}f_{i}(x-ty_{i})~{}\mathrm{d}\sigma(y)\Big{|}.$

We first establish multilinear local maximal estimates.

######  Theorem 1.2.

Let $s>\frac{(m-1)d}{2}+\frac{1}{2}$, $m\geq 2$, and a measure $\sigma$ be
supported in $\mathbb{B}^{md}(0,1)$ and satisfy

$|\widehat{\mathrm{d}\sigma}(\xi)|\lesssim(1+|\xi|)^{-s}.$

Then we have for $2/m\leq p\leq 2$,

$\|\mathrm{M}_{\sigma}^{loc}(\mathrm{F})\|_{L^{p}(\mathbb{B}^{d}(0,1))}\leq
C\prod_{i=1}^{m}\|f_{i}\|_{L^{2}(\mathbb{R}^{d})}.$

For $p=2/m$, we have

$\|\mathrm{M}_{\sigma}^{loc}(\mathrm{F})\|_{L^{2/m}(\mathbb{R}^{d})}\leq
C\prod_{i=1}^{m}\|f_{i}\|_{L^{2}(\mathbb{R}^{d})}.$

By making use of Theorem 1.2, we obtain estimates for global maximal functions
(1.2).

######  Theorem 1.3.

Let $m$, $d$, $\sigma$ be given as in Theorem 1.2. Then we have

$\|\mathrm{M}_{\sigma}(\mathrm{F})\|_{L^{2/m}(\mathbb{R}^{d})}\leq
C\prod_{i=1}^{m}\|f_{i}\|_{L^{2}(\mathbb{R}^{d})}.$

######  Remark 1.

In order to discuss optimal range of the above estimates, we consider two
operators. First, let $\mathrm{M}_{\mathbb{S}^{2}}$ denote the spherical
maximal function on $\mathbb{R}^{3}$ whose Fourier symbol has decay $1$. Then,
$m$-product of $\mathrm{M}_{\mathbb{S}^{2}}$ obeys that

$\Big{\|}\prod_{i=1}^{m}\mathrm{M}_{\mathbb{S}^{2}}(f_{i})\Big{\|}_{L^{p}(\mathbb{R}^{3})}\leq
C\prod_{i=1}^{m}\|f_{i}\|_{L^{p_{i}}(\mathbb{R}^{3})},$

whenever $p>3/(2m)$, $p_{i}>3/2$ with $1/p=\sum_{i=1}^{m}1/p_{i}$. This
certainly implies $L^{2}(\mathbb{R}^{3})\times\cdots\times
L^{2}(\mathbb{R}^{3})\to L^{2/m}(\mathbb{R}^{3})$ estimates. On the other
hand, for the circular maximal function $\mathrm{M}_{\mathbb{S}^{1}}$, its
Fourier decay of the circular measure is $1/2$ and its $m$-product does not
satisfy $L^{2}(\mathbb{R}^{2})\times\cdots\times L^{2}(\mathbb{R}^{2})\to
L^{2/m}(\mathbb{R}^{2})$ estimates since $\mathrm{M}_{\mathbb{S}^{1}}$ is
bounded on $L^{p}(\mathbb{R}^{2})$ if and only if $p>2$. In particular, in
bilinear case of Theorems 1.2 and 1.3, the condition
$s>\frac{(m-1)d}{2}+\frac{1}{2}$ becomes $s>\frac{d+1}{2}$. However, it should
be noted that $s>\frac{d+1}{2}$ is not sharp for Theorems 1.2 and 1.3, and in
terms of $k$ nonvanishing principal curvatures, $s>\frac{d+1}{2}$ is
equivalent to $k>d+1$ same as [4]. In fact, from the results for two maximal
operators, one may conjecture for the case of $m=2$ that optimal range of $s$
for Theorems 1.2 and 1.3 is $s>1/2$.

By multilinear real interpolation, we obtain
$L^{p}(\mathbb{R}^{d})\times\cdots\times L^{p}(\mathbb{R}^{d})\to
L^{p/m}(\mathbb{R}^{d})$ estimates of $\mathrm{M}^{loc}_{\sigma}$ for some
$p<2$.

######  Corollary 1.4.

Let $m\geq 2$ and $\Sigma$ be a compact and smooth hypersurface with non-
vanishing principal curvatures $k>(m-1)d+1$. Then, the multilinear local
maximal function $\mathrm{M}^{loc}_{\sigma}$ maps
$L^{p}(\mathbb{R}^{d})\times\cdots\times L^{p}(\mathbb{R}^{d})$ to
$L^{p/m}(\mathbb{R}^{d})$ for $p>\frac{k-(m-1)d+1}{k-(m-1)d}$.

By Corollary 1.4, it is shown for $m=2$ that $M_{\sigma}^{loc}$ satisfies
$L^{p}(\mathbb{R}^{d})\times L^{p}(\mathbb{R}^{d})\to L^{p/2}(\mathbb{R}^{d})$
estimates for $p>\frac{k-d+1}{k-d}$. However, the range of $p$ is not sharp as
we mentioned in Remark 1. It could be conjectured that $p>(k+1)/k$ is sharp,
which is obtained from validity of Theorems 1.2 and 1.3 for $s>1/2$.

## Notations

Let $\mathbb{B}^{d}(x,R)$ denote a $d$-dimensional ball of radius $R$ centered
at $x$. Let $\mathbb{A}^{d}(\lambda)$ be a $d$-dimensional annulus given by
$\\{x\in\mathbb{R}^{d}:2^{-1}\lambda<|x|<2\lambda\\}$.

##  2\. Preliminaries

###  2.1. $\alpha$-dimensional measures and weighted estimates

Contents of this subsection are mostly given by Ko, Lee, and Oh [14]. Let
$\mu$ be a positive Borel measure on $\mathbb{R}^{d}$. Then for
$\alpha\in(0,d]$, $\mu$ is said to be $\alpha$-dimensional if there exists a
$C>0$ such that

(2.1) $\displaystyle\mu(\mathbb{B}^{d}(x_{0},R))\leq CR^{\alpha},$

where $\mathbb{B}^{d}(x_{0},R)$ denotes a $d$-dimensional ball centered at
$x_{0}$ with radius $R$. For an $\alpha$-dimensional measure $\mu$, one can
define the following quantity:

(2.2)
$\displaystyle\langle\mu\rangle_{\alpha}=\sup_{x_{0}\in\mathbb{R}^{d},R>0}R^{-\alpha}\mu(\mathbb{B}^{d}(x_{0},R)).$

It is known in the literature that one can rewrite $\mathrm{M}_{\sigma}f$ by
making use of a measurable function $\mathbb{t}(\cdot)$:

(2.3) $\displaystyle\mathrm{M}_{\sigma}f(x)=\int
f(x-\mathbb{t}(x)y)~{}\mathrm{d}\sigma(y).$

We define a $d$-dimensional measure $\mu$ by

(2.4)
$\displaystyle\int_{\mathbb{R}^{d+1}}F(x,t)~{}\mathrm{d}\mu(x,t)=\int_{\mathbb{B}^{d}(0,1)}F(x,\mathbb{t}(x))~{}\mathrm{d}x$

for any function $F\in C_{c}(\mathbb{R}^{d+1})$. Then,
$L^{p}(\mathbb{B}^{d}(0,1),\mathrm{d}x)$ norm of $\mathrm{M}_{\sigma}$ is
equivalent to

(2.5)
$\displaystyle\Big{(}\int|\mathrm{A}_{\sigma}f(x,t)|^{p}~{}\mathrm{d}\mu(x,t)\Big{)}^{1/p}$

That is, $\mathrm{d}\mu$ is a $d$-dimensional measure on
$\mathbb{R}^{d}\times(0,\infty)$.

We restrict our consideration of $\alpha$-dimensional measures on $\mu$ given
by $\omega~{}\mathrm{d}x\mathrm{d}t$. Let $\Omega^{\alpha}$ be a collection of
non-negative measurable functions $\omega$ on $\mathbb{R}^{d+1}$ such that
$\omega~{}\mathrm{d}x\mathrm{d}t$ is $\alpha$-dimensional. We use a notation
$[\omega]_{\alpha}=\langle\omega~{}\mathrm{d}x\mathrm{d}t\rangle_{\alpha}$. It
is known in [14] that (2.5) is reduced to weighted norms associated with
$\omega\in\Omega^{d}$. The following lemma was already given in [14, Lemma
2.19] for dimension $d=4$. One can easily extend it to general dimension $d$
by modifying the proof of the case of $d=4$, so we omit it.

######  Lemma 2.1.

Let $p\in(0,\infty]$, $\alpha\in(0,d+1]$ and $\omega\in\Omega^{\alpha}$. For a
function $F\in L^{p}(\mathbb{R}^{d+1})$ whose Fourier support is a subset of
$\mathbb{B}^{d+1}(0,\lambda)$, we have

(2.6) $\displaystyle\|F\|_{L^{p}(\mathbb{R}^{d+1},\omega)}\leq
C[\omega]_{\alpha}^{\frac{1}{p}}\lambda^{\frac{d+1-\alpha}{p}}\|F\|_{L^{p}(\mathbb{R}^{d+1},\mathrm{d}x\mathrm{d}t)}.$

###  2.2. $L^{p}$ improving estimates for multilinear operators with the
limited decay condition

We recall simple estimates for multilinear operators whose symbols satisfy the
following decay conditions:

$|\mathfrak{m}(\xi)|\lesssim(1+|\xi|)^{-s}$

for some given $s>0$. We define the Littlewood-Paley decomposition for further
uses. Let $\Psi$ be a Schwartz function on $\mathbb{R}^{md}$ such that
$\widehat{\Psi}\equiv 1$ for $1<|\xi|<2$ and vanishes outside of
$\\{\xi\in\mathbb{R}^{md}:1/2<|\xi|<4\\}$. Define
$\Psi_{j}(\cdot)=2^{jmd}\Psi(2^{j}\cdot)$ and $\Phi(\cdot)=1-\sum_{j\geq
1}\Psi_{j}$. If we use the decomposition on $\mathbb{R}^{d}$, we say $P_{j}$
to denote the $j$-th Littlewood-Paley projection.

Then, we have the following lemma:

######  Lemma 2.2.

Let $m\geq 2$ and $T_{\mathfrak{m}}(\mathrm{F})$ be a multilinear operator
given by

$T_{\mathfrak{m}}(\mathrm{F})(x):=\int_{\mathbb{R}^{md}}\mathrm{e}^{2\pi
ix\cdot(\xi_{1}+\cdots+\xi_{m})}\mathfrak{m}(\xi)\prod_{i=1}^{m}\widehat{f}_{i}(\xi_{i})~{}\mathrm{d}\xi.$

Suppose that there is a constant $C>0$ such that $|\mathfrak{m}(\xi)|\leq
C(1+|\xi|)^{-s}$ for some $s>0$, and
$\mathrm{supp}(\mathfrak{m})\subset\mathbb{A}^{md}(2^{j})$. Then we have

(2.7)
$\displaystyle\|T_{\mathfrak{m}}(\mathrm{F})\|_{L^{2}(\mathbb{R}^{d})}\leq$
$\displaystyle
C2^{-j(s-\frac{(m-1)d}{2})}\prod_{i=1}^{m}\|f_{i}\|_{L^{2}(\mathbb{R}^{d})}.$

###### Proof.

For $L^{2}$ norm we begin with

(2.8) $\displaystyle T_{\mathfrak{m}}(\mathrm{F})(x)=\Psi_{j}\ast
T_{\mathfrak{m}}(\mathrm{F})(x,\dots,x),$

where $\widehat{\Psi}_{j}$ denotes the $j$-th Littlewood-Paley decomposition
in $\mathbb{R}^{md}$. Taking $L^{2}$ norm to (2.8), it yields

(2.9)
$\displaystyle\begin{split}\|T_{\mathfrak{m}}(\mathrm{F})\|_{L^{2}(\mathbb{R}^{d})}^{2}&=\int_{\mathbb{R}^{d}}\Big{|}\int_{\mathbb{R}^{md}}T_{\mathfrak{m}}(\mathrm{F})(y)\Psi_{j}(x-y_{1},\dots,x-y_{m})~{}\mathrm{d}y\Big{|}^{2}~{}\mathrm{d}x\\\
&\leq\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{md}}\Big{|}T_{\mathfrak{m}}(\mathrm{F})(y)\Big{|}^{2}|\Psi_{j}(x-y_{1},\dots,x-y_{m})|~{}\mathrm{d}y~{}\mathrm{d}x\\\
&\leq
2^{j(m-1)d}\|T_{\mathfrak{m}}(\mathrm{F})\|_{L^{2}(\mathbb{R}^{md})}^{2}.\end{split}$

Here we apply Hölder’s inequality in the first inequality and compute
$x$-integral first in the second inequality. By the Plancherel theorem and the
conditions of $\mathfrak{m}$, we obtain

$\|T_{\mathfrak{m}}(\mathrm{F})\|_{L^{2}(\mathbb{R}^{d})}\leq
2^{-j(s-\frac{(m-1)d}{2})}\prod_{i=1}^{m}\|f_{i}\|_{L^{2}(\mathbb{R}^{d})}.$

By (2.9) one can check that for general $p\geq 1$

$\|T_{\mathfrak{m}}(\mathrm{F})\|_{L^{p}(\mathbb{R}^{d})}\leq
2^{j\frac{(m-1)d}{p}}\|T_{\mathfrak{m}}(\mathrm{F})\|_{L^{p}(\mathbb{R}^{md})}.$

∎

##  3\. Proof of Theorem 1.2

We make use of the Littlewood-Paley decomposition to obtain

(3.1) $\displaystyle\mathrm{M}_{\sigma}^{loc}(\mathrm{F})(x)\leq\sum_{j\geq
0}\mathrm{M}_{\sigma_{j}}^{loc}(\mathrm{F})(x).$

Note that $\sigma_{j}=\sigma\ast\Psi_{j}$ for $j>0$ and
$\sigma_{0}=\sigma\ast\Phi$. It is worth noting that
$\mathrm{M}_{\sigma_{0}}^{loc}(\mathrm{F})$ is bounded pointwisely by product
of the Hardy-Littlewood maximal functions $\mathrm{M}_{HL}(f_{i})$. Thus we
only have to consider $j\geq 1$ cases.

Note that
$\mathrm{M}_{\sigma_{j}}^{loc}(\mathrm{F})(x)=\sup_{1<t<2}|\mathrm{A}_{\sigma_{j}}(\mathrm{F})(x,t)|$,
where $\mathrm{A}_{\sigma_{j}}(\mathrm{F})(x,t)$ is given by

$\mathrm{A}_{\sigma_{j}}(\mathrm{F})(x,t)=\int_{\mathbb{R}^{md}}\mathrm{e}^{2\pi
ix\cdot(\xi_{1}+\cdots+\xi_{m})}\widehat{\sigma}_{j}(t\xi)\prod_{i=1}^{m}\widehat{f}_{j}(\xi_{i})~{}\mathrm{d}\xi.$

Since $1<t<2$, one can say that $2^{j-2}\leq|\xi|\leq 2^{j+2}$ in the support
of $\widehat{\sigma}_{j}(t\cdot)$, hence $|\xi_{i}|\lesssim 2^{j}$ for each
$i=1,\dots,m$. Moreover, there is at least one $i=1,\dots,m$ such that
$2^{j-2}\leq|\xi_{i}|\leq 2^{j+2}$. We first obtain the following lemma:

######  Lemma 3.1.

Let $s(m,d)=s-\frac{(m-1)d}{2}-\frac{1}{2}$. Then we have

(3.2)
$\displaystyle\Big{\|}\sup_{1<t<2}\Big{|}\mathrm{A}_{\sigma_{j}}(\mathrm{F})\Big{|}\Big{\|}_{L^{2/m}(\mathbb{R}^{d})}\leq
C2^{-js(m,d)}\prod_{i=1}^{m}\|f_{i}\|_{L^{2}(\mathbb{R}^{d})}.$

As a consequence of the lemma 3.1, we show theorem 1.2. In fact, by the lemma
3.1, it follows that

(3.3)
$\displaystyle\begin{split}\|\mathrm{M}_{\sigma}^{loc}(\mathrm{F})\|_{L^{2/m}(\mathbb{R}^{d})}^{2/m}\leq&\sum_{j\geq
0}\Big{\|}\sup_{1<t<2}\Big{|}\mathrm{A}_{\sigma_{j}}(\mathrm{F})(\cdot,t)\Big{|}\Big{\|}_{L^{2/m}(\mathbb{R}^{d})}^{2/m}\\\
\leq&C\sum_{j\geq
0}2^{-js(m,d)2/m}\prod_{i=1}^{m}\|f_{i}\|_{L^{2}(\mathbb{R}^{d})}^{2/m}.\end{split}$

Since $s(m,d)>0$, this proves the theorem. Therefore it remains to prove Lemma
3.1.

###  3.1. Proof of Lemma 3.1

In order to show lemma 3.1, we use the following lemma in [14, Lemma 2.3].

######  Lemma 3.2.

Let $R=1+4\text{diam}(\mathrm{supp}(\sigma))$ and

$K_{j}(\mathbf{x},t)\coloneqq\int_{\mathbb{R}^{md}}\mathrm{e}^{2\pi
i\mathbf{x}\cdot\xi}\int\mathrm{e}^{-2\pi
it(y\cdot\xi)}~{}\mathrm{d}\sigma(y)\widehat{\Psi}(2^{-j}\xi)~{}\mathrm{d}\xi$

for $j\geq 0$ and $(\mathbf{x},t)\in\mathbb{R}^{md}\times\mathbb{R}$. If
$|\mathbf{x}|\geq R$ and $|t|\leq 2$, then $|K_{j}(\mathbf{x},t)|\lesssim
2^{-jN}(1+|\mathbf{x}|)^{-N}$ for any $N>0$.

###### Proof.

By Fubini’s theorem and change of variables, we have

(3.4)
$\displaystyle\begin{split}K_{j}(\mathbf{x},t)=2^{jmd}\int\int_{\mathbb{R}^{md}}\mathrm{e}^{2\pi
i2^{j}(\mathbf{x}\cdot\xi-
ty\cdot\xi)}\widehat{\Psi}(\xi)~{}\mathrm{d}\xi\mathrm{d}\sigma(y).\end{split}$

Note that $|\nabla_{\xi}(\mathbf{x}\cdot\xi-ty\cdot\xi)|\geq|\mathbf{x}|/2$
for $|\mathbf{x}|\geq R$ and $|t|\leq 2$. Then by iterated use of integration
by parts in $\xi$, we have

$|K_{j}(\mathbf{x},t)|\chi_{|\mathbf{x}|\geq R}(\mathbf{x})\chi_{|t|\leq
2}(t)\lesssim 2^{jmd}(1+2^{j}|\mathbf{x}|)^{-L}\chi_{|\mathbf{x}|\geq
R}(\mathbf{x})$

for any $L>0$. Taking $L\geq md+2N$ for a fixed number $N>0$, it follows that

$|K_{j}(\mathbf{x},t)|\chi_{|\mathbf{x}|\geq R}(\mathbf{x})\chi_{|t|\leq
2}(t)\leq 2^{-jN}(1+|\mathbf{x}|)^{-N}.$

∎

We split the integral range $\mathbb{R}^{d}$ into the Ball of radius $R$
centered at the origin and the complement, and apply Lemma 3.2 to get

(3.5)
$\displaystyle\begin{split}\Big{\|}&\sup_{1<t<2}\Big{|}\mathrm{A}_{\sigma_{j}}(\mathrm{F})\Big{|}\Big{\|}_{L^{2/m}(\mathbb{R}^{d})}\\\
\lesssim&\Big{\|}\sup_{1<t<2}\Big{|}\mathrm{A}_{\sigma_{j}}(\mathrm{F})\Big{|}\Big{\|}_{L^{2/m}(\mathbb{B}^{d}(0,R))}+\Big{\|}\sup_{1<t<2}\Big{|}\mathrm{A}_{\sigma_{j}}(\mathrm{F})\Big{|}\Big{\|}_{L^{2/m}(\mathbb{B}^{d}(0,R)^{\mathsf{c}})}\\\
\lesssim&\Big{\|}\sup_{1<t<2}\Big{|}\mathrm{A}_{\sigma_{j}}(\mathrm{F})\Big{|}\Big{\|}_{L^{2/m}(\mathbb{B}^{d}(0,R))}+2^{-jN}\Big{\|}\prod_{i=1}^{m}M_{HL}f_{i}\Big{\|}_{L^{2/m}(\mathbb{B}^{d}(0,R)^{\mathsf{c}})},\end{split}$

where $A^{\mathsf{c}}$ denotes the complement of a set $A$. Therefore to prove
Lemma 3.1, it suffices to show (3.2) with left-hand side replaced in to
$L^{2/m}(\mathbb{B}^{d}(0,R))$-norm since the other part can be controlled by
the right side norm from the Hardy-Littlewood maximal estimates.

We define $\widetilde{\mathrm{A}}_{\sigma_{j}}$ as

$\displaystyle\widetilde{\mathrm{A}}_{\sigma_{j}}(\mathrm{F})(x,t)$
$\displaystyle=\int_{\mathbb{R}}\int_{\mathbb{R}^{md}}\mathrm{e}^{2\pi
i(x\cdot(\xi_{1}+\cdots+\xi_{m})+t\tau)}m(\xi,\tau)\widehat{\Psi}(2^{-j}\xi)\widehat{\varphi}(2^{-j}R^{-1}\tau)\prod_{i=1}^{m}\widehat{f}_{i}(\xi_{i})~{}\mathrm{d}\xi\mathrm{d}\tau,$

where $m(\xi,\tau)=\int_{\mathbb{R}}\int e^{-2\pi
is(\tau+y\cdot\xi)}\rho(s)~{}\mathrm{d}\sigma(y)\mathrm{d}s$ with a smooth
function $\rho\in C_{c}^{\infty}(\mathbb{R})$ such that $\rho\equiv 1$ on
$[1,2]$ and $R=1+4\text{diam}(\mathrm{supp}(\sigma))$. Let $\mathcal{F}$
denote the space-time Fourier transform. For
$\mathrm{R}_{\sigma_{j}}=\mathrm{A}_{\sigma_{j}}-\widetilde{\mathrm{A}}_{\sigma_{j}}$,
$\mathcal{F}[\mathrm{R}_{\sigma_{j}}(\mathrm{F})]$ is supported outside of
$\mathbb{B}^{d+1}(0,m2^{j+1})$, and the kernel of $\mathrm{R}_{\sigma_{j}}$
has nice decay property. Then, we have the following lemma:

######  Lemma 3.3.

Let $m(\xi,\tau)\widehat{\Psi}_{j}(\xi)(1-\widehat{\varphi}_{j})(R^{-1}\tau)$
be a symbol of $\mathrm{R}_{\sigma_{j}}$ with respect to space-time Fourier
transform. Then for any $N\geq d$ we have

(3.6)
$\displaystyle\mathcal{F}^{-1}[m(\xi,\tau)\widehat{\Psi}_{j}(\xi)(1-\widehat{\varphi}_{j})(R^{-1}\tau)](\mathbf{x},t)\leq
C_{N}2^{-jN}(1+|\mathbf{x}|)^{-N}(1+|t|)^{-N},$

where $C_{N}$ is independent of $j$ and $\mathbf{x},\xi\in\mathbb{R}^{md}$.

###### Proof.

Note that

$m(\xi,\tau)=\int\widehat{\rho}(\tau+y\cdot\xi)~{}\mathrm{d}\sigma(y).$

For multi-indeces $\alpha$ with $|\alpha|=N$, it follows that

$\displaystyle|\partial_{\xi}^{\alpha}\partial_{\tau}^{N}\big{(}m(\xi,\tau)\widehat{\Psi}_{j}(\xi)(1-\widehat{\varphi}_{j})(R^{-1}\tau)\big{)}|$

is bounded by finite sum of the following term:

(3.7)
$\displaystyle\int|\widehat{\rho}^{(|\alpha_{1}|+N)}(\tau+y\cdot\xi)|~{}\mathrm{d}\sigma(y)\times
2^{-j|\alpha_{2}|}\widehat{\Psi}_{j}(\xi)(1-\widehat{\varphi}_{j})(R^{-1}\tau),$

where $\alpha=\alpha_{1}+\alpha_{2}$. Since $|\xi|\sim 2^{j}$,
$|\tau|>2^{j}R$, and $\rho\in C_{c}^{\infty}(\mathbb{R})$, for any $L\geq 0$
there exists a constant $C_{L}$ such that

$\displaystyle\eqref{eq_2308_1}$ $\displaystyle\leq
C_{L}2^{-j|\alpha_{2}|}\frac{1}{(1+|\tau|)^{L}}\widehat{\Psi}_{j}(\xi)(1-\widehat{\varphi}_{j})(R^{-1}\tau)$
(3.8) $\displaystyle\leq
C_{L}\frac{1}{(1+|\tau|)^{L/2}}(R2^{j})^{-L/2}\widehat{\Psi}_{j}(\xi).$

By (3.8) and the integration by parts, it follows that

$\displaystyle\mathcal{F}^{-1}[m(\xi,\tau)\widehat{\Psi}_{j}(\xi)$
$\displaystyle(1-\widehat{\varphi}_{j})(R^{-1}\tau)](\mathbf{x},t)$
$\displaystyle\lesssim_{R}\,$ $\displaystyle
C_{L}(1+|\mathbf{x}|)^{-N}(1+|t|)^{-N}2^{-j(L/2-md)}.$

If we choose $L\geq 2(N+md)$, then we complete the proof of the lemma. ∎

Now we return to the proof of Lemma 3.1. For $\mathrm{R}_{\sigma_{j}}$, apply
Lemma 3.3 to have

$\sup_{1<t<2}|\mathrm{R}_{\sigma_{j}}(\mathrm{F})(x,t)|\leq
2^{-jN}\prod_{i=1}^{m}\mathrm{M}_{HL}(f_{i})(x).$

$N$ could be sufficiently large, so we obtain the desired result.

To deal with $\widetilde{\mathrm{A}}_{\sigma_{j}}$, we observe that the
Fourier transform of $\mathrm{A}_{\sigma_{j}}$. The Fourier transform in $x$
is

(3.9)
$\displaystyle\begin{split}\widehat{\mathrm{A}_{\sigma_{j}}(\mathrm{F})}(\cdot,t)(\xi_{1})=&\int_{\mathbb{R}^{(m-1)d}}\widehat{\mathrm{d}\sigma}_{j}(t\xi_{1}-t\xi_{2},\dots,t\xi_{m-1}-t\xi_{m},t\xi_{m})\\\
&\quad\quad\times\widehat{f}_{1}(\xi_{1}-\xi_{2})\cdots\widehat{f}_{m-1}(\xi_{m-1}-\xi_{m})\widehat{f}_{m}(\xi_{m})~{}\mathrm{d}\xi_{2}\cdots\mathrm{d}\xi_{m}.\end{split}$

Since $\mathrm{supp}(\widehat{f_{i}})\subset\mathbb{B}^{d}(0,2^{j})$ and
$1<t<2$, it implies that $\xi_{1}\in\mathbb{B}^{d}(0,m2^{j+1})$, and
$\mathcal{F}[\widetilde{\mathrm{A}}_{\sigma_{j}}(\mathrm{F})](\xi_{1},\tau)$
is supported in $\mathbb{B}^{d+1}(0,m2^{j+1})$. Since
$|\widehat{\mathrm{d}\sigma}(\xi)|\lesssim(1+|\xi|)^{-s}$ and
$\widehat{\mathrm{d}\sigma_{j}}$ is supported in $\mathbb{A}^{md}(2^{j})$, we
use Young’s inequality in $t$-variable and Lemma 2.2 in $x$-variable to obtain

(3.10)
$\displaystyle\begin{split}\|\widetilde{\mathrm{A}}_{\sigma_{j}}(\mathrm{F})(\cdot,\cdot)\|_{L^{2}(\mathbb{R}^{d}\times[1,2])}&\leq\|{\mathrm{A}}_{\sigma_{j}}(\mathrm{F})(\cdot,\cdot)\|_{L^{2}(\mathbb{R}^{d}\times[1,2])}\\\
&\leq
C\,2^{-j(s-\frac{(m-1)d}{2})}\prod_{i=1}^{m}\|f_{i}\|_{L^{2}(\mathbb{R}^{d})}.\end{split}$

To estimate our maximal estimate, we use Hölder’s inequality and Lemma 2.1,
which is due to the equivalence (2.5). Then,

(3.11)
$\displaystyle\begin{split}\Big{\|}\sup_{1<t<2}\Big{|}\widetilde{\mathrm{A}}_{\sigma_{j}}(\mathrm{F})(\cdot,t)\Big{|}\Big{\|}_{L^{2/m}(\mathbb{B}^{d}(0,R))}&\leq
R^{\frac{(m-1)d}{2}}\Big{\|}\sup_{1<t<2}\Big{|}\widetilde{\mathrm{A}}_{\sigma_{j}}(\mathrm{F})(\cdot,t)\Big{|}\Big{\|}_{L^{2}(\mathbb{B}^{d}(0,R))}\\\
&\lesssim
2^{j/2}\|\widetilde{\mathrm{A}}_{\sigma_{j}}(\mathrm{F})(\cdot,\cdot)\|_{L^{2}(\mathbb{R}^{d}\times[1,2])}.\end{split}$

Combining two estimates (3.10) and (3.11), we obtain

$\Big{\|}\sup_{1<t<2}\Big{|}\widetilde{\mathrm{A}}_{\sigma_{j}}(\mathrm{F})(\cdot,t)\Big{|}\Big{\|}_{L^{2/m}(\mathbb{B}^{d}(0,R))}\lesssim
2^{-j(s-\frac{(m-1)d}{2}-\frac{1}{2})}\prod_{i=1}^{m}\|f_{i}\|_{L^{2}(\mathbb{R}^{d})}.$

Hence, the proof is completed.

We end this section with the proof of Corollary 1.4.

###### Proof of Corollary 1.4.

From the Lemma 3.1 we have

$\|\mathrm{M}^{loc}_{\sigma_{j}}\|_{L^{2}\times\cdots\times L^{2}\rightarrow
L^{2/m}}\lesssim 2^{-j(k/2-\frac{(m-1)d}{2}-\frac{1}{2})}.$

On the other hand, observe that

$\mathrm{M}^{loc}_{\sigma_{j}}(f_{1},\dots,f_{m})(x)\lesssim
2^{j}\prod_{i=1}^{m}M_{HL}f_{i}(x).$

Thus we also have
$\|\mathrm{M}^{loc}_{\sigma_{j}}\|_{L^{p_{1}}\times\cdots\times
L^{p_{m}}\rightarrow L^{p}}\lesssim 2^{j}$ for $1/m<p\leq\infty$,
$1<p_{1},\dots,p_{m}\leq\infty$ with $1/p=1/p_{1}+\cdots+1/p_{m}$. Therefore,
applying the real interpolation theory of multi(sub)linear operators and
summing over $j$ we get $L^{p}(\mathbb{R}^{d})\times\cdots\times
L^{p}(\mathbb{R}^{d})\rightarrow L^{p/m}(\mathbb{R}^{d})$ estimate of
$\mathrm{M}^{loc}_{\sigma}$ for $p>\frac{k-(m-1)d+1}{k-(m-1)d}$. ∎

##  4\. Proof of Theorem 1.3

We first recall that the global maximal function $\mathrm{M}_{\sigma}$ is
defined by

(4.1)
$\displaystyle\mathrm{M}_{\sigma}(\mathrm{F})(x)=\sup_{k\in\mathbb{Z}}\sup_{1<t<2}\Big{|}\int_{\Sigma}\,\prod_{i=1}f_{i}(x-2^{-k}ty_{i})~{}\mathrm{d}\sigma(y)\Big{|}.$

For any fixed $k\in\mathbb{Z}$, we define the Littlewood-Paley decomposition
on $\mathbb{R}^{d}$ as

(4.2) $\displaystyle P_{<k}=I-\sum^{\infty}_{n=0}P_{k+n},$

where $P_{k}$ denotes the $k$-th Littlewood-Paley projection and the identity
operator $I$. Then we may write

(4.3) $\displaystyle\prod_{i=1}^{m}f_{i}$
$\displaystyle=\prod_{i=1}^{m}\Big{(}P_{<k}f_{i}+P_{k\leq}f_{i}\Big{)}$
$\displaystyle=\Bigl{(}\prod_{\mu=1}^{m}P_{<k}f_{\mu}\Bigr{)}+\Bigl{(}\prod_{\nu=1}^{m}P_{k\leq}f_{\nu}\Bigr{)}$
$\displaystyle\quad+\sum_{\alpha=1}^{m-1}\frac{1}{\alpha!(m-\alpha)!}\sum_{\tau\in
S_{m}}\Bigl{(}\prod_{\mu=1}^{\alpha}P_{<k}f_{\tau(\mu)}\Bigr{)}\Bigl{(}\prod_{\nu=\alpha+1}^{m}P_{k\leq}f_{\tau(\nu)}\Bigr{)}.$

For
${\bf{n}}=(n_{1},\cdots,n_{m})\in\mathbb{N}_{0}^{m}=(\mathbb{N}\cup\\{0\\})^{m}$,
we define for $\alpha=1,\dots,m-1$ and $\tau\in S_{m}$,

$\displaystyle\mathfrak{A}_{k}^{\alpha,\tau}(\mathrm{F})(x)$
$\displaystyle:=\sup_{1<t<2}\Big{|}\int_{\Sigma}\Bigl{(}\prod_{\mu=1}^{\alpha}P_{<k}f_{\tau(\mu)}(x-2^{-k}ty_{\tau(\mu)})\Bigr{)}\Bigl{(}\prod_{\nu=\alpha+1}^{m}f_{\tau(\nu)}(x-2^{-k}ty_{\tau(\nu)})\Bigr{)}~{}~{}\mathrm{d}\sigma(y)\Big{|}$

and

$\displaystyle\mathfrak{A}_{k}^{m,\tau}(\mathrm{F})(x)=\mathfrak{A}_{k}^{m}(\mathrm{F})(x)$
$\displaystyle:=\sup_{1<t<2}\Big{|}\int_{\Sigma}\Bigl{(}\prod_{\mu=1}^{m}P_{<k}f_{\mu}(x-2^{-k}ty_{\mu})\Bigr{)}~{}\mathrm{d}\sigma(y)\Big{|}.$

Let us define

(4.4) $\displaystyle\mathfrak{M}_{\bf{n}}(\mathrm{F})$
$\displaystyle:=\sup_{k\in\mathbb{Z}}\sup_{1<t<2}\Big{|}\int_{\Sigma}\prod_{i=1}^{m}P_{k+n_{i}}f_{i}(x-2^{-k}ty_{i})~{}~{}\mathrm{d}\sigma(y)\Big{|},$
(4.5) $\displaystyle\mathfrak{S}_{\bf{n}}^{q}(\mathrm{F})$
$\displaystyle:=\Big{\|}\sup_{1<t<2}\Big{|}\int_{\Sigma}\prod_{i=1}^{m}P_{k+n_{i}}f_{i}(x-2^{-k}ty_{i})~{}~{}\mathrm{d}\sigma(y)\Big{|}\Big{\|}_{\ell^{q}({k\in\mathbb{Z}})}.$

Then, the global maximal function $\mathrm{M}_{\sigma}$ is bounded by a
constant multiple of

(4.6) $\displaystyle\sum_{\alpha=1}^{m}\sum_{\tau\in
S_{m}}\sup_{k\in\mathbb{Z}}|\mathfrak{A}_{k}^{\alpha,\tau}(\mathrm{F})|+\sum_{\mathbf{n}\in\mathbb{N}_{0}^{m}}\mathfrak{M}_{\mathbf{n}}(\mathrm{F}).$

Since the choice of $\tau$ for fixed $\alpha$ is harmless in estimating
$\mathfrak{A}_{k}^{\alpha,\tau}(\mathrm{F})$, we may consider
$\mathfrak{A}_{k}^{\alpha}(\mathrm{F})$ instead of
$\mathfrak{A}_{k}^{\alpha,\tau}(\mathrm{F})$, which is given by

(4.7) $\displaystyle\mathfrak{A}_{k}^{\alpha}(\mathrm{F})(x)$
$\displaystyle:=\sup_{1<t<2}\Big{|}\int_{\Sigma}\Bigl{(}\prod_{\mu=1}^{\alpha}P_{<k}f_{\mu}(x-2^{-k}ty_{\mu})\Bigr{)}\Bigl{(}\prod_{\nu=\alpha+1}^{m}f_{\nu}(x-2^{-k}ty_{\nu})\Bigr{)}~{}~{}\mathrm{d}\sigma(y)\Big{|}.$

In order to prove Theorem 1.3, we use an induction argument. We start with the
following lemma, which will play a crucial role to show the theorem when
$m=2$.

######  Lemma 4.1.

For $m=2$ and $\alpha=1$ we have

$\mathfrak{A}_{k}^{\alpha}(\mathrm{F})(x)\leq M_{HL}(f_{1})(x)\times
M_{\sigma}^{i}(f_{2})(x),$

where $M_{\sigma}^{i}$ is defined by

$M_{\sigma}^{i}(f)(x)=\sup_{t>0}\int_{\Sigma}|f(x-ty_{i})|~{}\mathrm{d}\sigma(y).$

###### Proof.

It suffices to show

$\sup_{1<t<2}\sup_{y\in\Sigma}|P_{<k}f(x-2^{-k}ty)|\lesssim M_{HL}(f)(x).$

Indeed,

$\displaystyle P_{<k}f(x-2^{-k}ty)$
$\displaystyle=\int_{\mathbb{R}^{d}}f(z)\,2^{kd}\,\varphi(2^{k}(x-2^{-k}ty-z))~{}~{}\mathrm{d}z$
$\displaystyle=\int_{\mathbb{R}^{d}}f(x+2^{-k}z)\,\varphi(ty-z)~{}~{}\mathrm{d}z.$

Since $y\in\Sigma$ and $1<t<2$, it follows that for any $N>0$

$|P_{<k}f(x-2^{-k}ty)|\lesssim\int_{\mathbb{R}^{d}}|f(x+2^{-k}z)|\frac{C_{N}}{(1+|z|)^{N}}~{}~{}\mathrm{d}z\leq
M_{HL}(f)(x).$

∎

Note that $M_{\sigma}^{i}$ is bounded on $L^{2}(\mathbb{R}^{d})$ whenever
$|\widehat{\mathrm{d}\sigma}(\xi)|\lesssim(1+|\xi|)^{-s}$ for $s>\frac{1}{2}$
(see [16, 17]). In fact, from the assumption $s>\frac{(m-1)d}{2}+\frac{1}{2}$,
it follows $s>\frac{1}{2}$ for $m=2$. Then, by Lemma 4.1, when $m=2$ we prove

$\Big{\|}\sum_{\alpha=1}^{2}\sup_{k\in\mathbb{Z}}|\mathfrak{A}_{k}^{\alpha}(\mathrm{F})|\Big{\|}_{L^{1}(\mathbb{R}^{d})}\lesssim\|f_{1}\|_{L^{2}(\mathbb{R}^{d})}\times\|f_{2}\|_{L^{2}(\mathbb{R}^{d})}.$

For the summation of $\mathfrak{M}_{\mathbf{n}}$ over
$\mathbf{n}\in\mathbb{N}_{0}^{m}$, we use the following lemma.

######  Lemma 4.2.

Let $\mathbf{n}\in\mathbb{N}^{m}$ and $q=2/m$. Then, we have

$\|\mathfrak{S}_{\mathbf{n}}^{q}(\mathrm{F})\|_{L^{2/m}(\mathbb{R}^{d})}\lesssim
2^{-\delta(\mathbf{n},m,d)}\prod_{i=1}^{m}\|f_{i}\|_{L^{2}(\mathbb{R}^{d})},$

where $\delta(\mathbf{n},m,d)=m^{-1/2}|\mathbf{n}|s(m,d)$.

Note that $\mathfrak{M}_{\mathbf{n}}\leq\mathfrak{S}^{q}_{\mathbf{n}}$ due to
$\ell^{\infty}\to\ell^{q}$ embedding with $q=2/m$. Hence, it follows that

(4.8)
$\displaystyle\|\mathfrak{M}_{\mathbf{n}}(\mathrm{F})\|_{L^{2/m}(\mathbb{R}^{d})}\lesssim
2^{-\delta(\mathbf{n},m,d)}\prod_{i=1}\|f_{i}\|_{L^{2}(\mathbb{R}^{d})}.$

Since $2^{-\delta(\mathbf{n},m,d)}$ is summable over
$\mathbf{n}\in\mathbb{N}_{0}^{m}$ for $s(m,d)>0$, this proves the theorem for
the case of $m=2$.

For the induction, we assume that Theorem 1.3 holds for $N$-linear operators
with $N=2,\cdots,m-1$. Note that we already show it holds that when $m=2$.
Under the assumption, we will show the following lemma.

######  Lemma 4.3.

For $\alpha=1,\dots,m$, we have

$\mathfrak{A}_{k}^{\alpha}(\mathrm{F})(x)\lesssim\prod_{\mu=1}^{\alpha}M_{HL}(f_{\mu})(x)\times\sup_{k\in\mathbb{Z}}\sup_{1<t<2}\int_{\Sigma}\Big{|}\prod_{\nu=\alpha+1}^{m}f_{\nu}(x-2^{-k}ty_{\nu})\Big{|}~{}\mathrm{d}\sigma(y).$

Moreover, if we assume Theorem 1.3 holds for $N$-linear operators with
$N=2,\cdots,m-1$, then we have

(4.9)
$\displaystyle\Big{\|}\sup_{k\in\mathbb{Z}}\sup_{1<t<2}\int_{\Sigma}\Big{|}\prod_{\nu=\alpha+1}^{m}f_{\nu}(x-2^{-k}ty_{\nu})\Big{|}~{}\mathrm{d}\sigma(y)\Big{\|}_{L^{2/(m-\alpha)}(\mathbb{R}^{d},\mathrm{d}x)}\lesssim\prod_{\nu=\alpha+1}^{m}\|f_{\nu}\|_{L^{2}(\mathbb{R}^{d})}.$

###### Proof.

Note that the first assertions of the lemma follows directly by the proof of
Lemma 4.1. For the second assertion, observe that the integrand of the left-
hand side of (4.9) is an $(m-\alpha)$-sublinear operator, and the symbol has
decay $s>\frac{(m-1)d}{2}+\frac{1}{2}$. For $\alpha=m$ we have nothing to
prove, for $\alpha=m-1$ we have a linear maximal average which is surely
bounded on $L^{2}(\mathbb{R}^{d})$, and for $\alpha=m-2$ we have a
bi(sub)linear operator which is already proved by previous steps.

Therefore it suffices to consider $1\leq\alpha\leq m-3$. In case of
$1\leq\alpha\leq m-3$, observe that the Fourier decay
$s>\frac{(m-1)d}{2}+\frac{1}{2}$ is clearly larger than
$\frac{(m-\alpha-1)d}{2}+\frac{1}{2}$ which is the condition of Theorem 1.3
for $(m-\alpha)$-sublinear operators. Since we assume that Theorem 1.3 holds
for $N=1,\dots,m-1$, it follows that

$\Big{\|}\sup_{k\in\mathbb{Z}}\sup_{1<t<2}\int_{\Sigma}\Big{|}\prod_{\nu=\alpha+1}^{m}f_{\nu}(x-2^{-k}ty_{\nu})\Big{|}~{}\mathrm{d}\sigma(y)\Big{\|}_{L^{2/(m-\alpha)}(\mathbb{R}^{d},\mathrm{d}x)}\leq
C\prod_{\nu=\alpha+1}^{m}\|f_{\nu}\|_{L^{2}(\mathbb{R}^{d})}.$

∎

Note that we assume Theorem 1.3 holds for $N$-linear operators with
$N=2,\cdots,m-1$ and prove the $N=2$ case. For general $m$, we make use of
Lemma 4.3 to obtain

(4.10)
$\displaystyle\begin{split}&\|\mathfrak{A}_{k}^{\alpha}(\mathrm{F})\|_{L^{2/m}(\mathbb{R}^{d})}\\\
\lesssim&\Big{\|}\prod_{\mu=1}^{\alpha}M_{HL}(f_{\mu})(x)\times\sup_{0<t}\int_{\Sigma}\Big{|}\prod_{\nu=\alpha+1}^{m}f_{\nu}(\cdot-
ty_{\nu})\Big{|}~{}\mathrm{d}\sigma(y)\Big{\|}_{L^{2/m}(\mathbb{R}^{d})}\\\
\leq&\Big{\|}\prod_{\mu=1}^{\alpha}M_{HL}(f_{\mu})\Big{\|}_{L^{2/\alpha}(\mathbb{R}^{d})}\times\Big{\|}\sup_{0<t}\int_{\Sigma}\Big{|}\prod_{\nu=\alpha+1}^{m}f_{\nu}(\cdot-
ty_{\nu})\Big{|}~{}\mathrm{d}\sigma(y)\Big{\|}_{L^{2/(m-\alpha)}(\mathbb{R}^{d})}\\\
\lesssim&\prod_{\mu=1}^{\alpha}\|f_{\mu}\|_{L^{2}(\mathbb{R}^{d})}\times\prod_{\nu=\alpha+1}^{m}\|f_{\nu}\|_{L^{2}(\mathbb{R}^{d})}.\end{split}$

By (4.10) and Lemma 4.2, Theorem 1.3 is true under the assumption that $N$
cases hold for $N=2,\dots,m-1$. This closes the induction. Hence, it remains
to verify Lemma 4.2 to complete the proof.

###  4.1. Proof of Lemma 4.2

We make use of the following scaling:

$\displaystyle\Big{\|}\sup_{1<t<2}\Big{|}\int_{\Sigma}\prod_{i=1}^{m}\,$
$\displaystyle
P_{k+n_{i}}f_{i}(x-2^{-k}ty_{i})~{}~{}\mathrm{d}\sigma(y)\Big{|}\Big{\|}_{L^{2/m}(\mathbb{R}^{d})}$
$\displaystyle=2^{-kmd/2}\,\|\mathrm{M}_{\sigma}^{loc}(P_{n_{1}}f_{1,k},\dots,P_{n_{m}}f_{m,k})\|_{L^{2/m}(\mathbb{R}^{d})},$

where $f_{i,k}(x)=f_{i}(x/2^{k})$. Then, by Lemma 3.1 for $q=2/m$

$\displaystyle\|\mathfrak{S}_{\mathbf{n}}^{q}(\mathrm{F})\|_{L^{2/m}(\mathbb{R}^{d})}^{2/m}$
$\displaystyle=\sum_{k\in\mathbb{Z}}2^{-kd}\,\|\mathrm{M}_{\sigma}^{loc}(P_{n_{1}}f_{1,k},\dots,P_{n_{m}}f_{m,k})\|_{L^{2/m}(\mathbb{R}^{d})}^{2/m}$
$\displaystyle\lesssim\sum_{k\in\mathbb{Z}}2^{-kd}\,2^{-(\max_{1\leq i\leq
m}|n_{i}|)(2s(m,d))/{m}}\prod_{i=1}^{m}\|P_{n_{i}}f_{i,k}\|_{L^{2}(\mathbb{R}^{d})}^{2/m}$
$\displaystyle\leq\sum_{k\in\mathbb{Z}}2^{-kd}\,2^{-(2\delta(\mathbf{n},m,d))/{m}}\prod_{i=1}^{m}2^{(dk)/{m}}\|P_{n_{i}+k}f_{i}\|_{L^{2}(\mathbb{R}^{d})}^{2/m}$
$\displaystyle=\sum_{k\in\mathbb{Z}}2^{-(2\delta(\mathbf{n},m,d))/{m}}\prod_{i=1}^{m}\|P_{n_{i}+k}f_{i}\|_{L^{2}(\mathbb{R}^{d})}^{2/m},$

where $\delta(\mathbf{n},m,d)=m^{-1/2}|\mathbf{n}|s(m,d)$ since $\max_{1\leq
i\leq m}|n_{i}|\geq m^{-1/2}|\mathbf{n}|$. We apply Hölder’s inequality to the
last line in the above inequalities to obtain

$\displaystyle\|\mathfrak{S}_{\mathbf{n}}^{q}(\mathrm{F})\|_{L^{2/m}(\mathbb{R}^{d})}\lesssim
2^{-\delta(\mathbf{n},m,d)}\prod_{i=1}^{m}\Bigl{(}\sum_{k\in\mathbb{Z}}\|P_{n_{i}+k}f_{j}\|_{L^{2}(\mathbb{R}^{d})}^{2}\Bigr{)}^{{1}/{2}}.$

By the Littlewood-Paley decomposition and Plancherel theorem, one can see that

$\Big{(}\sum_{j}\big{\|}P_{j}f\big{\|}_{p}^{p}\Big{)}^{1/p}\lesssim\|f\|_{p}$

for $p\geq 2$. Combining these two estimates, we have

$\|\mathfrak{S}_{\mathbf{n}}^{q}(\mathrm{F})\|_{L^{2/m}(\mathbb{R}^{d})}\lesssim
2^{-\delta(\mathbf{n},m,d)}\prod_{i=1}^{m}\|f_{i}\|_{L^{2}(\mathbb{R}^{d})}.$

This proves the lemma.

## Acknowledgement

All three authors have been partially supported by NRF grant no.
2022R1A4A1018904 funded by the Korea government(MSIT) . They are supported
individually by NRF no. RS-2023-00239774(C. Cho), no. 2021R1C1C2008252(J. B.
Lee), and BK21 Postdoctoral fellowship of Seoul National University(K. Shuin).
The authors are sincerely grateful to Prof. Saurabh Shrivastava, Ankit Bhojak
and Surjeet Singh Choudhary for their kind comments pointing out some errors.

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