#  $L^{p}$ improving properties and maximal estimates for certain multilinear
averaging operators

Chu-hee Cho  Chuhee Cho
Research Institute of Mathematics
Seoul National University
08826 Gwanak-ro 1, Seoul, Republic of Korea<EMAIL_ADDRESS>,  Jin Bong Lee
Jin Bong Lee
Research Institute of Mathematics
Seoul National University
08826 Gwanak-ro 1, Seoul, Republic of Korea<EMAIL_ADDRESS>and  Kalachand
Shuin  Kalachand Shuin
Research Institute of Mathematics
Seoul National University
08826 Gwanak-ro 1, Seoul, Republic of Korea<EMAIL_ADDRESS>
###### Abstract.

In this article we focus on $L^{p}$-improving estimates of two types of
multilinear averaging operators. We also suggest that different
$L^{p}$-improving estimates can be obtained by means of the nonlinear
Brascamp-Lieb inequality. Moreover, we prove improved $L^{p_{1}}\times
L^{p_{2}}\times\cdots\times L^{p_{m}}\rightarrow L^{p}$ boundedness of the
corresponding $m$-(sub)linear lacunary maximal operators when
$1/p=\sum_{j=1}^{m}1/p_{j}$. In showing that, we make use of the
$L^{1}$-improving estimates of averaging operators.

###### Key words and phrases:

Multilinear lacunary maximal operators, $L^{p}$ improving, Nonlinear Brascamp-
Lieb inequality

###### 2020 Mathematics Subject Classification:

Primary 42B25, 47H60

###### Contents

  1. 1 Introduction
  2. 2 Proofs of Propositions 1.1 and 1.2
    1. 2.1 Proof of Proposition 1.1
    2. 2.2 Proof of Proposition 1.2
      1. 2.2.1 $L^{1}$\- improving estimates
      2. 2.2.2 $L^{p}$\- improving estimates
      3. 2.2.3 smoothing estimates
  3. 3 A Nonlinear Brascamp-Lieb inequality approach to $L^{p}$-improving estimates for ${\mathcal{A}}_{\mathcal{S}}^{\Theta}$
    1. 3.1 Nonlinear Brascamp-Lieb inequality
    2. 3.2 Proof of Theorem 1.3
      1. 3.2.1 $d>d_{K}\geq k$
      2. 3.2.2 $k>d_{K}\geq 1$
  4. 4 Proof of Theroem 1.5
  5. 5 Proofs of Lemmas 4.2 and 4.3
    1. 5.1 Proof of Lemma 4.2
      1. 5.1.1 Estimates for $\mathcal{M}_{\mathbf{n}}(b_{1},\dots,b_{m})$
      2. 5.1.2 Estimates for other terms
      3. 5.1.3 Proof of Lemma 5.1
      4. 5.1.4 Proof of Lemma 5.2
    2. 5.2 Proof of Lemma 4.3
  6. 6 Proof of Theorem 1.6

##  1\. Introduction

Let $\mathcal{S}$ be a smooth hypersurface with non-vanishing $k$ principal
curvatures, and $\Theta_{j}$ be rotation matrices in
$\mathbf{M}_{d,d}({\mathbb{R}})$ for $j=1,2,\dots,m$. We assume that
$\\{\Theta_{j}\\}_{j=1}^{m}$ is linearly independent. Then for
$f_{1},f_{2},\dots,f_{m}\in\mathscr{S}(\mathbb{R}^{d})$, we define

(1.1) $\displaystyle\ \mathcal{A}_{\mathcal{S}}^{\Theta}(\mathrm{F})(x)$
$\displaystyle:=\int_{\mathcal{S}}\prod_{j=1}^{m}f_{j}(x+\Theta_{j}y)~{}\mathrm{d}\sigma_{\mathcal{S}}(y),$

where $F=(f_{1},f_{2},\cdots,f_{m})$ and $d\sigma_{\mathcal{S}}$ is the
normalized surface measure on $\mathcal{S}$. We also consider another
$m$-linear averaging operator defined by

(1.2) $\displaystyle\textsl{A}_{\Sigma}(\mathrm{F})(x)$
$\displaystyle:=\int_{\Sigma}\prod_{j=1}^{m}f_{j}(x+y_{j})~{}\mathrm{d}\sigma_{\Sigma}(\textsl{y}),\,\,(y_{1},\dots,y_{m})=\textsl{y}\in{\mathbb{R}}^{md},$

where $\Sigma$ is a compact $(md-1)$ dimensional smooth hypersurface in
$\mathbb{R}^{md}$ with $k$ non-vanishing principal curvatures. Note that $k$
arising in (1.1) satisfies $1\leq k\leq d-1$, while $k$ in (1.2) is $1\leq
k\leq md-1$. Moreover, we are interested in the following lacunary maximal
operators associated with (1.1) and (1.2):

(1.3) $\displaystyle\mathcal{M}^{\Theta}_{\mathcal{S}}(\mathrm{F})(x)$
$\displaystyle=\sup_{\ell\in{\mathbb{Z}}}\Big{|}\int_{\mathcal{S}}\prod_{j=1}^{m}f_{j}(x-2^{\ell}\Theta_{j}y)~{}~{}\mathrm{d}\sigma_{\mathcal{S}}(y)\Big{|},$
(1.4) $\displaystyle\mathfrak{M}_{\Sigma}(\mathrm{F})(x)$
$\displaystyle=\sup_{\ell\in{\mathbb{Z}}}\Big{|}\int_{\Sigma}\prod_{j=1}^{m}f_{j}(x-2^{\ell}y_{j})~{}~{}\mathrm{d}\sigma_{\Sigma}(\textsl{y})\Big{|}.$

The purpose of this article is to prove $L^{p}$-improving estimates of
multilinear averaging operators defined by (1.1) and (1.2). Further, using
this $L^{p}$-improving estimates we show $L^{p_{1}}\times
L^{p_{2}}\times\cdots\times L^{p_{m}}\rightarrow L^{p}$ boundedness for
$1/p=\sum^{m}_{j=1}1/p_{j}$ of the multi-(sub)linear lacunary maximal
functions ${\mathcal{M}}_{\mathcal{S}}^{\Theta}$ and $\mathfrak{M}_{\Sigma}$.

Averaging operators given in (1.1) and (1.2) and related maximal operators
arise in many studies in multilinear harmonic analysis. Since Coifman and
Meyer [11] opened the path of multilinear harmonic analysis in 1975, there
have been significant developments in the area of multilinear harmonic
analysis over the last few decades. Among those achievements, we introduce
works of Lacey and Thiele [21, 22] in which they proved $L^{p}$-boundedness of
the bilinear Hilbert transform given as

$BHT_{\alpha}(f,g)(x):=p.v.\int_{-\infty}^{\infty}f(x-t)g(x-\alpha
t)\frac{\mathrm{d}t}{t},\quad\alpha\not=0,1.$

Their seminal work settled the long standing conjecture of Calderón. Later,
Lacey [20] studied $L^{p}$\- boundedness of bilinear maximal operator

$M_{\alpha}(f,g)(x):=\sup_{t>0}\frac{1}{2t}\int_{-t}^{t}|f(x-y)g(x-\alpha
y)|~{}\mathrm{d}y,\quad\alpha\not=0,1,$

which is related to the bilinear Hilbert transform. One may regard averaging
operators ${\mathcal{A}}_{\mathcal{S}}^{\Theta}$ as a generalization of
$M_{\alpha}$ without the supremum because the condition $\alpha\not=0,1$
corresponds to the linearly independent condition of $\\{\Theta_{j}\\}$.

On the other hand, $\textsl{A}_{\Sigma}$ given in (1.2) is a direct analogue
of a spherical averages $A^{t}_{\mathbb{S}^{d-1}}f(x)$ for $t=1$, which is
defined by

$A^{t}_{\mathbb{S}^{d-1}}f(x):=\int_{\mathbb{S}^{d-1}}f(x-ty)~{}\mathrm{d}\sigma(y).$

Thus we write
$\textsl{A}_{\mathbb{S}^{md-1}}(\mathrm{F})(x)=A^{1}_{\mathbb{S}^{md-1}}(f_{1}\otimes\cdots\otimes
f_{m})(x,\dots,x)$. For studies on
$\textsl{A}_{\mathbb{S}^{md-1}}(\mathrm{F})$, we recommend [27, 1, 31, 12] and
references therein. In the literature, $A^{t}_{\mathbb{S}^{d-1}}f(x)$ have
been extensively studied in terms of maximal operators. For the (sub)linear
spherical maximal operaor $M^{*}_{\mathbb{S}^{d-1}}$ defined by

$M^{*}_{\mathbb{S}^{d-1}}f(x)=\sup_{t>0}|A^{t}_{\mathbb{S}^{d-1}}f(x)|:=\sup_{t>0}\Big{|}\int_{\mathbb{S}^{d-1}}f(x-ty)~{}\mathrm{d}\sigma(y)\Big{|}$

with $d\sigma$ is the normalized surface measure on the sphere
$\mathbb{S}^{d-1}$, Stein [32] proved that for $d\geq 3$, the spherical
maximal operator $M^{*}_{\mathbb{S}^{d-1}}$ is bounded in $L^{p}$, if and only
if $p>\frac{d}{d-1}$ in 1976. Later, Bourgain [8] obtained $L^{p}$ boundedness
of $M^{*}_{\mathbb{S}^{1}}$ for $p>2$. Those restricted boundedness of
$M^{*}_{\mathbb{S}^{d-1}}$ can be improved if one considers the lacunary
spherical maximal operator, which is given by
$M_{\mathbb{S}^{d-1}}f(x):=\sup_{j\in\mathbb{Z}}|A_{\mathbb{S}^{d-1}}^{2^{j}}f(x)|$.
Calderón [9] proved $L^{p}$ estimates of the operator $M_{\mathbb{S}^{d-1}}$
for $1<p\leq\infty$ and $d\geq 2$. After then, Seeger and Wright [30] showed
$L^{p}$ estimates of general lacunary maximal operators $M_{\mathcal{S}}$ for
$1<p\leq\infty$, when the Fourier transform of the surface measure $\sigma$ of
$\mathcal{S}$ satisfies $|\hat{\sigma}(\xi)|\lesssim|\xi|^{-\epsilon}$, for
any $\epsilon>0$. There are also $L^{p}-L^{q}$ estimates for $p\leq q$ (we
call this $L^{p}$-improving estimates) of the spherical average
$A^{1}_{\mathbb{S}^{d-1}}$ [25, 33].

Lacey [19] used the $L^{p}$-improving estimates of spherical averages to prove
sparse domination of the corresponding lacunary and full spherical maximal
functions. It is well known that sparse domination of an operator implies
vector valued boundedness and weighted boundedness of that operator with
respect to Muckenhoupt $A_{p}$ weights [26, 23]. This idea has been
extensively used to obtain sparse domination of several linear and sub-linear
operators in the field of Harmonic analysis (See [3]). The idea of Lacey [19],
together with $L^{p}$-improving estimates of certain bilinear averaging
operators, can be used to study sparse domination of maximal operators
associated with the bilinear operators. We recommend [6, 28, 29] and
references therein, which contains results of bilinear spherical maximal
operator, bilinear maximal triangle averaging operators and bilinear product-
type spherical maximal operators, respectively.

Recently, Christ and Zhou [10] studied $L^{p_{1}}\times L^{p_{2}}\rightarrow
L^{p}$ with $1/p_{1}+1/p_{2}=1/p$ boundedness of bi-(sub)linear lacunary
maximal functions defined on a class of singular curves, which might be
understood in the sense of both (1.3) and (1.4).

$\mathcal{M}(f_{1},f_{2})(x):=\sup_{\ell\in\mathbb{Z}}|B_{2^{\ell}}(f_{1},f_{2})(x)|=\sup_{\ell\in\mathbb{Z}}\Big{|}\int_{{\mathbb{R}}^{1}}\prod_{j=1}^{2}f_{j}(x-2^{\ell}\gamma_{j}(t))\eta(t)~{}\mathrm{d}t\Big{|},$

where $\gamma=(\gamma_{1},\gamma_{2}):(-1,1)\to{\mathbb{R}}^{2}$, and $\eta\in
C_{0}^{\infty}((-1,1))$. In consequence, they have proved $L^{p_{1}}\times
L^{p_{2}}\rightarrow L^{p}$ estimates for $1<p_{1},p_{2}\leq\infty$,
$1/p_{1}+1/p_{2}=1/p$ of the bi-(sub)linear lacunary spherical maximal
operator $\mathfrak{M}_{\mathbb{S}^{2d-1}}$, for dimension $d=1$ which is
defined by

$\mathfrak{M}_{\mathbb{S}^{1}}(f_{1},f_{2})(x):=\sup_{\ell\in\mathbb{Z}}\Big{|}\textsl{A}_{\mathbb{S}^{1}}^{2^{\ell}}(f_{1},f_{2})(x)\Big{|}=\sup_{\ell\in\mathbb{Z}}\Big{|}\int_{\mathbb{S}^{1}}\prod_{j=1,2}f_{j}(x-2^{\ell}y_{j})~{}\mathrm{d}\sigma(\textsl{y})\Big{|},$

where $d\sigma(y,z)$ is the normalized surface measure on the sphere
$\mathbb{S}^{2d-1}$. For $d\geq 2$, the complete $(L^{p_{1}}\times
L^{p_{2}}\rightarrow L^{p})$-estimate of the operator
$\mathfrak{M}_{\mathbb{S}^{2d-1}}$ was not known. However, there are some
partial results of the operator $\mathfrak{M}_{\mathbb{S}^{2d-1}}$ [28, 6],
and very recently Borges and Foster [7] have obtained almost sharp results
including some endpoint estimates. In this paper, we give a different proof of
the same $(L^{p_{1}}\times L^{p_{2}}\rightarrow L^{p})$-estimate for
$\mathfrak{M}_{\mathbb{S}^{2d-1}}$.

There is another important bi-(sub)linear maximal function

$\mathfrak{M}^{*}_{\mathbb{S}^{2d-1}}(f_{1},f_{2})(x):=\sup_{t>0}|\textsl{A}_{\mathbb{S}^{2d-1}}^{t}(f_{1},f_{2})(x)|,$

which is known as bilinear spherical maximal function. Study of this operator
was generated in [2]. Later, in [18] Jeong and Lee proved almost complete
$L^{p_{1}}\times L^{p_{2}}\rightarrow L^{p}$ estimates for
$1/p_{1}+1/p_{2}=1/p$, $p_{1},p_{2}>1$ and $p>\frac{d}{2d-1}$ when $d\geq 2$.
This result is extended to $d=1$ by Chirst and Zhou [10]. For full maximal
operators associated with a general surface with non-vanishing Gaussian
curvature, however, only $L^{2}({\mathbb{R}}^{d})\times
L^{2}({\mathbb{R}}^{d})\to L^{1}({\mathbb{R}}^{d})$ estimate is known for
$d\geq 4$ [13]. It would be interesting to study $L^{p_{1}}\times L^{p_{2}}\to
L^{p}$ estimate of such maximal averages for $p\leq 1$ in all dimensions, but
we do not discuss further in this paper.

We first state $L^{1}$-improving and quasi-Banach estimates of the $m$-linear
averaging operators $\mathcal{A}^{\Theta}_{\mathcal{S}}$ and
$\textsl{A}_{\Sigma}$. Note that the following two propositions are derived by
simple Fourier analysis and multilinear interpolation.

######  Proposition 1.1.

Let $\mathcal{A}_{\mathcal{S}}^{\Theta}(\mathrm{F})$ be given in (1.1) and
$\mathcal{S}$ be a smooth hypersurface with $k$ nonvanishing principal
curvatures with $1\leq k\leq d-1$. Note that
$\Theta=\\{\Theta_{j}\\}_{j=1}^{m}$ is chosen so that
$\Theta_{j_{1}},\Theta_{j_{2}}$ are linearly independent for
$j_{1}\not=j_{2}$. Let also $\mathcal{V}_{k}$ be a set given by
$\\{z=(z_{1},\dots,z_{m})\in[0,1]^{m}:z_{i}=z_{j}=\frac{k+1}{k+2},z_{k}=0,k\not=i,j\\}$
and $\mathrm{conv}(\mathcal{V}_{k})$ be its convex hull. Then for
$(\frac{1}{p_{1}},\dots,\frac{1}{p_{m}})\in\mathrm{conv}(\mathcal{V}_{k})$ we
have the following inequalities:

$\displaystyle\|\mathcal{A}_{\mathcal{S}}^{\Theta}(\mathrm{F})\|_{L^{1}({\mathbb{R}}^{d})}$
$\displaystyle\lesssim\prod_{j=1}^{m}\|f_{j}\|_{L^{p_{j}}({\mathbb{R}}^{d})},$
$\displaystyle\|\mathcal{A}_{\mathcal{S}}^{\Theta}(\mathrm{F})\|_{L^{p}({\mathbb{R}}^{d})}$
$\displaystyle\lesssim\prod_{j=1}^{m}\|f_{j}\|_{L^{p_{j}}({\mathbb{R}}^{d})},$

whenever $\frac{1}{p}=\sum_{j=1}^{m}\frac{1}{p_{j}}=\frac{2(k+1)}{k+2}$.

######  Proposition 1.2.

Let $d\geq 2$ and $\textsl{A}_{\Sigma}(\mathrm{F})$ be an average given by
(1.2) over a smooth hypersurface $\Sigma$ with $k$ nonvanishing principal
curvatures with $(m-1)d<k\leq md-1$. Then for $1\leq p_{j}\leq 2$,
$j=1,2,\dots,m$ and
$\frac{m+1}{2}\leq\sum_{j=1}^{m}\frac{1}{p_{j}}<\frac{2d+k}{2d}$, the
following $L^{1}$-improving estimates hold:

$\|\textsl{A}_{\Sigma}(\mathrm{F})\|_{L^{1}({\mathbb{R}}^{d})}\lesssim\prod_{j=1}^{m}\|f_{j}\|_{L^{p_{j}}({\mathbb{R}}^{d})}.$

Moreover, we have for
$\frac{m+1}{2}\leq\frac{1}{p}=\sum_{j=1}^{m}\frac{1}{p_{j}}<\frac{2d+k}{2d}$

(1.5)
$\displaystyle\|\textsl{A}_{\Sigma}(\mathrm{F})\|_{L^{p}({\mathbb{R}}^{d})}\lesssim\prod_{j=1}^{m}\|f_{j}\|_{L^{p_{j}}({\mathbb{R}}^{d})}.$

For $p_{j}\geq 2$ we have the following smoothing estimates:

(1.6)
$\displaystyle\left\|\textsl{A}_{\Sigma}(\mathrm{F})\right\|_{L^{2}({\mathbb{R}}^{d})}\lesssim\prod_{j=1}^{m}\|f_{j}\|_{B_{p_{j}}^{-k/p_{j}}({\mathbb{R}}^{d})},$

where $\frac{1}{2}=\sum_{j=1}^{m}\frac{1}{p_{j}}$.

When $p>1$, one can obtain different $L^{p}$-improving estimates for
${\mathcal{A}}_{\mathcal{S}}^{\Theta}$ under specific choice of
$\\{\Theta_{j}\\}$ and $\mathcal{S}$. In this case, we do not need any
curvature condition on ${\mathcal{S}}$ and only the dimension of surfaces is
matter. Let $\mathcal{S}^{k}$ be a $k$-dimensional $C^{2}$ surface in
${\mathbb{R}}^{d}$. We choose linearly independent $\\{\Theta_{j}\\}$ so that
for any choice of $\\{j_{i}\\}_{i=1}^{\ell}$ with $2\leq\ell\leq k+1\leq m$

(1.7)
$\displaystyle\dim\Big{(}\bigcap_{i=1}^{\ell}\\{\Theta_{j_{i}}(y^{\prime},0)\in{\mathbb{R}}^{d}:y^{\prime}\in{\mathbb{R}}^{k}\\}\Big{)}\leq
k+1-\ell.$

That is, any subset $\\{\Theta_{j_{i}}\\}_{i=1}^{k+1}$ of
$\\{\Theta_{j}\\}_{j=1}^{m}$ yields a non-degenerate $k$ simplex in
${\mathbb{R}}^{d}$. Note that $(y^{\prime},0)$ in (1.7) can be replaced by any
$k$ dimensional plane in ${\mathbb{R}}^{d}$. Then we have the following
results:

######  Theorem 1.3.

Suppose $d\geq 2$ and $\frac{m-1}{m}\geq\frac{(d-k)k}{d}$. Then
$\mathcal{A}_{\mathcal{S}}^{\Theta}$ is of strong type
$(m,\dots,m,\frac{d}{d-k})$. That is, we have

(1.8)
$\displaystyle\|\mathcal{A}_{\mathcal{S}^{k}}^{\Theta}(\mathrm{F})\|_{L^{\frac{d}{d-k}}({\mathbb{R}}^{d})}\lesssim\prod_{j=1}^{m}\|f_{j}\|_{L^{m}({\mathbb{R}}^{d})}.$

In our proof of Theorem 1.3, we mainly use the nonlinear Brascamp-Lieb
inequality introduced in [5]. We give details on the inequality and the proof
of Theorem 1.3 in Section 3. As a direct consequence of Theorem 1.3 we have
the following corollary.

######  Corollary 1.4.

Let $m\geq d\geq 2$. Then for a $C^{2}$ hypersurface $\mathcal{S}^{d-1}$,
$\mathcal{A}_{\mathcal{S}^{d-1}}^{\Theta}$ is of strong-type $(m,\dots,m,d)$.

One can find similar results in [17, Theorem 1.2], which yields restricted
strong-type $(m,\dots,m,m)$ and
$\Big{(}m\frac{d+1}{d},\dots,m\frac{d+1}{d},d+1\Big{)}$ estimates for
${\mathcal{A}}_{{\mathcal{S}}^{d-1}}^{\Theta}$ when $\mathcal{S}^{d-1}$ is a
sphere. Note that the authors [17] considers $m\leq d$ cases with linearly
independent $\\{\Theta_{j}\\}$, so it cannot be directly compared to Corollary
1.4 in which $m\geq d$ and (1.7) are considered. When $m=d$, however,
Corollary 1.4 gives strong-type $(m,\dots,m,m)$ estimates.

To study further how [17, Theorem 1.2] and Corollary 1.4 are related, we
introduce a quantity $\mathfrak{D}$ which is given for each
$(p_{1},\dots,p_{m},p)$-estimate by

$\mathfrak{D}(p_{1},\dots,p_{m};p):=\Big{(}\frac{1}{p_{1}}+\cdots+\frac{1}{p_{m}}\Big{)}-\frac{1}{p}.$

One can measure extent of $L^{p}$-improving by means of the difference
$\mathfrak{D}$. Then we have

$\mathfrak{D}(m,\dots,m;m)=\frac{m-1}{m},\quad\mathfrak{D}\Big{(}m\frac{d+1}{d},\dots,m\frac{d+1}{d};d+1\Big{)}=\frac{d-1}{d+1},$

where $m\leq d$. On the other hand, Corollary 1.4 yields

$\mathfrak{D}(m,\dots,m;d)=\frac{d-1}{d},\quad m\geq d.$

Thus Corollary 1.4 yields wider range of $L^{p}$-improving than
$(m(d+1)/d,\dots,m(d+1)/d,d+1)$-estimate of [17, Theorem 1.2] under certain
choice of $\\{\Theta_{j}\\}$. We also note that the difference
$\frac{d-1}{d+1}$ is the best possible for linear spherical averages, since
${\mathcal{A}}_{\mathbb{S}^{d-1}}$ satisfies
$L^{\frac{d+1}{d}}({\mathbb{R}}^{d})\to L^{d}({\mathbb{R}}^{d})$ boundedness.
Even for $L^{1}$-improving estimates in Proposition 1.1, we obtain
$\mathfrak{D}(p_{1},\dots,p_{m};1)=\frac{d-1}{d+1}$. Hence one can say that
the number $\frac{d-1}{d}$ only occurs for multilinear averaging operators
with certain transversality of $\\{\Theta_{j}\\}$. Moreover, we only assume a
surface ${\mathcal{S}}$ is of class $C^{2}$ without any curvature condition,
and it would be very interesting to study boundedness of maximal operators
associated with $\mathcal{A}_{\mathcal{S}^{k}}^{\Theta}$.

By making use of the quasi-Banach space estimates, Propositions 1.1 and 1.2,
we obtain multilinear estimates for lacunary maximal operators
${\mathcal{M}}_{\mathcal{S}}^{\Theta}$ and $\mathfrak{M}_{\Sigma}$.

######  Theorem 1.5.

Let $p>\frac{k+2}{2(k+1)}$ for $d\geq 2$. Then the lacunary maximal function
$\mathcal{M}^{\Theta}_{\mathcal{S}}$ maps
$L^{p_{1}}({\mathbb{R}}^{d})\times\cdots\times
L^{p_{m}}({\mathbb{R}}^{d})\rightarrow L^{p}({\mathbb{R}}^{d})$ for
$(\frac{1}{p_{1}},\dots,\frac{1}{p_{m}})\in\mathrm{conv}(\mathcal{V}_{k})$. In
particular, if one consider a lacunary maximal operator associated with
$\mathbb{S}^{d-1}$, then the range of $p$ becomes $p>\frac{d+1}{2d}$.

Observe that the multilinear averaging operator (1.1) is an analogous
multilinear averaging operator to the bilinear operator $\textsl{B}_{\theta}$
considered by Greenleaf $et.$ $al.$ in [16].

$B_{\theta}(f,g)(x)=\int_{\mathbb{S}^{1}}f(x-y)g(x-\theta
y)~{}\mathrm{d}\sigma(y),$

where $\theta$ denotes a counter-clockwise rotation. Therefore, Theorem 1.5
(when $m=2$) yields boundedness of the lacunary maximal function corresponding
to the averaging operator $\textsl{B}_{\theta}$.

######  Theorem 1.6.

Let
$\frac{m+1}{2}\leq\frac{1}{p}=\sum_{j=1}^{m}\frac{1}{p_{j}}<\frac{2d+k}{2d}$
for $1\leq p_{j}\leq 2$ and $k>(m-1)d$. Then the lacunary maximal operator
$\mathfrak{M}_{\Sigma}$ maps $L^{p_{1}}({\mathbb{R}}^{d})\times\cdots\times
L^{p_{m}}({\mathbb{R}}^{d})\rightarrow L^{p}({\mathbb{R}}^{d})$.

Note that one can extend above $(L^{p_{1}}\times\cdots\times L^{p_{m}}\to
L^{p})$-estimates for $1\leq p_{j}\leq\infty$ via multilinear interpolation,
since $\mathfrak{M}_{\Sigma}$ is bounded from $L^{\infty}\times\cdots\times
L^{\infty}$ to $L^{\infty}$.

######  Remark 1.7.

What we really proved in Sections 4 and 5 is that multi-linear estimates of
lacunary maximal operators are derived from $L^{1}$-improving estimates and
Sobolev regularity estimates of corresponding averaging operators. Precisely,
if one obtains $L^{p_{1}^{\circ}}\times\cdots\times L^{p_{m}^{\circ}}\to
L^{1}$ estimates of averaging operators with
$\sum_{j=1}^{m}1/p_{j}^{\circ}>1$, then we obtain
$L^{p_{1}^{\circ}}\times\cdots\times L^{p_{m}^{\circ}}\to
L^{p^{\circ},\infty}$ estimates of the lacunary maximal operators for
$\sum_{j=1}^{m}1/p_{j}^{\circ}=1/p^{\circ}$ together with certain polynomial
growth, which is Lemma 4.2. The polynomial growth of Lemma 4.1 will be handled
by interpolation with an exponential decay estimates of Lemma 4.3, which is
originated by the Sobolev regularity estimates of averaging operators. In
result, we obtain $L^{p_{1}}\times\cdots\times L^{p_{m}}\to L^{p}$ estimates
for $\sum_{j=1}^{m}1/p_{j}=1/p$ with $p_{1},\dots,p_{m}\geq 1$ and
$p>p^{\circ}$.

As a simple application of Remark 1.7, we obtain the following result:

######  Remark 1.8.

Theorem 1.6 also yields the following boundedness of the bilinear lacunary
spherical maximal function $\mathfrak{M}_{\mathbb{S}^{2d-1}}$. Let $d\geq 1$,
$1<p_{1},p_{2}\leq\infty$ and $1/p_{1}+1/p_{2}=1/p$. Then

(1.9)
$\displaystyle\|\mathfrak{M}_{\mathbb{S}^{2d-1}}(f_{1},f_{2})\|_{L^{p}}\lesssim\|f_{1}\|_{L^{p_{1}}}\|f_{2}\|_{L^{p_{2}}}.$

Note that we make use of $L^{1}\times L^{1}\to L^{1/2}$ estimates of
$\textsl{A}_{\mathbb{S}^{2d-1}}^{1}$ given by [17] and machinery of Section 4
to obtain (1.9) for $p>1/2$. This estimate is already given in [7] and we give
a different proof at the end of this paper.

######  Remark 1.9.

It is known in [14] that $\mathfrak{M}_{\Sigma}$ satisfies
$(L^{2}\times\cdots\times L^{2}\to L^{2/m})$-estimates for certain $k$. One
can check that even for the worst indices, our estimates (2.13) is better than
the $(L^{2}\times\cdots\times L^{2}\to L^{2/m})$-estimates in the sense that
Theorem 1.6 holds for $L^{p}$ spaces with lower indices since
$\frac{2}{m}>\frac{2}{m+1}>\frac{2d}{2d+k}$. When $k\leq(m-1)d$, we do not
know anything yet.

## Notations and definitions

$\bullet$ For a cube $Q$ or a ball $B$ in ${\mathbb{R}}^{d}$, we define $CQ$
and $CB$ whose sidelength and radius are $C$ times those of $Q$ and $B$ with
the same centers, respectively. For a measureable set $E$, we say
$\mathrm{meas}(E)$ as a measure of $E$.

$\bullet$ For $1<p<\infty$ and real $s$, we introduce the Bessel potential
space $H^{s,p}({\mathbb{R}}^{d})$ whose norm is given by

$\|f\|_{H^{s,p}({\mathbb{R}}^{d})}=\left\|\mathcal{F}^{-1}\left[(1+|\xi|^{2})^{\frac{s}{2}}\mathcal{F}[f]\right]\right\|_{L^{p}({\mathbb{R}}^{d})},$

where $\mathcal{F}$, $\mathcal{F}^{-1}$ denote the Fourier transform and its
inverse. We also use $\widehat{f}$, $\check{f}$ to mean a Fourier transform
and its inverse of $f$.

$\bullet$ Choose a Schwartz class function $\phi$ such that
$\operatorname{supp}(\widehat{\phi})\subset B(0,2)$ and
$\widehat{\phi}(\xi)=1$ for $\xi\in B(0,1)$. Also consider
$\widehat{\psi}(\xi)=\widehat{\phi}(\xi)-\widehat{\phi}(2\xi)$ so that
$\operatorname{supp}(\widehat{\psi})\subset\\{2^{-1}<|\xi|<2\\}$. By symbols
$\widehat{\phi}_{\ell}(\xi)=\widehat{\phi}(2^{-\ell}\xi)$ and
$\widehat{\psi}_{\ell}(\xi)=\widehat{\psi}(2^{-\ell}\xi)$ we define frequency
projection operators.

(1.10) $\displaystyle\widehat{P_{<\ell}f}(\xi)$
$\displaystyle=\widehat{f}(\xi)\widehat{\phi}_{\ell}(\xi),$ (1.11)
$\displaystyle\widehat{P_{\ell}f}(\xi)$
$\displaystyle=\widehat{f}(\xi)\widehat{\psi}_{\ell}(\xi).$

Then we define a norm of Besov spaces by

$\|f\|_{B_{p,q}^{s}({\mathbb{R}}^{d})}=\|P_{<1}f\|_{L^{p}({\mathbb{R}}^{d})}+\Big{(}\sum_{\ell\geq
1}2^{jsq}\|P_{\ell}f\|_{L^{p}({\mathbb{R}}^{d})}^{q}\Big{)}^{\frac{1}{q}}.$

##  2\. Proofs of Propositions 1.1 and 1.2

###  2.1. Proof of Proposition 1.1

We first assume the following lemma which states the first assertion of
Proposition 1.1 and exploit a standard technique from [15, 17] to prove the
second assertion.

######  Lemma 2.1.

$\mathcal{A}_{\mathcal{S}}^{\Theta}$ is bounded from
$L^{p_{1}}({\mathbb{R}}^{d})\times\cdots\times L^{p_{m}}({\mathbb{R}}^{d})$ to
$L^{1}({\mathbb{R}}^{d})$ for
$(\frac{1}{p_{1}},\dots,\frac{1}{p_{m}})\in\mathrm{conv}(\mathcal{V}_{k})$. In
particular, if $k=d-1$, then one example of
$\mathcal{A}_{\mathcal{S}}^{\Theta}$ is the spherical averaging operator
$\mathcal{A}_{\mathbb{S}^{d-1}}^{\Theta}$.

Let $p=\frac{k+2}{2(k+1)}$ and
$(\frac{1}{p_{1}},\dots,\frac{1}{p_{m}})\in\mathrm{conv}(\mathcal{V}_{k})$. We
begin with

$\displaystyle\|\mathcal{A}_{\mathcal{S}}^{\Theta}(\mathrm{F})\|_{L^{p}({\mathbb{R}}^{d})}^{p}=\int_{{\mathbb{R}}^{d}}\Big{|}\int_{\mathcal{S}}\prod_{j=1}^{m}f_{j}(x+\Theta_{j}y)~{}\mathrm{d}\sigma(y)\Big{|}^{p}~{}\mathrm{d}x.$

Decompose ${\mathbb{R}}^{d}$ into countable union of unit cubes
${Q}_{\bf{n}}={\bf{n}}+[0,1)^{d}$, ${\bf{n}}\in\mathbb{Z}^{d}$. Together with
compactness of $\mathcal{S}$, we have

(2.1)
$\displaystyle\|\mathcal{A}_{\mathcal{S}}^{\Theta}(\mathrm{F})\|_{L^{p}({\mathbb{R}}^{d})}^{p}=\sum_{{\bf{n}}\in\mathbb{Z}^{d}}\int_{{Q}_{\bf{n}}}\Big{|}\int_{\mathcal{S}}\prod_{j=1}^{m}f_{j}(x+\Theta_{j}y)~{}\mathrm{d}\sigma(y)\Big{|}^{p}~{}\mathrm{d}x.$

Now we apply Hölder’s inequality to obtain

(2.2)
$\displaystyle\int_{{Q}_{\bf{n}}}\Big{|}\int_{\mathcal{S}}\prod_{j=1}^{m}f_{j}(x+\Theta_{j}y)~{}\mathrm{d}\sigma(y)\Big{|}^{p}~{}\mathrm{d}x$
$\displaystyle\lesssim\Bigl{(}\int_{Q_{\bf{n}}}\int_{\mathcal{S}}\Big{|}\prod_{j=1}^{m}f_{j}(x+\Theta_{j}y)\Big{|}~{}\mathrm{d}\sigma(y)~{}\mathrm{d}x\Bigr{)}^{p}.$

Since $x\in{Q}_{\bf{n}}$ and $y\in\operatorname{supp}(S)$, we have the
following equality

(2.3) $\displaystyle
f_{j}(x+\Theta_{j}y)=(f_{j}\mathbbm{1}_{\widetilde{Q}_{\bf{n}}})(x+\Theta_{j}y),$

where $\widetilde{Q}$ denotes a cube whose sidelength is 3 times of that of
${Q}$ with the same center. With help of (2.3) and Lemma 2.1, we have

(2.4)
$\displaystyle\Bigl{(}\int_{{Q}_{\bf{n}}}\int_{\mathcal{S}}\Big{|}\prod_{j=1}^{m}f_{j}(x+\Theta_{j}y)\Big{|}~{}\mathrm{d}\sigma(y)~{}\mathrm{d}x\Bigr{)}^{p}$
$\displaystyle=\Bigl{(}\int_{{Q}_{\bf{n}}}\int_{\mathcal{S}}\Big{|}\prod_{j=1}^{m}(f_{j}\mathbbm{1}_{\widetilde{Q}_{\bf{n}}})(x+\Theta_{j}y)\Big{|}~{}\mathrm{d}\sigma(y)~{}\mathrm{d}x\Bigr{)}^{p}$
$\displaystyle\lesssim\Big{(}\prod_{j=1}^{m}\|(f_{j}\mathbbm{1}_{\widetilde{Q}_{\bf{n}}})\|_{L^{p_{j}}({\mathbb{R}}^{d})}\Big{)}^{p}=\prod_{j=1}^{m}\|f_{j}\mathbbm{1}_{\widetilde{Q}_{\bf{n}}}\|_{L^{p_{j}}({\mathbb{R}}^{d})}^{p}$

whenever $(\frac{1}{p_{1}},\dots,\frac{1}{p_{m}})$ is in
$\mathrm{conv}(\mathcal{V}_{k})$.

By (2.1), (2.2), and (2.4), we have

(2.5)
$\displaystyle\|\mathcal{A}_{\mathcal{S}}^{\Theta}(\mathrm{F})\|_{L^{p}({\mathbb{R}}^{d})}^{p}\lesssim\sum_{{\bf{n}}\in\mathbb{Z}^{d}}\prod_{j=1}^{m}\|f_{j}\mathbbm{1}_{\widetilde{Q}_{\bf{n}}}\|_{L^{p_{j}}({\mathbb{R}}^{d})}^{p}.$

We make use of Hölder’s inequality on (2.5) to obtain

(2.6)
$\displaystyle\|\mathcal{A}_{\mathcal{S}}^{\Theta}(\mathrm{F})\|_{L^{p}({\mathbb{R}}^{d})}^{p}\lesssim\prod_{j=1}^{m}\Bigl{(}\sum_{{\bf{n}}\in\mathbb{Z}^{d}}\|f_{j}\mathbbm{1}_{\widetilde{Q}_{\bf{n}}}\|_{L^{p_{j}}({\mathbb{R}}^{d})}^{p_{j}}\Bigr{)}^{\frac{p}{p_{j}}}.$

Note that $\\{\widetilde{Q}_{\bf{n}}\\}_{{\bf{n}}\in\mathbb{Z}^{d}}$ is a
finitely overlapping cover of ${\mathbb{R}}^{d}$. Therefore we have

(2.7)
$\displaystyle\|\mathcal{A}_{\mathcal{S}}^{\Theta}(\mathrm{F})\|_{L^{p}({\mathbb{R}}^{d})}\lesssim\prod_{j=1}^{m}\|f_{j}\|_{L^{p_{j}}({\mathbb{R}}^{d})},$

where $\frac{1}{p}=\frac{2(k+1)}{k+2}$,
$\frac{1}{p}=\sum_{j=1}^{m}\frac{1}{p_{j}}$, and
$(\frac{1}{p_{1}},\dots,\frac{1}{p_{m}})\in\mathrm{conv}(\mathcal{V}_{k})$.

Thus, by showing Lemma 2.1, we complete the proof of Proposition 1.1.

###### Proof of Lemma 2.1.

Let $\mathfrak{a}$ be a symbol satisfying
$|\mathfrak{a}(\xi)|\lesssim(1+|\xi|)^{-\rho}$ for some $\rho>0$. Then for
$m=1$, it is well-known in [33, 25] that
$T_{\mathfrak{a}}(f)=(\mathfrak{a}\widehat{f}\,)\check{\,}$ is bounded from
$L^{p}({\mathbb{R}}^{d})$ to $L^{p^{\prime}}({\mathbb{R}}^{d})$ for
$\frac{1}{p}+\frac{1}{p^{\prime}}=1$, $p\in[1,2]$, and
$\frac{1}{p}-\frac{1}{2}\leq\frac{1}{2}(\frac{\rho}{\rho+1})$. Let
$\mathcal{S}$ be a surface with $k$ nonvanishing principal curvatures. For
bilinear case $m=2$, by change of variables we have

$\displaystyle\|\mathcal{A}_{\mathcal{S}}^{\Theta}(f,g)\|_{1}$
$\displaystyle\leq\int_{{\mathbb{R}}^{d}}\int_{\mathcal{S}}\Big{|}f(x+\Theta_{1}y)g(x+\Theta_{2}y)\Big{|}~{}\mathrm{d}\sigma(y)~{}\mathrm{d}x$
$\displaystyle=\int_{{\mathbb{R}}^{d}}|f(x)|\int_{\mathcal{S}}\Big{|}g(x+(\Theta_{2}-\Theta_{1})y)\Big{|}~{}\mathrm{d}\sigma(y)~{}\mathrm{d}x$
$\displaystyle\leq\|f\|_{p}\|g\|_{p},$

where the last inequality follows from Hölder’s inequality and
$\frac{1}{p}=\frac{k+1}{k+2}$. Thus for $m$-linear case, it follows that

$\displaystyle\|\mathcal{A}_{\mathcal{S}}^{\Theta}(\mathrm{F})\|_{1}$
$\displaystyle\leq\int_{{\mathbb{R}}^{d}}\int_{\mathcal{S}}\Big{|}\prod_{j=1}^{m}f_{j}(x+\Theta_{j}y)\Big{|}~{}\mathrm{d}\sigma(y)~{}\mathrm{d}x$
$\displaystyle\leq\int_{{\mathbb{R}}^{d}}\int_{\mathcal{S}}\Big{|}f_{j_{1}}(x+\Theta_{j_{1}}y)f_{j_{2}}(x+\Theta_{j_{2}}y)\Big{|}~{}\mathrm{d}\sigma(y)~{}\mathrm{d}x\times\prod_{l\not=1,2}\|f_{j_{l}}\|_{\infty}$
$\displaystyle\leq\|f_{j_{1}}\|_{p}\|f_{j_{2}}\|_{p}\prod_{l\not=1,2}\|f_{j_{l}}\|_{\infty},$

where $\frac{1}{p}=\frac{k+1}{k+2}$. Since $j_{1},j_{2}$ are arbitrary, we
make use of multilinear interpolation and the lemma is proved. ∎

###  2.2. Proof of Proposition 1.2

####  2.2.1. $L^{1}$\- improving estimates

By translation $x\to x+y_{m}$, we reduce the $L^{1}$-norm of
$\textsl{A}_{\Sigma}$ into $L^{\infty}$-norm of the following $(m-1)$-linear
operator:

(2.8)
$\displaystyle\int_{\Sigma}\prod_{j=1}^{m-1}f_{j}(x+y_{m}-y_{j})~{}\mathrm{d}\sigma_{\Sigma}(y).$

By using the Fourier transform, we rewrite (2.8) as

(2.9) $\displaystyle\int_{{\mathbb{R}}^{(m-1)d}}e^{2\pi
ix\cdot(\xi_{1}+\cdots+\xi_{m-1})}\widehat{\mathrm{d}\sigma_{\Sigma}}(\xi^{\prime},-\xi_{1}-\cdots-\xi_{m-1})\prod_{j=1}^{m-1}\widehat{f}_{j}(\xi_{j})~{}\mathrm{d}\xi^{\prime},$

where $\xi^{\prime}=(\xi_{1},\dots,\xi_{m-1})\in{\mathbb{R}}^{(m-1)d}$.

Since the surface $\Sigma$ has $k$ nonvanishing principal curvatures,
therefore using the result of Littman [24] we get
$|\widehat{\mathrm{d}\sigma_{\Sigma}}(\xi)|\lesssim(1+|\xi|)^{-k/2}$ for
$\xi\in{\mathbb{R}}^{md}$. This implies that the symbol of (2.9) satisfies

$\left|\widehat{\mathrm{d}\sigma_{\Sigma}}(\xi^{\prime},-\xi_{1}-\cdots-\xi_{m-1})\right|\lesssim(1+|\xi^{\prime}|)^{-k/2}.$

By applying Hölder’s inequality on (2.9), it is bounded above by

$\Big{(}\int_{{\mathbb{R}}^{(m-1)d}}\prod_{j=1}^{m-1}|\widehat{f}_{j}(\xi_{j})|^{p^{\prime}}~{}\mathrm{d}\xi^{\prime}\Big{)}^{1/p^{\prime}}\times\Big{(}\int_{{\mathbb{R}}^{(m-1)d}}(1+|\xi^{\prime}|)^{-\frac{kp}{2}}~{}\mathrm{d}\xi^{\prime}\Big{)}^{1/p}$

and the last term is finite if $p^{\prime}<\frac{2d(m-1)}{2d(m-1)-k}$. Thus,
for $2\leq p^{\prime}<\frac{2d(m-1)}{2d(m-1)-k}$ we have

(2.10)
$\displaystyle\Big{|}\int_{\Sigma}\prod_{j=1}^{m-1}f(x+y_{m}-y_{j})~{}\mathrm{d}\sigma_{\Sigma}(y)\Big{|}\lesssim\prod_{j=1}^{m-1}\|\widehat{f_{j}}\|_{L^{p^{\prime}}({\mathbb{R}}^{d})}.$

Together with $L^{1}$-norm of $f_{m}$, for $\frac{2d(m-1)}{k}<p\leq 2$ we have

(2.11)
$\displaystyle\Big{\|}\textsl{A}_{\Sigma}(\mathrm{F})\Big{\|}_{L^{1}({\mathbb{R}}^{d})}\lesssim\prod_{j=1}^{m-1}\|f_{j}\|_{L^{p}({\mathbb{R}}^{d})}\times\|f_{m}\|_{L^{1}({\mathbb{R}}^{d})}.$

Symmetry of estimates (2.11) and multilinear interpolation yield that

(2.12)
$\displaystyle\Big{\|}\textsl{A}_{\Sigma}(\mathrm{F})\Big{\|}_{L^{1}({\mathbb{R}}^{d})}\lesssim\prod_{j=1}^{m}\|f_{j}\|_{L^{p_{j}}({\mathbb{R}}^{d})},$

where $\frac{m+1}{2}\leq\sum_{j=1}^{m}\frac{1}{p_{j}}<\frac{2d+k}{2d}$ and
$1\leq p_{j}\leq 2$.

####  2.2.2. $L^{p}$\- improving estimates

Since we obtain $L^{1}$-improving estimates for $\textsl{A}_{\Sigma}$, one can
apply the argument of Subsection 2.1 to show that $\textsl{A}_{\Sigma}$
satisfies a Hölder-type multilinear estimates on $L^{p}({\mathbb{R}}^{d})$ for
$\frac{1}{p}=\sum_{j=1}^{m}\frac{1}{p_{j}}$ with $p_{j}$ in (2.12). That is,
we have

(2.13)
$\displaystyle\Big{\|}\textsl{A}_{\Sigma}(\mathrm{F})\Big{\|}_{L^{p}({\mathbb{R}}^{d})}\lesssim\prod_{j=1}^{m}\|f_{j}\|_{L^{p_{j}}({\mathbb{R}}^{d})},$

where $\frac{m+1}{2}\leq\sum_{j=1}^{m}\frac{1}{p_{j}}<\frac{2d+k}{2d}$ and
$1\leq p_{j}\leq 2$. This proves the quasi-Banach space estimates.

####  2.2.3. smoothing estimates

For the Sobolev regularity estimates, note that $\textsl{A}_{\Sigma}$ is a
positive operator, hence we have for positive $f_{1},\dots,f_{m}$

(2.14)
$\displaystyle\left|\textsl{A}_{\Sigma}(\mathrm{F})(x)\right|\leq\int_{\Sigma}f_{j}(x-y_{j})~{}\mathrm{d}\sigma_{\Sigma}(y)\times\prod_{\ell\not=j}\|f_{\ell}\|_{\infty}.$

Note that the first term in the right-hand side can be rewrite as

$\int_{{\mathbb{R}}^{d}}e^{2\pi
ix\cdot\xi_{j}}\widehat{\mathrm{d}\sigma_{\Sigma}}(\dots,0,\xi_{j},0,\dots)\widehat{f}_{j}(\xi_{j})~{}\mathrm{d}\xi_{j}.$

Due to the curvature condition, it follows that
$|\widehat{\mathrm{d}\sigma_{\Sigma}}(\xi)|\lesssim(1+|\xi|)^{-k/2}$, hence we
have

(2.15)
$\displaystyle\left\|\int_{\Sigma}f_{j}(x-y_{j})~{}\mathrm{d}\sigma_{\Sigma}(y)\right\|_{L^{2}({\mathbb{R}}^{d})}\lesssim\|f_{j}\|_{H^{-\frac{k}{2},2}({\mathbb{R}}^{d})}.$

Then by using (2.15), (2.14) and multilinear interpolation, we have

(2.16)
$\displaystyle\left\|\textsl{A}_{\Sigma}(\mathrm{F})\right\|_{L^{2}({\mathbb{R}}^{d})}\lesssim\prod_{j=1}^{m}\|f_{j}\|_{B_{p_{j}}^{-k/p_{j}}({\mathbb{R}}^{d})},$

where $\frac{1}{2}=\sum_{j=1}^{m}\frac{1}{p_{j}}$ and $p_{j}\geq 2$. This
proves the smoothing estimates.

##  3\. A Nonlinear Brascamp-Lieb inequality approach to $L^{p}$-improving
estimates for ${\mathcal{A}}_{\mathcal{S}}^{\Theta}$

###  3.1. Nonlinear Brascamp-Lieb inequality

Let $f_{j}$ be nonnegative integrable funtions,
$L_{j}:{\mathbb{R}}^{d}\to{\mathbb{R}}^{d_{j}}$ be linear surjections, and
$c_{j}\in[0,1]$ for $j=1,\cdots,m$. We also identify a finite dimensional
Hilbert sapce $H$ and a Euclidean space ${\mathbb{R}}^{n}$, for instance we
let $H={\mathbb{R}}^{d}$ and $H_{j}={\mathbb{R}}^{d_{j}}$. Then we can
consider the linear Brascamp-Lieb inequality we mean

(3.1)
$\displaystyle\int_{{\mathbb{R}}^{d}}\prod_{j=1}^{m}\Bigl{(}f_{j}(L_{j}x)\Bigr{)}^{c_{j}}\,~{}\mathrm{d}x\leq\mathrm{BL}(\mathbf{L},\mathbf{c})\prod_{j=1}^{m}\Bigl{(}\int_{{\mathbb{R}}^{d_{j}}}f_{j}(x_{j})~{}\mathrm{d}x_{j}\Bigr{)}^{c_{j}},$

where $\mathbf{L}=(H,\\{H_{j}\\}_{1\leq j\leq m},\\{L_{j}\\}_{1\leq j\leq
m})$, $\mathbf{c}=(c_{1},\dots,c_{m})$, and
$\mathrm{BL}(\mathbf{L},\mathbf{c})$ is the smallest such constant. Here, we
call $(\mathbf{L},\mathbf{c})$ a Brascamp-Lieb datum, and
$\mathrm{BL}(\mathbf{L},\mathbf{c})$ a Brascamp-Lieb constant. Under these
notions we introduce the nonlinear Brascamp-Lieb inequality in [5].

######  Theorem 3.1.

[5, Theorem 1.1] Let $(\mathbf{L},\mathbf{c})$ be a Brascamp-Lieb datum.
Suppose that $B_{j}:{\mathbb{R}}^{d}\to{\mathbb{R}}^{d_{j}}$ are $C^{2}$
submersions in a neighborhood of a point $x_{0}$ and
$~{}\mathrm{d}B_{j}(x_{0})=L_{j}$, $j=1,1\cdots m$. Then for each
$\varepsilon>0$ there exists a neighborhood $U$ of $x_{0}$ such that

(3.2)
$\displaystyle\int_{U}\prod_{j=1}^{m}\Bigl{(}f_{j}(B_{j}(x))\Bigr{)}^{c_{j}}\,~{}\mathrm{d}x\leq(1+\varepsilon)\mathrm{BL}(\mathbf{L},\mathbf{c})\prod_{j=1}^{m}\Bigl{(}\int_{{\mathbb{R}}^{d_{j}}}f_{j}(x_{j})~{}\mathrm{d}x_{j}\Bigr{)}^{c_{j}}.$

Note that $\mathrm{BL}(\mathbf{L},\mathbf{c})$ is finite if and only if the
following conditions hold.

(3.3) $\displaystyle\dim(V)$
$\displaystyle\leq\sum_{j=1}^{m}c_{j}\dim(L_{j}V)\quad\text{for all subspaces
$V$ of ${\mathbb{R}}^{d}$},$ (3.4) $\displaystyle d$
$\displaystyle=\sum_{j=1}^{m}c_{j}d_{j}.$

These conditions are called the transversality condition and the scaling
condition, respectively. We also present necessary conditions for finiteness
of $\mathrm{BL}(\mathbf{L},\mathbf{c})$:

(3.5)
$\displaystyle\bigcap_{j=1}^{m}\ker(L_{j})=\\{0\\},\quad\sum_{j=1}^{m}c_{j}\geq
1,\quad c_{j}\leq 1.$

But, it is not simple to check (3.3), (3.4) in direct manner. The following
lemma may be useful in such verification. First, we say a proper subspace
$V_{c}$ of ${\mathbb{R}}^{d}$ is a critical subspace if it satisfies

$\dim(V_{c})=\sum_{j=1}^{m}c_{j}\dim(L_{j}V_{c}).$

For a given subspace $V_{c}$, we split the Brascamp-Lieb datum into two parts,
$(\mathbf{L}_{V_{c}},\mathbf{c})$ and
$(\mathbf{L}_{V_{c}^{\perp}},\mathbf{c})$ as follows:

$\displaystyle\mathbf{L}_{V_{c}}$
$\displaystyle=(V_{c},\\{L_{j}V_{c}\\}_{1\leq j\leq
m},\\{L_{j,{V_{c}}}\\}_{1\leq j\leq m}),$
$\displaystyle\mathbf{L}_{V_{c}^{\perp}}$
$\displaystyle=(H/V,\\{H_{j}/(L_{j}V_{c})\\}_{1\leq j\leq
m},\\{L_{j,{H_{j}/V_{c}}}\\}_{1\leq j\leq m}),$

where $H/V_{c}=V_{c}^{\perp}$ and

$\displaystyle L_{j,V_{c}}$ $\displaystyle:V_{c}\to L_{j}V_{c},$
$\displaystyle L_{j,H_{j}/V_{c}}$ $\displaystyle:H_{j}/V_{c}\to
H/(L_{j}V_{c}).$

In this paper, we choose $H={\mathbb{R}}^{d}\times{\mathbb{R}}^{k}$ and
$H_{j}={\mathbb{R}}^{d}$.

######  Lemma 3.2 (Lemma 4.6, [4]).

Let $V_{c}$ be a critical subspace. Then $\mathrm{BL}(\mathbf{L},\mathbf{c})$
is finite if and only if $(\mathbf{L}_{V_{c}},\mathbf{c})$ and
$(\mathbf{L}_{V_{c}^{\perp}},\mathbf{c})$ satisfy (3.3) and (3.4) for any
subspace $V$ of $V_{c}$ and $V_{c}^{\perp}$, respectively.

Now we will prove Theorem 1.3. We first decompose $\mathcal{S}^{k}$ into a
finite cover $\\{\mathcal{S}_{\tau}^{k}\\}$ for which
$\mathcal{A}_{\mathcal{S}^{k}}(\mathrm{F})(x)$ can be written as a finite
summation of the following operators.

(3.6)
$\displaystyle\mathcal{A}_{\mathcal{S}_{\tau}^{k}}^{\Theta}(\mathrm{F})(x)=\int_{{\mathbb{R}}^{k}}\prod_{j=1}^{m}f_{j}(x+\Theta_{j}(y^{\prime},\Phi_{\tau}(y^{\prime}))\chi_{\tau}(y^{\prime})~{}\mathrm{d}y^{\prime},$

where $\Phi_{\tau}:{\mathbb{R}}^{k}\to{\mathbb{R}}^{d-k}$ is a
$C^{2}$-submersion and $\chi_{\tau}$ is a smooth cut-off function.

To simplify our proof, we consider a more general $m$-linear operator
$T_{K}^{\vec{B}}$. Suppose
$B_{j}:{\mathbb{R}}^{d}\times{\mathbb{R}}^{k}\to{\mathbb{R}}^{d}$ be $C^{2}$
submersions and $L_{j}=~{}\mathrm{d}B_{j}(0,0)$ with $j=1,\dots,m$. Then
$T_{K}^{\vec{B}}(\mathrm{F})(x)$ is given by

(3.7) $\displaystyle
T_{K}^{\vec{B}}(\mathrm{F})(x)=\int_{{\mathbb{R}}^{k}}\prod_{j=1}^{m}f_{j}(B_{j}(x,y))\,K(y)~{}\mathrm{d}y,\quad
y\in{\mathbb{R}}^{k},x\in{\mathbb{R}}^{d},$

where $K$ is a nonnegative bounded function supported in a ball
$B(0,\varepsilon)\subset{\mathbb{R}}^{k}$. Note that
$\mathcal{A}_{\mathcal{S}_{\tau}^{k}}^{\Theta}(\mathrm{F})$ is an example of
$T_{K}^{\vec{B}}$ for $K=\chi_{\tau}$ and
$B_{j}(x,y^{\prime})=x+\Theta_{j}(y^{\prime},\Phi_{\tau}(y^{\prime}))$ Then we
prove the following proposition.

######  Proposition 3.3.

Let $\frac{1}{p}=\sum_{j=1}^{m}\frac{1}{p_{j}}-\frac{k}{d}$, $1\leq
p\leq\frac{d}{d-k}$, $p_{j}\in[1,\infty)$, $j=1,\dots,m$ satisfy
$\sum_{j=1}^{m}\frac{1}{p_{j}}\geq 1$. Suppose $(\mathbf{L},\mathbf{p})$ is a
Brascamp-Lieb datum for

$\displaystyle\mathbf{L}$
$\displaystyle=({\mathbb{R}}^{d}\times{\mathbb{R}}^{k},\\{L_{j}\\}_{j=1}^{m+1},\\{{\mathbb{R}}^{d}\\}_{j=1}^{m+1}),\quad
L_{m+1}=~{}\mathrm{d}\pi_{{\mathbb{R}}^{d}},$ $\displaystyle\mathbf{p}$
$\displaystyle=\Big{(}\frac{1}{p_{1}},\dots,\frac{1}{p_{m}},\frac{1}{p^{\prime}}\Big{)}.$

Then we have

$\|T_{K}^{\vec{B}}(\mathrm{F})\|_{L^{p}({\mathbb{R}}^{d})}\lesssim\mathrm{BL}(\mathbf{L},\mathbf{p})\prod_{j=1}^{m}\|f_{j}\|_{L^{p_{j}}({\mathbb{R}}^{d})}.$

Proposition 3.3 states that the Brascamp-Lieb inequality implies an $L^{p}$
improving estimates.

###### Proof.

Since $p\geq 1$, we choose $g\in L^{p^{\prime}}(\mathbb{R}^{d})$ such that
$\|g\|_{L^{p^{\prime}}(\mathbb{R}^{d})}\leq 1$. By making use of duality,

$\displaystyle\|T_{K}^{\vec{B}}(\mathrm{F})\|_{L^{p}(\mathbb{R}^{d})}=\sup_{\|g\|_{p^{\prime}}\leq
1}\int_{\mathbb{R}^{d}}T_{K}^{\vec{B}}(\mathrm{F})(x)g(x)~{}\mathrm{d}x.$

Decompose ${\mathbb{R}}^{d}$ into countable union of unit cubes ${Q}_{\bf{n}}$
again and Apply Theorem 3.1 to $f_{j}^{p_{j}},g^{p^{\prime}}$ together with
additional mapping $L_{m+1}=~{}\mathrm{d}\pi_{{\mathbb{R}}^{d}}$. Then we have

$\displaystyle\int_{\mathbb{R}^{d}}T_{K}^{\vec{B}}(\mathrm{F})(x)g(x)~{}\mathrm{d}x$
$\displaystyle=\sum_{{\bf{n}}\in\mathbb{Z}^{d}}\int_{{Q}_{\bf{n}}}\int_{{\mathbb{R}}^{k}}\Big{(}\prod_{j=1}^{m}f_{j}(B_{j}(x,y))\Big{)}g(x)K(y)~{}\mathrm{d}y~{}\mathrm{d}x$
$\displaystyle\lesssim\mathrm{BL}(\mathbf{L},\mathbf{p})\sum_{{\bf{n}}\in\mathbb{Z}^{d}}\Big{(}\prod_{j=1}^{m}\|f_{j}\|_{L^{p_{j}}(\widetilde{Q}_{\bf{n}})}\Big{)}\|g\|_{L^{p^{\prime}}(\widetilde{Q}_{\bf{n}})}$

whenever

(3.8)
$\displaystyle\dim(V)\leq\sum_{j=1}^{m}\frac{\dim(dB_{j}(0_{x},0_{y})(V))}{p_{j}}+\frac{\dim(~{}\mathrm{d}\pi_{\mathbb{R}^{d}}(V))}{p^{\prime}}$

for every subspace $V$ of ${\mathbb{R}}^{d}\times{\mathbb{R}}^{k}$ together
with

(3.9) $\displaystyle\frac{k}{d}+\frac{1}{p}=\sum_{j=1}^{m}\frac{1}{p_{j}}.$

Since $g$ is chosen such that $\|g\|_{p^{\prime}}\leq 1$, we simply ignore
$\|g\|_{L^{p^{\prime}}(\widetilde{Q}_{\bf{n}})}$. For each $j=1,\cdots,m$, it
follows that by Hölder’s inequality

$\displaystyle\sum_{{\bf{n}}\in\mathbb{Z}^{d}}\|f_{j}\|_{L^{p_{j}}(\widetilde{Q}_{\bf{n}})}$
$\displaystyle\lesssim\Big{(}\sum_{{\bf{n}}\in\mathbb{Z}^{d}}\|f_{j}\|_{L^{p_{j}}(\widetilde{Q}_{\bf{n}})}^{m_{j}}\Big{)}^{\frac{1}{m_{j}}}.$

Plugging these in the above inequality, for $\sum_{j=1}^{m}\frac{1}{p_{j}}\geq
1$, $\sum_{j=1}^{m}\frac{1}{m_{j}}=1$ and $\frac{1}{m_{j}}\leq\frac{1}{p_{j}}$
we have

$\displaystyle\int_{\mathbb{R}^{d}}T_{K}^{\vec{B}}(\mathrm{F})(x)g(x)~{}\mathrm{d}x$
$\displaystyle\lesssim\mathrm{BL}(\mathbf{L},\mathbf{p})\prod_{j=1}^{m}\Big{(}\sum_{{\bf{n}}\in\mathbb{Z}^{d}}\|f_{j}\|_{L^{p_{j}}(\widetilde{Q}_{\bf{n}})}^{m_{j}}\Big{)}^{\frac{1}{m_{j}}}$
$\displaystyle\lesssim\mathrm{BL}(\mathbf{L},\mathbf{p})\prod_{j=1}^{m}\Big{(}\sum_{{\bf{n}}\in\mathbb{Z}^{d}}\|f_{j}\|_{L^{p_{j}}(\widetilde{Q}_{\bf{n}})}^{p_{j}}\Big{)}^{\frac{1}{p_{j}}}.$

Since $\widetilde{Q}_{\bf{n}}$ are finitely overlapped, taking supremum over
$\|g\|_{p^{\prime}}\leq 1$ gives

(3.10)
$\displaystyle\|T_{K}^{\vec{B}}(\mathrm{F})\|_{L^{p}(\mathbb{R}^{d})}\lesssim\mathrm{BL}(\mathbf{L},\mathbf{p})\prod_{j=1}^{m}\|f_{j}\|_{L^{p_{j}}(\mathbb{R}^{d})}.$

∎

###  3.2. Proof of Theorem 1.3

Since ${\mathcal{S}}^{k}$ is a $k$-dimensional $C^{2}$ surface lying in
${\mathbb{R}}^{d}$, there is a $C^{2}$ mapping
$\Phi:{\mathbb{R}}^{k}\to{\mathbb{R}}^{d_{c}}$ for $d_{c}=d-k$ such that
${\mathcal{S}}^{k}$ is locally a graph
$\\{(y^{\prime},\Phi(y^{\prime}))\in{\mathbb{R}}^{d}\\}$. Then in Proposition
3.3 we let $B_{j}(x,y^{\prime})=x+\Theta_{j}(y^{\prime},\Phi(y^{\prime}))$ for
$\Phi=(\phi^{1},\dots,\phi^{d_{c}})$, $j=1,\dots,m$. For $j=m+1$, let
$B_{m+1}=\pi_{{\mathbb{R}}^{d}}$, where
$\pi_{{\mathbb{R}}^{d}}:{\mathbb{R}}^{d+k}\to{\mathbb{R}}^{d}$ is a projection
onto ${\mathbb{R}}^{d}$. For $j=1,\dots,m$, we define
$L_{j}:=~{}\mathrm{d}B_{j}(0,0)$, which is given by

(3.11)
$\displaystyle\begin{bmatrix}I_{d}&\Theta_{j}\nabla|_{y^{\prime}=0}\begin{bmatrix}y^{\prime}_{1}\\\
\vdots\\\ y^{\prime}_{k}\\\ \phi^{1}(y^{\prime})\\\ \vdots\\\
\phi^{d-k}(y^{\prime})\end{bmatrix}\end{bmatrix}=\begin{bmatrix}I_{d}&\Theta_{j}\begin{bmatrix}1&0&\dots&0\\\
0&1&\dots&0\\\ \vdots&\vdots&\ddots&\vdots\\\ 0&0&\dots&1\\\
\phi_{y^{\prime}_{1}}^{1}(0)&\phi_{y^{\prime}_{2}}^{1}(0)&\dots&\phi_{y^{\prime}_{k}}^{1}(0)\\\
\vdots&\vdots&\ddots&\vdots\\\
\phi_{y^{\prime}_{1}}^{d_{c}}(0)&\phi_{y^{\prime}_{2}}^{d_{c}}(0)&\dots&\phi_{y^{\prime}_{k}}^{d_{c}}(0)\end{bmatrix}\end{bmatrix},$

where $I_{d}$ denotes the $d\times d$ identity matrix. If there is no
confusion, we simply write

(3.12) $\displaystyle
L_{j}=\begin{bmatrix}I_{d}&\Theta_{j}\begin{bmatrix}I_{k}\\\
~{}\mathrm{d}\Phi(0)\end{bmatrix}\end{bmatrix},\quad~{}\mathrm{d}\Phi=(~{}\mathrm{d}\phi^{1},\dots,~{}\mathrm{d}\phi^{d_{c}}).$

Without loss of generality we assume that $~{}\mathrm{d}\Phi(0)$ is a
$d_{c}\times k$ zero matrix. That is, we have for $j=1,\dots,m$

(3.13) $\displaystyle
L_{j}=\begin{bmatrix}I_{d}&\Theta_{j}^{1}\end{bmatrix},\quad\Theta_{j}=\begin{bmatrix}\Theta_{j}^{1}&\Theta_{j}^{2}\end{bmatrix},$

where $\Theta_{j}^{1}$ and $\Theta_{j}^{2}$ are $d\times k$ and $d\times
d_{c}$ matrices, respectively. In the case of $j=m+1$, we have

(3.14)
$\displaystyle~{}\mathrm{d}\pi_{{\mathbb{R}}^{d}}=\begin{bmatrix}I_{d}&0_{d}\end{bmatrix},$

where $0_{d}$ means all $d\times d$ elements are zero. We show that
$\mathbf{L}=(L_{1},\dots,L_{m},~{}\mathrm{d}\pi_{{\mathbb{R}}^{d}})$ and
$\mathbf{p}=(\frac{1}{m},\dots,\frac{1}{m},\frac{k}{d})$ are Brascamp-Lieb
data by making use of Lemma 3.2.

Let $K_{\pi}=\ker(\pi_{{\mathbb{R}}^{d}})$ so that
$K_{\pi}=\\{(0,y^{\prime})\in{\mathbb{R}}^{d}\times{\mathbb{R}}^{k}:y^{\prime}\in{\mathbb{R}}^{k}\\}$,
which is $k$-dimensional. Then it is clear that
$K_{\pi}^{\perp}=\\{(x,0)\in{\mathbb{R}}^{d}\times{\mathbb{R}}^{k}:x\in{\mathbb{R}}^{d}\\}$.
For $K_{\pi}$ we have

$\displaystyle\sum_{j=1}^{m}\frac{\dim(L_{j}K_{\pi})}{p_{j}}+\frac{\dim(\pi_{{\mathbb{R}}^{d}}K_{\pi})}{p^{\prime}}=\sum_{j=1}^{m}\frac{k}{p_{j}}=k=\dim(K_{\pi}).$

Since $\dim(L_{j}K)$ equals to $\dim(K)$ for any subspace $K$ of $K_{\pi}$
with $j=1,\dots,m$, we also have

$\displaystyle\sum_{j=1}^{m}\frac{\dim(L_{j}K)}{p_{j}}+\frac{\dim(\pi_{{\mathbb{R}}^{d}}K)}{p^{\prime}}=\sum_{j=1}^{m}\frac{\dim(K)}{p_{j}}=\dim(K).$

Thus $K_{\pi}$ is a critical subspace and $(\mathbf{L}_{K_{\pi}},\mathbf{p})$
is a Brascamp-Lieb datum.

On the other hand, for
$K_{\pi}^{\perp}=\\{(x,0)\in{\mathbb{R}}^{d}\times{\mathbb{R}}^{k}:x\in{\mathbb{R}}^{d}\\}$
we consider $(\mathbf{L}_{K_{\pi}^{\perp}},\mathbf{p})$

$\displaystyle\mathbf{L}_{K_{\pi}^{\perp}}$
$\displaystyle=(K_{\pi}^{\perp},\\{{\mathbb{R}}^{d}/(L_{j}K_{\pi})\\}_{1\leq
j\leq m+1},\\{L_{j,{K_{\pi}^{\perp}}}\\}_{1\leq j\leq m+1}),$
$\displaystyle\mathbf{p}$
$\displaystyle=\Big{(}\frac{1}{p_{1}},\dots,\frac{1}{p_{m}},\frac{1}{p^{\prime}}\Big{)}=\Big{(}\frac{1}{m},\dots,\frac{1}{m},\frac{k}{d}\Big{)}.$

Note that $\pi_{{\mathbb{R}}^{d},K_{\pi}^{\perp}}=\pi_{{\mathbb{R}}^{d}}$.
Then we have

$\displaystyle\sum_{j=1}^{m}\frac{\dim(L_{j,K_{\pi}^{\perp}}K_{\pi}^{\perp})}{p_{j}}+\frac{\dim(\pi_{{\mathbb{R}}^{d}}K_{\pi}^{\perp})}{p^{\prime}}$
$\displaystyle=\sum_{j=1}^{m}\frac{d-k}{p_{j}}+\frac{d}{p^{\prime}}$
$\displaystyle=d-k+\frac{dk}{d}=d=\dim(K_{\pi}^{\perp}).$

Now it remains to show $(\mathbf{L},\mathbf{p})$ is a Brascamp-Lieb data. In
order to show this, we consider a subspace $K$ of $K_{\pi}^{\perp}$ whose
dimension $d_{K}$ satisfies $d>d_{K}\geq k$ or $k>d_{K}\geq 1$, where $d_{K}$
denotes the dimension of $K$.

####  3.2.1. $d>d_{K}\geq k$

When $d>d_{K}\geq k$, we can check that there is $K$ whose image under
$L_{\mu}$ contains $L_{\mu}K_{\pi}$ as a subspace for $\mu=1,\dots,m$. Indeed,
it follows that

$\displaystyle L_{\mu}(x,0)$ $\displaystyle=x,\quad\forall(x,0)\in K,$
$\displaystyle L_{\mu}(0,y^{\prime})$
$\displaystyle=\Theta_{\mu}^{\prime}y^{\prime},\quad\forall(0,y^{\prime})\in
K_{\pi}.$

Then
$K_{\mu}:=\\{(\Theta_{\mu}^{1}y^{\prime},0)\in{\mathbb{R}}^{d}\times{\mathbb{R}}^{k}:y^{\prime}\in{\mathbb{R}}^{k}\\}$
is a $k$ dimensional subspace of $K_{\pi}^{\perp}$ and equals to
$L_{\mu}K_{\pi}$. Since our choice of $\\{\Theta_{j}\\}$ allows us to have
$K_{j}\cap K_{j^{\prime}}$ is at most $k-1$ dimensional whenever
$j\not=j^{\prime}$, we have

$\displaystyle\dim(L_{\mu,K_{\pi}^{\perp}}K)$ $\displaystyle\geq d_{K}-k,$
$\displaystyle\dim(L_{j,K_{\pi}^{\perp}}K)$ $\displaystyle\geq d_{K}-k+1,\quad
j\not=\mu.$

Hence it follows that

$\displaystyle\sum_{j=1}^{m}\frac{\dim(L_{j,K_{\pi}^{\perp}}K)}{p_{j}}+\frac{\dim(\pi_{{\mathbb{R}}^{d}}K)}{p^{\prime}}$
$\displaystyle\geq\sum_{j\not=\mu}\frac{d_{K}-k+1}{m}+\frac{d_{K}-k}{m}+\frac{k}{d}d_{K}$
$\displaystyle=\frac{(m-1)(d_{K}-k+1)}{m}+\frac{d_{K}-k}{m}+\frac{k}{d}d_{K}$
$\displaystyle=d_{K}-k+\frac{m-1}{m}+\frac{k}{d}d_{K}$ $\displaystyle\geq
d_{K}+\frac{m-1}{m}-\frac{d-k}{d}k.$

Note that the last line is greater than or equal to $d_{K}$ whenever

(3.15) $\displaystyle\frac{m-1}{m}\geq\frac{d-k}{d}k.$

If $k=1$ or $d-1$, then (3.15) is reduced to $m\geq d$. So far, it suffices to
choose linearly independent $\\{\Theta_{j}\\}$.

####  3.2.2. $k>d_{K}\geq 1$

In this case, we begin with $k-1$ dimensional subspace of $K_{j}$ for some
$j=1,\dots,m$. We assume there is $\mu,\nu=1,\dots,m$ such that $\mu\not=\nu$
and $K=K_{\mu}\cap K_{\nu}$ is $k-1$ dimensional. Note that we have
$\dim(K\cap K_{j})\geq 1$ for $j\not=\mu,\nu$. Thus it follows that

$\displaystyle\sum_{j=1}^{m}\frac{\dim(L_{j,K_{\pi}^{\perp}}K)}{p_{j}}+\frac{\dim(\pi_{{\mathbb{R}}^{d}}K)}{p^{\prime}}$
$\displaystyle\geq\sum_{j\not=\mu,\nu}\frac{1}{m}+\frac{k}{d}(k-1)$
$\displaystyle=\frac{(m-2)}{m}+\frac{k}{d}(k-1)$
$\displaystyle=(k-1)+\frac{(m-2)}{m}-\frac{d-k}{d}(k-1).$

Note that the last line is greater than or equal to $k-1$ whenever

(3.16) $\displaystyle\frac{(m-2)}{m}\geq\frac{d-k}{d}(k-1).$

If $k=2$ or $d-1$, then (3.16) is reduced to $m\geq d$. For an arbitrary
$k-\ell$ dimensional subspace $K$, one can check that $K$ is contained in
$K_{j_{i}}$ for $j_{1},\dots,j_{\ell}$. Thus we have

$\displaystyle\sum_{j=1}^{m}\frac{\dim(L_{j,K_{\pi}^{\perp}}K)}{p_{j}}+\frac{\dim(\pi_{{\mathbb{R}}^{d}}K)}{p^{\prime}}$
$\displaystyle\geq\sum_{j\not=\mu,j_{1},\dots,j_{\ell}}\frac{1}{m}+\frac{k}{d}(k-\ell)$
$\displaystyle=\frac{(m-\ell-1)}{m}+\frac{k}{d}(k-\ell)$
$\displaystyle=(k-\ell)+\frac{(m-\ell-1)}{m}-\frac{d-k}{d}(k-\ell).$

Then the last line is greater than or equal to $k-\ell$ whenever

(3.17) $\displaystyle\frac{(m-\ell-1)}{m}\geq\frac{d-k}{d}(k-\ell),$

which is equivalent to $m\geq d$ when $k=d-1$ or $k=\ell+1$. In conclusion,
$\mathrm{BL}(\mathbf{L},\mathbf{p})$ is finite for given data
$(\mathbf{L},\mathbf{p})$ whenever
$\frac{m-\ell-1}{m}\geq\frac{d-k}{d}(k-\ell)$ for all $0\leq\ell\leq k-1$.
Note that we can rewrite the condition as

(3.18)
$\displaystyle\frac{m-1}{m}\geq\frac{d-k}{d}k-\Big{(}\frac{d-k}{d}-\frac{1}{m}\Big{)}\ell,\quad
0\leq\ell\leq k-1.$

Note that (3.18) is reduced to $\frac{m-1}{m}\geq\frac{d-k}{d}k$. Thus,
$(\mathbf{L},\mathbf{p})$ is a Brascamp-Lieb data. Hence, by Proposition 3.3
it proves Theorem 1.3.

##  4\. Proof of Theroem 1.5

Recall that the lacunary maximal function $\mathcal{M}^{\Theta}_{\mathcal{S}}$
is defined by

(4.1)
$\displaystyle\mathcal{M}^{\Theta}_{\mathcal{S}}(\mathrm{F})(x)=\sup_{\ell\in\mathbb{Z}}\Big{|}\int_{\mathcal{S}}\prod_{j=1}f_{j}(x-2^{-\ell}\Theta_{j}y)~{}~{}\mathrm{d}\sigma(y)\Big{|},$

where $\mathcal{S}$ has $k$-nonvanishing principal curvatures and
$\Theta=\\{\Theta_{j}\\}$ is a family of linearly independent rotation
matrices.

Observe that for any fixed $\ell\in\mathbb{Z}$, we can write the identity
operator $I$ as follows

(4.2) $\displaystyle
I=P_{<\ell}+\sum^{\infty}_{n=0}P_{\ell+n}=P_{<\ell}+P_{\ell\leq}.$

Then we have

(4.3) $\displaystyle\prod_{j=1}^{m}f_{j}$
$\displaystyle=\prod_{j=1}^{m}\Big{(}P_{<\ell}f_{j}+P_{\ell\leq}f_{j}\Big{)}$
$\displaystyle\sim\sum_{\alpha=0}^{m}\Bigl{(}\prod_{\mu=1}^{\alpha}P_{<\ell}f_{\mu}\Bigr{)}\Bigl{(}\prod_{\nu=\alpha+1}^{m}P_{\ell\leq}f_{\nu}\Bigr{)}.$

For
${\bf{n}}=(n_{1},\cdots,n_{m})\in\mathbb{N}_{0}^{m}=(\mathbb{N}\cup\\{0\\})^{m}$,
we define

(4.4) $\displaystyle\mathcal{A}_{\ell}^{\alpha}(\mathrm{F})(x)$
$\displaystyle:=\int_{\mathcal{S}}\Bigl{(}\prod_{\mu=1}^{\alpha}P_{<\ell}f_{\mu}(x-2^{-\ell}\Theta_{\mu}y)\Bigr{)}\Bigl{(}\prod_{\nu=\alpha+1}^{m}f_{\nu}(x-2^{-\ell}\Theta_{\nu}y)\Bigr{)}~{}~{}\mathrm{d}\sigma(y),$
(4.5) $\displaystyle\mathcal{M}_{\bf{n}}(\mathrm{F})$
$\displaystyle:=\sup_{\ell\in\mathbb{Z}}\Big{|}\int_{\mathcal{S}}\prod_{j=1}^{m}P_{\ell+n_{j}}f_{j}(x-2^{-\ell}\Theta_{j}y)~{}~{}\mathrm{d}\sigma(y)\Big{|},$
(4.6) $\displaystyle\mathcal{S}_{\bf{n}}(\mathrm{F})$
$\displaystyle:=\sum_{\ell\in\mathbb{Z}}\Big{|}\int_{\mathcal{S}}\prod_{j=1}^{m}P_{\ell+n_{j}}f_{j}(x-2^{-\ell}\Theta_{j}y)~{}~{}\mathrm{d}\sigma(y)\Big{|}.$

Therefore, the lacunary maximal function $\mathcal{M}^{\Theta}_{\mathcal{S}}$
can be controlled by a constant mutiple of

(4.7)
$\displaystyle\sum_{\alpha=1}^{m}\sup_{\ell\in{\mathbb{Z}}}|\mathcal{A}_{\ell}^{\alpha}(\mathrm{F})|+\sum_{\mathbf{n}\in\mathbb{N}_{0}^{m}}\mathcal{M}_{\mathbf{n}}(\mathrm{F}).$

Then the proof will be completed by combination of the following lemmas and
induction on $m$-linearity:

######  Lemma 4.1.

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

$\mathcal{A}_{\ell}^{\alpha}(\mathrm{F})(x)\leq M_{HL}(f_{1})(x)\times
M_{\mathcal{S}}(f_{2})(x),$

where $\mathrm{F}=(f_{1},f_{2})$ and
$M_{\mathcal{S}}(f)(x)=\sup_{\ell\in{\mathbb{Z}}}|\int_{\mathcal{S}}f(x-2^{-\ell}y)~{}~{}\mathrm{d}\sigma(y)|$.

###### Proof.

It suffices to show $\sup_{y\in\mathcal{S}}|P_{<\ell}f(x-2^{-\ell}y)|\lesssim
M_{HL}(f)(x)$. Since $\phi_{\ell}(x)=2^{\ell d}\phi(2^{\ell}x)$, we have

$\displaystyle P_{<\ell}f(x-2^{-\ell}y)$
$\displaystyle=\int_{{\mathbb{R}}^{d}}f(z)2^{\ell
d}\phi(2^{\ell}(x-2^{-\ell}y-z))~{}~{}\mathrm{d}z$
$\displaystyle=\int_{{\mathbb{R}}^{d}}f(x+2^{-\ell}z)\phi(y-z)~{}~{}\mathrm{d}z.$

Since $y$ is contained in a compact surface $\mathcal{S}$, we have for any
$N>0$

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

∎

Since $M_{HL},M_{\mathcal{S}}$ are bounded on $L^{p}$ for $p\in(1,\infty]$, we
need to handle the summation of $M_{\mathbf{n}}$ over
$\mathbf{n}\in\mathbb{N}_{0}^{m}$.

######  Lemma 4.2.

Let $\mathbf{n}\in\mathbb{N}^{m}$ and $\frac{1}{p}=\frac{2(k+1)}{k+2}$. Then
for $(\frac{1}{p_{1}},\cdots,\frac{1}{p_{m}})\in\text{conv}(\mathcal{V}_{k})$
with $\sum_{j=1}^{m}\frac{1}{p_{j}}=\frac{2(k+1)}{k+2}$, we have

$\|\mathcal{M}_{\mathbf{n}}(\mathrm{F})\|_{L^{p,\infty}}\leq
C(1+|\mathbf{n}|^{m})\prod_{j=1}^{m}\|f_{j}\|_{L^{p_{j}}}.$

In particular, we have $\frac{1}{p}=\frac{2d}{d+1}$ when we consider averages
over $\mathcal{S}=\mathbb{S}^{d-1}$.

######  Lemma 4.3.

Let $\mathbf{n}\in\mathbb{N}^{m}$ and $1=\sum_{j=1}^{m}\frac{1}{r_{j}}$. Then
we have

$\|\mathcal{S}_{\mathbf{n}}(\mathrm{F})\|_{L^{1}}\lesssim
2^{-\delta|\mathbf{n}|}\prod_{j=1}^{m}\|f_{j}\|_{L^{r_{j}}}.$

Proofs of Lemmas 4.2 and 4.3 will be given in Section 5. Since
$\mathcal{M}_{\mathbf{n}}\leq\mathcal{S}_{\mathbf{n}}$ by definition, it
follows from interpolation between Lemmas 4.2 and 4.3 that

(4.8) $\displaystyle\|\mathcal{M}_{\mathbf{n}}(\mathrm{F})\|_{L^{p}}\lesssim
2^{-\delta^{\prime}|\mathbf{n}|}\prod_{j=1}\|f_{j}\|_{L^{pm}},\quad\delta^{\prime}>0,$

whenever $\frac{1}{p}<\frac{2(k+1)}{k+2}$. Since
$2^{-\delta^{\prime}|\mathbf{n}|}$ is summable over
$\mathbf{n}\in\mathbb{N}_{0}^{m}$, this proves the theorem for $m=2$ case.

For the induction, we assume that Theorem 1.5 holds for $N$-linear operators
with $N=2,\cdots,m-1$. Note that we already show $m=2$-case holds. By the
assumption, we have the following lemma:

######  Lemma 4.4.

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

$\mathcal{A}_{\ell}^{\alpha}(\mathrm{F})(x)\lesssim\prod_{\mu=1}^{\alpha}M_{HL}(f_{\mu})(x)\times\sup_{\ell\in{\mathbb{Z}}}\Big{|}\int_{\mathcal{S}}\prod_{\nu=\alpha+1}^{m}f_{\nu}(x-2^{-\ell}\Theta_{\nu}y)~{}~{}\mathrm{d}\sigma(y)\Big{|}.$

Moreover,

$\sup_{\ell\in{\mathbb{Z}}}\Big{|}\int_{\mathcal{S}}\prod_{\nu=\alpha+1}^{m}f_{\nu}(x-2^{-\ell}\Theta_{\nu}y)~{}~{}\mathrm{d}\sigma(y)\Big{|}$

satisfies multilinear estimates of Theorem 1.5.

Note that the assertions of the lemma follows directly by the proof of Lemma
4.1 and the assumption, respectively. Since we already proved Lemmas 4.2 and
4.3 for general $m$, Theorem 1.5 holds under the assumption that $N$ cases
hold for $N=2,\dots,m-1$. This closes the induction hence proves the theorem.

##  5\. Proofs of Lemmas 4.2 and 4.3

###  5.1. Proof of Lemma 4.2

For the $L^{p,\infty}$-estimate of Lemma 4.2, we assume $\|f_{j}\|_{p_{j}}=1$
and show the following inequality:

(5.1)
$\displaystyle\mathrm{meas}\Bigl{(}\\{x:\mathcal{M}_{\mathbf{n}}(\mathrm{F})(x)>\lambda\\}\Bigr{)}\lesssim|\mathbf{n}|^{m}\lambda^{-p}.$

To obtain (5.1), we exploit the approach of Chirst and Zhou [10], which is
based on the multilinear Calderón-Zygmund decomposition. We apply the
Calderón-Zygmund decomposition at height $C\lambda^{\frac{p}{p_{j}}}$ to each
$f_{j}$, $j=1,\dots,m$ for some $C>0$ so that for each $j$ we have
$f_{j}=g_{j}+b_{j}$ such that

(5.2) $\displaystyle\|g_{j}\|_{\infty}\leq C\lambda^{\frac{p}{p_{j}}},$ (5.3)
$\displaystyle
b_{j}=\sum_{\gamma}b_{j,\gamma},\quad\operatorname{supp}(b_{j,\gamma})\subset
Q_{j,\gamma},$ (5.4)
$\displaystyle\|b_{j,\gamma}\|_{L^{p_{j}}}^{p_{j}}\lesssim\lambda^{p}\mathrm{meas}(Q_{j,\gamma}),\quad\sum_{\gamma}\mathrm{meas}(Q_{j,\gamma})\lesssim\lambda^{-p},$
(5.5) $\displaystyle\int b_{j,\gamma}=0.$

Note that $Q_{j,\gamma}$ denotes a dyadic cube. Then we have

$\displaystyle\mathrm{meas}\Bigl{(}\\{x:\mathcal{M}_{\mathbf{n}}(\mathrm{F})(x)>\lambda\\}\Bigr{)}\lesssim$
$\displaystyle\mathrm{meas}\Bigl{(}\\{x:\mathcal{M}_{\mathbf{n}}(g_{1},\dots,g_{m})(x)>2^{-m}\lambda\\}\Bigr{)}$
$\displaystyle+\mathrm{meas}\Bigl{(}\\{x:\mathcal{M}_{\mathbf{n}}(g_{1},\dots,g_{m-1},b_{m})(x)>2^{-m}\lambda\\}\Bigr{)}$
$\displaystyle+\cdots+\mathrm{meas}\Bigl{(}\\{x:\mathcal{M}_{\mathbf{n}}(b_{1},\dots,b_{m})(x)>2^{-m}\lambda\\}\Bigr{)}.$

For $C_{\mathcal{S}}=5\max(1,\mathop{\rm diam}(\mathcal{S}))$ we define
$\mathcal{E}=\cup_{j=1}^{m}\cup_{\gamma}C_{\mathcal{S}}Q_{j,\gamma}$ so that
$\mathrm{meas}(\mathcal{E})\lesssim\lambda^{-p}$. Note that $C_{\mathcal{S}}Q$
is a cube whose side-length is $C_{\mathcal{S}}$ times of that of $Q$ with the
same center as $Q$. Thus we estimate each levelset for
$x\in{\mathbb{R}}^{d}\setminus\mathcal{E}$.

####  5.1.1. Estimates for $\mathcal{M}_{\mathbf{n}}(b_{1},\dots,b_{m})$

Let $b_{j}^{i}=\sum_{\gamma:s(Q_{j,\gamma})=2^{-i}}b_{j,\gamma}$ where $s(Q)$
denotes a side length of $Q$. Then
$|\mathcal{M}_{\mathbf{n}}(b_{1},\dots,b_{m})|^{p}$ with
$p=\frac{k+2}{2(k+1)}$ is bounded by

(5.6)
$\displaystyle\sum_{i_{1},\dots,i_{m}\in{\mathbb{Z}}}\sum_{\ell\in{\mathbb{Z}}}|\mathcal{A}_{\ell}^{\mathbf{n}}(b_{1}^{i_{1}},\dots,b_{m}^{i_{m}})|^{p},$

where

$\mathcal{A}_{\ell}^{\mathbf{n}}(f_{1},\dots,f_{m})(x)=\int_{\mathcal{S}}\prod_{j=1}^{m}P_{n_{j}+\ell}f_{j}(x-2^{-\ell}\Theta_{j}y)~{}~{}\mathrm{d}\sigma(y).$

To proceed further, we introduce the following lemmas whose proofs will be
given in the last part of this subsection:

######  Lemma 5.1.

For $(\frac{1}{p_{1}},\dots,\frac{1}{p_{m}})\in\text{conv}(\mathcal{V}_{k})$
with $\frac{1}{p}=\sum_{j=1}^{m}\frac{1}{p_{j}}$, we have

$\|\mathcal{A}_{\ell}^{\mathbf{n}}(b_{1}^{i_{1}},\dots,b_{m}^{i_{m}})\|_{L^{p}({\mathbb{R}}^{d}\setminus\mathcal{E})}^{p}\lesssim\min_{j=1,\dots,m}\min(1,2^{(n_{j}+\ell-
i_{j})(1+\frac{d}{p_{j}^{\prime}})},2^{i_{j}-\ell})\prod_{j=1}^{m}\|b_{j}^{i_{j}}\|_{p_{j}}^{p}.$

For $m=2$ and $p=\frac{1}{2}$, Lemma 5.1 is given in [10]. The proof for
general $m\geq 2$ and $p=\frac{k+2}{2(k+1)}$ case is given in similar manner.

######  Lemma 5.2.

Under the same condition in Lemma 5.1,

$\sum_{\ell\in{\mathbb{Z}}}\min_{i_{1},\dots,i_{m}}\min(1,2^{(n_{j}+\ell-
i_{j})(1+\frac{d}{p_{j}^{\prime}})},2^{i_{j}-\ell})\lesssim|\mathbf{n}|\prod_{j,j^{\prime},j\not=j^{\prime}}\min(1,2^{|\mathbf{n}|-|i_{j}-i_{j^{\prime}}|})^{\frac{1}{m(m-1)}}.$

By using Lemmas 5.1 and 5.2, we have

$\displaystyle\|\mathcal{M}_{\mathbf{n}}(b_{1},\dots,b_{m})\|_{L^{p}({\mathbb{R}}^{d}\setminus\mathcal{E})}^{p}$
$\displaystyle\lesssim\sum_{i_{1}\in{\mathbb{Z}}}\cdots\sum_{i_{m}\in{\mathbb{Z}}}\sum_{\ell\in{\mathbb{Z}}}\|\mathcal{A}_{\ell}^{\mathbf{n}}(b_{1}^{i_{1}},\dots,b_{m}^{i_{m}})\|_{L^{p}({\mathbb{R}}^{d}\setminus\mathcal{E})}^{p}$
$\displaystyle\lesssim\sum_{i_{1},\dots,i_{m}}|\mathbf{n}|\prod_{j,j^{\prime},j\not=j^{\prime}}\min(1,2^{|\mathbf{n}|-|i_{j}-i_{j^{\prime}}|})^{\frac{p}{m(m-1)}}\prod_{j=1}^{m}\|b_{j}^{i_{j}}\|_{p_{j}}^{p}.$

We apply Hölder’s inequality to the last line and obtain

$\displaystyle\|\mathcal{M}_{\mathbf{n}}(b_{1},\dots,b_{m})\|_{L^{p}({\mathbb{R}}^{d}\setminus\mathcal{E})}^{p}\lesssim|\mathbf{n}|\prod_{j=1}^{m}\Bigl{(}\sum_{i_{1},\dots,i_{m}}\prod_{l\not=j}\min(1,2^{|\mathbf{n}|-|i_{j}-i_{l}|})^{\frac{p_{j}}{m(m-1)}}~{}\|b_{j}^{i_{j}}\|_{p_{j}}^{p_{j}}\Bigr{)}^{\frac{p}{p_{j}}}.$

Oberserve that the summation over $i_{1},\dots,i_{j-1},i_{j+1},\dots,i_{m}$
yields

$\sum_{i_{j}\in{\mathbb{Z}}}|\mathbf{n}|^{m-1}\|b_{j}^{i_{j}}\|_{p_{j}}^{p_{j}}.$

It is because we have
$\sum_{i_{l}}\min(1,2^{|\mathbf{n}|-|i_{j}-i_{l}|})^{\frac{p_{j}}{m(m-1)}}\lesssim|\mathbf{n}|$
since

$\displaystyle\min(1,2^{|\mathbf{n}|-|i_{j}-i_{l}|})=\left\\{\begin{array}[]{lll}2^{|\mathbf{n}|+i_{j}-i_{l}},&i_{l}>i_{j}+|\mathbf{n}|;\\\
1,&i_{j}-|\mathbf{n}|\leq i_{l}\leq i_{j}+|\mathbf{n}|;\\\
2^{|\mathbf{n}|-i_{j}+i_{l}},&i_{l}<i_{j}-|\mathbf{n}|.\end{array}\right.$

Therefore
$\|\mathcal{M}_{\mathbf{n}}(b_{1},\dots,b_{m})\|_{L^{p}({\mathbb{R}}^{d}\setminus\mathcal{E})}$
is bounded by a constant multiple of

(5.8)
$\displaystyle|\mathbf{n}|^{\frac{m}{p}}\prod_{j=1}^{m}\Bigl{(}\sum_{i_{j}\in{\mathbb{Z}}}\|b_{j}^{i_{j}}\|_{p_{j}}^{p_{j}}\Bigr{)}^{\frac{1}{p_{j}}}=|\mathbf{n}|^{\frac{m}{p}}\prod_{j=1}^{m}\|b_{j}\|_{p_{j}}.$

With help of (5.8), we finally estimate the levelset of
$\mathcal{M}_{\mathbf{n}}(b_{1},\dots,b_{m})$

$\displaystyle\mathrm{meas}\Bigl{(}\\{x:\mathcal{M}_{\mathbf{n}}(b_{1},\dots,b_{m})(x)>2^{-m}\lambda\\}\Bigr{)}$
$\displaystyle\lesssim\lambda^{-p}\|\mathcal{M}_{\mathbf{n}}(b_{1},\dots,b_{m})\|_{L^{p}({\mathbb{R}}^{d}\setminus\mathcal{E})}^{p}$
$\displaystyle\leq\lambda^{-p}|\mathbf{n}|^{m}\prod_{j=1}^{m}\|b_{j}\|_{p_{j}}^{p}.$

Since $\|b_{j}\|_{p_{j}}\lesssim 1$, we obtain (5.1) for $b_{1},\dots,b_{m}$.

####  5.1.2. Estimates for other terms

The cases $\mathcal{M}_{\mathbf{n}}(g_{1},\dots,g_{m})$,
$\mathcal{M}_{\mathbf{n}}(g_{1},\dots,g_{m-1},b_{m})$, $\dots$,
$\mathcal{M}_{\mathbf{n}}(b_{1},\dots,b_{m-1},g_{m})$ follow from simplified
arguments given in 5.1.1. We first consider the cases except for
$\mathcal{M}_{\mathbf{n}}(g_{1},\dots,g_{m})$. Thus we define for
$\alpha+\beta=m$ and $1\leq\alpha,\beta\leq m-1$

(5.9)
$\displaystyle\mathcal{M}_{\mathbf{n}}(\mathbf{g}^{\alpha},\mathbf{b}^{\beta})=\mathcal{M}_{\mathbf{n}}(g_{1},\dots,g_{\alpha},b_{\alpha+1},\dots,b_{m}).$

Note that one can modify the proofs of Lemmas 5.1, 5.2 to obtain
$\mathbf{b}^{\beta}$-analogue. Let
$(0,\dots,0,\frac{1}{r_{\alpha+1}},\dots,\frac{1}{r_{m}})\in\text{conv}(\mathcal{V}_{k})$
with $\frac{1}{p}=\sum_{\nu=\alpha+1}^{m}\frac{1}{r_{\nu}}$. Then the proofs
of Lemma 5.1 yields that

(5.10)
$\displaystyle\|\mathcal{A}_{\ell}^{\mathbf{n}}(\mathbf{g}^{\alpha},\mathbf{b}^{\beta})\|_{L^{p}({\mathbb{R}}^{d}\setminus\mathcal{E})}$
$\displaystyle\lesssim\prod_{\mu=1}^{\alpha}\|g_{\mu}\|_{L^{\infty}}\min_{\nu=\alpha+1,\dots,m}\min(1,2^{(n_{\nu}+\ell-
i_{\nu})(1+\frac{d}{r_{\nu}^{\prime}})},2^{i_{\nu}-\ell})\prod_{\nu=\alpha+1}^{m}\|b_{\nu}^{i_{\nu}}\|_{r_{\nu}}.$

We have

(5.11)
$\displaystyle\sum_{\ell\in{\mathbb{Z}}}\min_{i_{\alpha+1},\dots,i_{m}}\min(1,2^{(n_{\nu}+\ell-
i_{\nu})(1+\frac{d}{r_{\nu}^{\prime}})},2^{i_{\nu}-\ell})$
$\displaystyle\lesssim|\mathbf{n}|\prod_{\alpha+1\leq\nu,\nu^{\prime}\leq
m,j\not=j^{\prime}}\min(1,2^{|\mathbf{n}|-|i_{\nu}-i_{\nu^{\prime}}|})^{\frac{1}{\beta(\beta-1)}}.$

With help of (5.10) and (5.11), we estimate
$\|\mathcal{M}_{\mathbf{n}}(\mathbf{g}^{\alpha},\mathbf{b}^{\beta})\|_{L^{p}({\mathbb{R}}^{d}\setminus\mathcal{E})}$
as following:

(5.12)
$\displaystyle\|\mathcal{M}_{\mathbf{n}}(\mathbf{g}^{\alpha},\mathbf{b}^{\beta})\|_{L^{p}({\mathbb{R}}^{d}\setminus\mathcal{E})}\lesssim|\mathbf{n}|^{\frac{\beta}{p}}\prod_{\mu=1}^{\alpha}\|g_{\mu}\|_{L^{\infty}}\times\prod_{\nu=\alpha+1}^{m}\|b_{\nu}\|_{L^{r_{\nu}}}.$

Since $\operatorname{supp}(b_{\nu})\subset\cup_{\gamma}Q_{\nu,\gamma}$ and
$\sum_{\gamma}\mathrm{meas}(Q_{\nu,\gamma})\lesssim\lambda^{-p}$ due to (5.3),
(5.4), the right-hand side of (5.12) is bounded by a constant multiple of

(5.13)
$\displaystyle|\mathbf{n}|^{\frac{\beta}{p}}\lambda^{\sum_{\mu=1}^{\alpha}\frac{p}{p_{j}}}\times\lambda^{-p(\sum_{\nu=\alpha+1}^{m}\frac{1}{r_{\nu}}-\frac{1}{p_{\nu}})}=|\mathbf{n}|^{\frac{\beta}{p}}.$

Here the left-hand side of (5.13) is a consequence of Hölder’s inequality on
$\|b_{\nu}\|_{L^{r_{\nu}}}$. Finally, by making use of (5.12) and (5.13), we
have

$\displaystyle\mathrm{meas}\Bigl{(}\\{x:\mathcal{M}_{\mathbf{n}}(\mathbf{g}^{\alpha},\mathbf{b}^{\beta})(x)>2^{-m}\lambda\\}\Bigr{)}$
$\displaystyle\lesssim\lambda^{-p}\|\mathcal{M}_{\mathbf{n}}(\mathbf{g}^{\alpha},\mathbf{b}^{\beta})\|_{L^{p}({\mathbb{R}}^{d}\setminus\mathcal{E})}^{p}$
$\displaystyle\leq\lambda^{-p}|\mathbf{n}|^{\beta}.$

For $\mathcal{M}_{n}(g_{1},\dots,g_{m})$, we simply choose $C<2^{-1}$ in (5.2)
so that

$|\mathcal{M}_{n}(g_{1},\dots,g_{m})|\leq\prod_{j=1}^{m}\|g_{j}\|_{L^{\infty}}\leq
C^{m}\lambda<2^{-m}\lambda.$

Thus we have

$\mathrm{meas}\Bigl{(}\\{x:\mathcal{M}_{\mathbf{n}}(g_{1},\dots,g_{m})(x)>2^{-m}\lambda\\}\Bigr{)}=0.$

####  5.1.3. Proof of Lemma 5.1

For simplicity, let $n_{j}+\ell=\tau$ and $b_{j}^{i_{j}}=b=\sum_{Q}b_{Q}$
where $Q=Q_{j,\gamma}$ whose sidelength is $2^{-i}$. Then thanks to
Proposition 1.1, it suffices for the first and second term in the minimum to
show that

(5.14)
$\displaystyle\|P_{\tau}b_{Q}\|_{p}\lesssim\min(1,(2^{\tau}s(Q))^{1+\frac{d}{p^{\prime}}})\|b_{Q}\|_{p}.$

The first term, $1$, is directly given by the fact that $\|P_{\tau}\|_{1}=1$
and Young’s inequality.

For the second term, we make use of the vanishing property of $b_{Q}$. Let
$c_{Q}$ be a center of $Q$.

$\displaystyle P_{\tau}b_{Q}(x)$
$\displaystyle=\int_{{\mathbb{R}}^{d}}(\psi_{\tau}(x-y)-\psi_{\tau}(x-c_{Q}))b_{Q}(y)~{}~{}\mathrm{d}y$
$\displaystyle=\int_{{\mathbb{R}}^{d}}\int_{0}^{1}\langle\nabla_{y}(\psi_{\tau})(x-c_{Q}-t(y-c_{Q})),y-c_{Q}\rangle~{}~{}\mathrm{d}t~{}b_{Q}(y)~{}~{}\mathrm{d}y.$

Since $\psi$ is of Schwartz class, it follows that
$\frac{1}{|y-c_{Q}|}\int_{0}^{1}\langle\nabla_{y}(\psi_{\tau})(x-c_{Q}-t(y-c_{Q})),y-c_{Q}\rangle~{}~{}\mathrm{d}t$
is bounded by a constant multiple of

$\frac{2^{\tau(d+1)}}{(1+2^{\tau}|x-c_{Q}-t(y-c_{Q})|)^{N}},$

for any $N>0$. Thus we apply Minkowski’s integral inequality to obtain

$\displaystyle\|P_{\tau}b_{Q}\|_{L^{p}({\mathbb{R}}^{d})}$
$\displaystyle\lesssim\Big{(}\int_{{\mathbb{R}}^{d}}\frac{2^{\tau(d+1)p}}{(1+2^{\tau}|x|)^{pN}}~{}~{}\mathrm{d}x\Big{)}^{\frac{1}{p}}\times\int_{Q}|y-c_{Q}||b_{Q}(y)|~{}~{}\mathrm{d}y$
$\displaystyle\leq
2^{\tau(d+1)}2^{-\tau\frac{d}{p}}s(Q)s(Q)^{d-\frac{d}{p}}\|b_{Q}\|_{p}.$

This establishes

$\|P_{\tau}b_{Q}\|_{L^{p}({\mathbb{R}}^{d})}\lesssim
2^{\tau(1+\frac{d}{p^{\prime}})}s(Q)^{1+\frac{d}{p^{\prime}}}\|b_{Q}\|_{p}.$

Therefore we have

(5.15) $\displaystyle\|P_{\tau}b\|_{L^{p}({\mathbb{R}}^{d})}$
$\displaystyle=\Bigl{(}\sum_{Q}\|P_{\tau}b_{Q}\|_{p}^{p}\Bigr{)}^{\frac{1}{p}}$
$\displaystyle\lesssim(2^{\tau}s(Q))^{1+\frac{d}{p^{\prime}}}\Bigl{(}\sum_{Q}\|b_{Q}\|_{p}^{p}\Bigr{)}^{\frac{1}{p}}=(2^{\tau}s(Q))^{1+\frac{d}{p^{\prime}}}\|b\|_{p}.$

The first and the last equalities follow from the disjointness of $Q$’s. This
gives a decay estimate when $n_{j}+\ell<i_{j}$.

Lastly, we assume that $\ell>i_{j}$ so that for
$x\in(C_{\mathcal{S}}Q)^{\complement}$ and $z\in Q$, we have ${\rm
dist}(x-2^{-\ell}y,z)\geq s(Q)$ uniformly in $y\in\mathcal{S}$, because we
choose $C_{\mathcal{S}}=5\max(1,\mathop{\rm diam}({\mathcal{S}}))$. Thus it
follows that

$\displaystyle
P_{n_{j}+\ell}b_{j,\gamma}^{i_{j}}(x-2^{-\ell}\Theta_{j}y)=\widetilde{P}_{n_{j}+\ell}b_{j,\gamma}^{i_{j}}(x-2^{-\ell}\Theta_{j}y),$

where the kernel of $\widetilde{P}_{\tau}$ is given by

$\psi_{\tau}(y)\mathbbm{1}_{|y|\geq s(Q)}(y).$

Therefore, we obtain with help of (5.15) that

$\displaystyle\|\mathcal{A}_{\ell}^{\mathbf{n}}(b_{1}^{i_{1}},\dots,b_{m}^{i_{m}})\|_{L^{p}({\mathbb{R}}^{d}\setminus\mathcal{E})}\lesssim\min_{j=1,\dots,m}\|\widetilde{P}_{n_{j}+\ell}\|_{p\to
p}\prod_{j=1}^{m}\|b_{j}^{i_{j}}\|_{p_{j}}.$

Observe that for any $N>0$

$\displaystyle\|\widetilde{P}_{n_{j}+\ell}\|_{p\to
p}\leq\|\widetilde{P}_{n_{j}+\ell}\|_{1}=\int_{{\mathbb{R}}^{d}}|\psi_{n_{j}+\ell-
i_{j}}(x)|\mathbbm{1}_{|x|\geq 1}(x)~{}~{}\mathrm{d}x\lesssim
2^{-N(n_{j}+\ell-i_{j})}.$

Thus we have $\|\widetilde{P}_{n_{j}+\ell}\|_{p\to p}\leq 2^{-\ell+i_{j}}$
regardless of $n_{j}\geq 0$. This proves the lemma.

####  5.1.4. Proof of Lemma 5.2

It suffices to show that for any $i_{1}$, $i_{2}$

(5.16)
$\displaystyle\sum_{\ell\in{\mathbb{Z}}}\min_{i_{1},i_{2}}\min(1,2^{(n_{j}+\ell-
i_{j})(1+\frac{d}{p_{j}^{\prime}})},2^{i_{j}-\ell})\lesssim|\mathbf{n}|\min(1,2^{|\mathbf{n}|-|i_{1}-i_{2}|}).$

Note that

$\displaystyle\min(1,2^{(n_{j}+\ell-
i_{j})(1+\frac{d}{p_{j}^{\prime}})},2^{i_{j}-\ell})\leq\left\\{\begin{array}[]{lll}2^{i_{j}-\ell},&i_{j}<\ell;\\\
1,&i_{j}-|\mathbf{n}|\leq\ell\leq i_{j};\\\
2^{(|\mathbf{n}|-i_{j}+\ell)(1+\frac{d}{p_{j}^{\prime}})},&\ell<i_{j}-|\mathbf{n}|.\end{array}\right.$

When $i_{1}\sim i_{2}$, then the left side of (5.16) is bounded by a costant
multiple of $|\mathbf{n}|$. Thus we consider the case of $i_{2}$ is greater
than $i_{1}+|\mathbf{n}|$. Since $i_{2}>i_{1}+|\mathbf{n}|$, it follows that
for $i_{1}-|\mathbf{n}|\leq\ell\leq i_{1}$

(5.18) $\displaystyle\min_{i_{1},i_{2}}\min(1,2^{(n_{j}+\ell-
i_{j})(1+\frac{d}{p_{j}^{\prime}})},2^{i_{j}-\ell})\leq
2^{(|\mathbf{n}|-i_{2}+\ell)(1+\frac{d}{p_{j}^{\prime}})}\leq
2^{(|\mathbf{n}|-|i_{1}-i_{2}|)(1+\frac{d}{p_{j}^{\prime}})}.$

Similarly, we have for $i_{2}-|\mathbf{n}|\leq\ell\leq i_{2}$

(5.19) $\displaystyle\min_{i_{1},i_{2}}\min(1,2^{(n_{j}+\ell-
i_{j})(1+\frac{d}{p_{j}^{\prime}})},2^{i_{j}-\ell})\leq 2^{i_{1}-\ell}\leq
2^{|\mathbf{n}|-|i_{1}-i_{2}|}.$

One can obtain the same bounds when $i_{1}>i_{2}+|\mathbf{n}|$ by changing
roles of $i_{1},i_{2}$ in (5.18), (5.19). Therefore we conclude that

$\sum_{\ell\in{\mathbb{Z}}}\min_{i_{1},i_{2}}\min(1,2^{(n_{j}+\ell-
i_{j})(1+\frac{d}{p_{j}^{\prime}})},2^{i_{j}-\ell})\lesssim|\mathbf{n}|\min(1,2^{|\mathbf{n}|-|i_{1}-i_{2}|}).$

###  5.2. Proof of Lemma 4.3

Note that by the positivity of averaging operators
$\mathcal{A}_{\mathcal{S}}^{\Theta}$ and the Plancherel theorem we have

(5.20)
$\displaystyle\|\mathcal{A}_{\mathcal{S}}^{\Theta}(\mathrm{F})\|_{L^{2}}\lesssim\prod_{j=1}\|f_{j}\|_{B_{p_{j}}^{-k/p_{j}}},$

where $B_{p}^{s}$ denotes a Besov space and
$\sum_{j=1}^{m}\frac{1}{p_{j}}=\frac{1}{2}$, $\frac{1}{p_{j}}\leq\frac{1}{2}$.
Thus it follows that

(5.21)
$\displaystyle\|\mathcal{A}_{\mathcal{S}}^{\Theta}(P_{n_{1}}f_{1},\dots,P_{n_{m}}f_{m})\|_{L^{2}}\lesssim
2^{-\frac{k|\mathbf{n}|}{2}}\prod_{j=1}\|P_{n_{j}}f_{j}\|_{L^{p_{j}}}.$

By Proposition 1.1, for
$(\frac{1}{q_{1}},\cdots,\frac{1}{q_{m}})\in\text{conv}(\mathcal{V}_{k})$ we
also have

(5.22)
$\displaystyle\|\mathcal{A}_{\mathcal{S}}^{\Theta}(P_{n_{1}}f_{1},\dots,P_{n_{m}}f_{m})\|_{L^{\frac{k+2}{2(k+1)}}}\lesssim\prod_{j=1}\|P_{n_{j}}f_{j}\|_{L^{q_{j}}}.$

Then interpolation between (5.21) and (5.22) yields that for
$(\frac{1}{r_{1}},\dots,\frac{1}{r_{m}})$ contained in a neighborhood of
$(\frac{1}{m},\dots,\frac{1}{m})$ such that $1=\sum_{j=1}^{m}\frac{1}{r_{j}}$

(5.23)
$\displaystyle\|\mathcal{A}_{\mathcal{S}}^{\Theta}(P_{n_{1}}f_{1},\dots,P_{n_{m}}f_{m})\|_{L^{1}}\lesssim
2^{-\delta|\mathbf{n}|}\prod_{j=1}^{m}\|P_{n_{j}}f_{j}\|_{L^{r_{j}}},\quad
0<\delta<\frac{k}{2}.$

We make use of the following scaling invariance of
$\mathcal{A}_{\mathcal{S}}^{\Theta}$:

$\displaystyle\Big{\|}\int_{\mathcal{S}}\prod_{j=1}^{m}P_{\ell+n_{j}}f_{j}(x-2^{-\ell}\Theta_{j}y)~{}~{}\mathrm{d}\sigma(y)\Big{\|}_{L^{1}(\mathrm{d}x)}=2^{-\ell
d}\|\mathcal{A}_{\mathcal{S}}^{\Theta}(P_{n_{1}}f_{1,-\ell},\dots,P_{n_{m}}f_{m,-\ell})\|_{L^{1}},$

where $f_{j,-\ell}(x)=f_{j}(2^{-\ell}x)$. Then it follows that

$\displaystyle\|\mathcal{S}_{\mathbf{n}}(\mathrm{F})\|_{L^{1}}$
$\displaystyle\leq\sum_{\ell\in{\mathbb{Z}}}2^{-\ell
d}\|\mathcal{A}_{\mathcal{S}}^{\Theta}(P_{n_{1}}f_{1,-\ell},\dots,P_{n_{m}}f_{m,-\ell})\|_{L^{1}}$
$\displaystyle\lesssim\sum_{\ell\in{\mathbb{Z}}}2^{-\ell
d}2^{-\delta|\mathbf{n}|}\prod_{j=1}^{m}\|P_{n_{j}}f_{j,-\ell}\|_{L^{r_{j}}}$
$\displaystyle=\sum_{\ell\in{\mathbb{Z}}}2^{-\ell
d}2^{-\delta|\mathbf{n}|}\prod_{j=1}^{m}2^{\frac{\ell
d}{r_{j}}}\|P_{n_{j}+\ell}f_{j}\|_{L^{r_{j}}}$
$\displaystyle=\sum_{\ell\in{\mathbb{Z}}}2^{-\delta|\mathbf{n}|}\prod_{j=1}^{m}\|P_{n_{j}+\ell}f_{j}\|_{L^{r_{j}}}.$

We apply Hölder’s inequality to the last line and obtain

$\displaystyle\|\mathcal{S}_{\mathbf{n}}(\mathrm{F})\|_{L^{1}}\lesssim
2^{-\delta|\mathbf{n}|}\prod_{j=1}^{m}\Bigl{(}\sum_{\ell\in{\mathbb{Z}}}\|P_{n_{j}+\ell}f_{j}\|_{L^{r_{j}}}^{r_{j}}\Bigr{)}^{\frac{1}{r_{j}}}.$

Note that $(\sum_{j}\|P_{j}f\|_{p}^{p})^{1/p}\lesssim\|f\|_{p}$ for $p\geq 2$,
which gives

$\|\mathcal{S}_{\mathbf{n}}(\mathrm{F})\|_{L^{1}}\lesssim
2^{-\delta|\mathbf{n}|}\prod_{j=1}^{m}\|f_{j}\|_{L^{r_{j}}}.$

This proves the lemma.

##  6\. Proof of Theorem 1.6

Recall that for an $(md-1)$-dimensional surface $\Sigma$ in
${\mathbb{R}}^{md}$ with $k$ non-vanishing principal curvatures and
$k>(m-1)d$, we define $\textsl{A}_{\Sigma}(\mathrm{F})(x)$ as following:

(6.1)
$\displaystyle\int_{\Sigma}\prod_{j=1}^{m}f_{j}(x-y_{j})~{}~{}\mathrm{d}\sigma_{\Sigma}(\textsl{y}),\quad\textsl{y}=(y_{1},\dots,y_{m})\in{\mathbb{R}}^{md}.$

By making use of the dyadic decomposition of Section 4 satisfying (1.10),
(1.11), (4.2), and (4.3), we define the following quantities similar to (4.4),
(4.5), (4.6):

(6.2) $\displaystyle\textsl{A}_{\ell}^{\alpha}(\mathrm{F})(x)$
$\displaystyle:=\int_{\Sigma}\Bigl{(}\prod_{\mu=1}^{\alpha}P_{<\ell}f_{\mu}(x-2^{-\ell}y_{\mu})\Bigr{)}\Bigl{(}\prod_{\nu=\alpha+1}^{m}f_{\nu}(x-2^{-\ell}y_{\nu})\Bigr{)}~{}~{}\mathrm{d}\sigma(\textsl{y}),$
(6.3) $\displaystyle\textsl{M}_{\bf{n}}(\mathrm{F})$
$\displaystyle:=\sup_{\ell\in\mathbb{Z}}\Big{|}\int_{\Sigma}\prod_{j=1}^{m}P_{\ell+n_{j}}f_{j}(x-2^{-\ell}y_{j})~{}\mathrm{d}\sigma(\textsl{y})\Big{|},$
(6.4) $\displaystyle\textsl{S}_{\bf{n}}(\mathrm{F})$
$\displaystyle:=\sum_{\ell\in\mathbb{Z}}\Big{|}\int_{\Sigma}\prod_{j=1}^{m}P_{\ell+n_{j}}f_{j}(x-2^{-\ell}y_{j})~{}\mathrm{d}\sigma(\textsl{y})\Big{|}.$

Therefore, the lacunary maximal operator $\mathfrak{M}_{\Sigma}$ is bounded by
a constant mutiple of

(6.5)
$\displaystyle\sum_{\alpha=1}^{m}\sup_{\ell\in{\mathbb{Z}}}|\textsl{A}_{\ell}^{\alpha}(\mathrm{F})|+\sum_{\mathbf{n}\in\mathbb{N}_{0}^{m}}\textsl{M}_{\mathbf{n}}(\mathrm{F}),\quad\mathbb{N}_{0}=\mathbb{N}\cup\\{0\\}.$

Then the proof will be completed by combination of the following lemmas and an
induction argument which is slightly different from the argument in Section 4:

######  Lemma 6.1.

Let $F=(f_{1},f_{2},1,\dots,1)$ and $\alpha=1$. Then we have

$\textsl{A}_{\ell}^{\alpha}(\mathrm{F})(x)\leq
M_{HL}(f_{1})(x)\times{M}_{\Sigma}(f_{2})(x),$

where
${M}_{\Sigma}(f)(x)=\sup_{\ell\in{\mathbb{Z}}}|\int_{\Sigma}f(x-2^{-\ell}y_{2})~{}\mathrm{d}\sigma(\textsl{y})|$.

The proof of Lemma 6.1 is same with that of Lemma 4.1, so we omit it. Note
that $M_{HL},{M}_{\Sigma}$ are bounded on $L^{p}$ for $p\in(1,\infty]$, hence
we need the boundedness of the second term in (6.5).

######  Lemma 6.2.

Let $\mathbf{n}\in\mathbb{N}^{m}$ and
$\frac{m+1}{2}\leq\frac{1}{p}<\frac{2d+k}{2d}$. For
$p_{j}\in[1,2],j=1\cdots,m$ with $\sum_{j=1}^{m}\frac{1}{p_{j}}=\frac{1}{p}$,
we have

$\|\textsl{M}_{\mathbf{n}}(\mathrm{F})\|_{L^{p,\infty}}\leq
C(1+|\mathbf{n}|^{m})\prod_{j=1}^{m}\|f_{j}\|_{L^{p_{j}}}.$

The proof fo Lemma 6.2 is to repeat the proof of Lemma 4.2. The only
difference occurs in showing Lemma 5.1 in terms of
$\textsl{A}_{\ell}^{\mathbf{n}}$ which corresponds to
$\mathcal{A}_{\ell}^{\mathbf{n}}$, since $\textsl{A}_{\ell}^{\mathbf{n}}$ is
an average over $\Sigma$ which is $(md-1)$-dimensional and each $f_{j}$
depends on $x-y_{j}$ not $x-\Theta_{j}y$. This difference is harmless,
however, because only the compactness of ${\mathcal{S}}$ does matter in the
proof of Lemma 5.1 and $\Sigma$ is a compact surface. On the other hand, the
range $\frac{m+1}{2}\leq\frac{1}{p}<\frac{2d+k}{2d}$ follows from Proposition
1.2.

######  Lemma 6.3.

Let $\mathbf{n}\in\mathbb{N}^{m}$ and $1=\sum_{j=1}^{m}\frac{1}{r_{j}}$. Then
it holds that

$\|\textsl{S}_{\mathbf{n}}(\mathrm{F})\|_{L^{1}}\lesssim
2^{-\delta|\mathbf{n}|}\prod_{j=1}^{m}\|f_{j}\|_{L^{r_{j}}}.$

The proof of Lemma 6.3 is same with that of Lemma 4.3. The only difference is
that instead of (5.21) and (5.22) we put respectively

(6.6)
$\displaystyle\|\textsl{A}_{\Sigma}(P_{n_{1}}f_{1},\dots,P_{n_{m}}f_{m})\|_{L^{2}}$
$\displaystyle\lesssim
2^{-\frac{k|\mathbf{n}|}{2}}\prod_{j=1}^{m}\|P_{n_{j}}f_{j}\|_{L^{p_{j}}},$
(6.7)
$\displaystyle\|\textsl{A}_{\Sigma}(P_{n_{1}}f_{1},\dots,P_{n_{m}}f_{m})\|_{L^{q}}$
$\displaystyle\lesssim\prod_{j=1}^{m}\|P_{n_{j}}f_{j}\|_{L^{q_{j}}},$

where
$\frac{m+1}{2}\leq\frac{1}{q}=\sum_{j=1}^{m}\frac{1}{q_{j}}<\frac{2d+k}{2d}$,
$1\leq q_{j}\leq 2$, and $\frac{1}{2}=\sum_{j=1}^{m}\frac{1}{p_{j}}$. Note
that (6.6) and (6.7) follow from (1.6) and (1.5) of Proposition 1.2,
respectively.

Due to $\textsl{M}_{\mathbf{n}}\leq\textsl{S}_{\mathbf{n}}$ by definitions
(6.3), (6.4), it follows from interpolation between Lemmas 6.2 and 6.3 that

(6.8) $\displaystyle\|\textsl{M}_{\mathbf{n}}(\mathrm{F})\|_{L^{p}}\lesssim
2^{-\delta^{\prime}|\mathbf{n}|}\prod_{j=1}\|f_{j}\|_{L^{pm}},\quad\delta^{\prime}>0,$

whenever $\frac{1}{p}<\frac{2d+k}{2d}$. It should be noted that Lemmas 6.2 and
6.3 are still valid for $F=(f_{1},\dots,f_{N},1,\dots,1)$ with $m$ replaced by
$N$. Thus (6.8) also holds for $F=(f_{1},\dots,f_{N},1,\dots,1)$. Since
$2^{-\delta^{\prime}|\mathbf{n}|}$ is summable over
$\mathbf{n}\in\mathbb{N}_{0}^{m}$, this together with Lemma 6.1 proves the
theorem for $F=(f_{1},f_{2},1,\dots,1)$.

For the induction, we assume that Theorem 1.6 holds for
$F=(f_{1},\dots,f_{N},1,\dots,1)$ for $N=2,\dots,m-1$. Note that we have shown
$N=2$-case. By the assumption, we have the following lemma:

######  Lemma 6.4.

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

$\textsl{A}_{\ell}^{\alpha}(\mathrm{F})(x)\lesssim\prod_{\mu=1}^{\alpha}M_{HL}(f_{\mu})(x)\times\sup_{\ell\in{\mathbb{Z}}}\Big{|}\int_{\Sigma}\prod_{\nu=\alpha+1}^{m}f_{\nu}(x-2^{-\ell}y_{\nu})~{}\mathrm{d}\sigma_{\Sigma}(\textsl{y})\Big{|}.$

Moreover,

$\sup_{\ell\in{\mathbb{Z}}}\Big{|}\int_{\Sigma}\prod_{\nu=\alpha+1}^{m}f_{\nu}(x-2^{-\ell}y_{\nu})~{}~{}\mathrm{d}\sigma_{\Sigma}(\textsl{y})\Big{|}$

satisfies multilinear estimates of Theorem 1.5 for $(m-\alpha)$-linear
operators.

Note that the assertions of the lemma follows directly by the proof of Lemma
4.1 and the assumption on $F=(f_{1},\dots,f_{N},1,\dots,1)$, respectively.
Since we already proved Lemmas 4.2 and 4.3 for general $m$, Theorem 1.6 for
$m$-linear operators holds under the assumption that $N=2,\dots,m-1$ cases
hold. This closes the induction hence proves the theorem.

###### Proof of Remark 1.8.

Note that, for dimension $d=1$, proof of this remark is already given in [10].
Although the proof for $d\geq 2$ case is given in [7], we present a different
proof by exploiting ideas of [10] to prove Remark 1.8 in higher dimensions
$d\geq 2$, which can work for general compact hypersurface with nonvanishing
$k$ principal curvatures. In fact, the proof follows from Theorem (1.6) with
minor modifications in the Lemma 6.2 and 6.3. Indeed, note that in [17] the
authors proved $L^{1}\times L^{1}\rightarrow L^{1/2}$ estimate of the bilinear
spherical average $\textsl{A}^{1}_{\mathbb{S}^{2d-1}}$. Using this estimate in
Lemma 6.2 we get

$\|\textsl{M}_{\mathbf{n}}(\mathrm{F})\|_{L^{\frac{1}{2},\infty}}\leq
C(1+|\mathbf{n}|^{2})\prod_{j=1}^{2}\|f_{j}\|_{L^{1}}.$

Further, using the estimate (2.16) with $\Sigma=\mathbb{S}^{2d-1}$ in Lemma
6.3 we get

$\|\textsl{S}_{\mathbf{n}}(\mathrm{F})\|_{L^{1}}\lesssim
2^{-\delta|\mathbf{n}|}\prod_{j=1}^{m}\|f_{j}\|_{L^{2}}$

for some $\delta>0$. The rest of the proof follows by imitating the machinery
of Theorem 1.6. ∎

## Acknowledgement

This work has been supported by the South Korea NRF grant no.
2022R1A4A1018904(Cho, Lee & Shuin). C. Cho is supported by the NRF grant No.
2020R1I1A1A01072942. J. B. Lee is supported by the NRF grant funded by the
Korea government(MSIT) (No.2021R1C1C2008252). K. Shuin is supported by BK21
Post doctoral fellowship of Seoul National University.

## References

  * [1] J.-G. Bak, Y.-S. Shim, Endpoint Inequalities for Spherical Multilinear Convolutions, J. Funct. Anal. 157(2) (1998) 534–553.
  * [2] A. J. Barrionuevo, L. Grafakos, D. He, P. Honzik and L. Oliveira, Bilinear spherical maximal function, Math. Res. Lett. 25(5) (2018) 1369–1388.
  * [3] D. Beltran, J. Roos and A. Seeger, Multi-scale sparse domination, arXiv:2009.00227.
  * [4] J. Bennett, A. Carbery, M. Christ, and T. Tao, The Brascamp–Lieb Inequalities: Finiteness, Structure and Extremals, Geom. Funct. Anal. 17 (2008) 1343–1415.
  * [5] J. Bennett, N. Bez, S. Buschenhenke, M. Cowling and T. C. Flock, On the nonlinear Brascamp-Lieb inequality, Duke Math. J. 169(17) (2020) 3291–3338.
  * [6] T. Borges, B. Foster, Y. Ou, J. Pipher and Zirui Zhou, Sparse bounds for the bilinear spherical maximal function, J. London Math. Soc. 107(4) (2023) 1409–1449.
  * [7] T. Borges and B. Foster, _Bounds for lacunary bilinear spherical and triangle maximal functions_ , arXiv:2305.12269.
  * [8] J. Bourgain, Averages in the plane over convex curves and maximal operators, J. Anal. Math. 47 (1986) 69–85.
  * [9] C. P. Calderón, Lacunary spherical means, Illinois J. Math. 23(3) (1979) 476–484.
  * [10] M. Christ and Z. Zhou, A class of singular bilinear maximal functions, arXiv:2203.16725.
  * [11] R. R. Coifman and Y. Meyer, On commutators of singular integrals and bilinear singular integrals, Trans. Amer. Math. Soc. 212 (1975), 315–331.
  * [12] G. Dosidis, Multilinear spherical maximal function, Proc. Amer. Math. Soc. 149(4) (2021), 1471–1480.
  * [13] L. Grafakos, D. He and P. Honzík, Maximal operators associated with bilinear multipliers of limited decay, J. Anal. Math. 143 (2021) 231–251.
  * [14] L. Grafakos, D. He, P. Honzík and B. J. Park, On pointwise a.e. convergence of multilinear operators, arXiv: 2012.10837v2.
  * [15] L. Grafakos and N. Kalton, Some remarks on multilinear maps and interpolation, Math. Ann. 319 (2001) 151–180.
  * [16] A. Greenleaf, A. Iosevich, B. Krause and A. Liu, Bilinear generalized Radon transforms in the plane, arXiv: 1704.00861v1.
  * [17] A. Iosevich, E. A. Palsson and S. R. Sovine, Simplex Averaging Operators: Quasi-Banach and $L^{p}$-Improving Bounds in Lower Dimensions, J. Geom. Anal. 32 (2022) Article No. 87, 16 pp.
  * [18] E. Jeong and S. Lee, Maximal estimates for the bilinear spherical averages and the bilinear Bochner-Riesz operators, J. Funct. Anal. 279(7) (2020) 108629.
  * [19] M. T. Lacey, Sparse bounds for spherical maximal functions, J. Anal. Math. 139 (2019), 613–635.
  * [20] M. T. Lacey, The bilinear maximal functions map into $L^{p}$ for $2/3<p\leq 1$, Ann. Math. 151(1) (2000) 35–57.
  * [21] M. T. Lacey and C. Thiele, $L^{p}$ estimates on the bilinear Hilbert transform for $2<p<\infty$, Ann. Math. 146(3) (1997) 693–724.
  * [22] M. T. Lacey and C. Thiele, On Calderón’s conjecture, Ann. Math. 149(2) (1999) 475–496.
  * [23] A. K. Lerner, On pointwise estimates involving sparse operators, New York J. Math. 22 (2016), 341–349.
  * [24] W. Littman, Fourier transforms of surface-carried measures and differentiability of surface averages, Bull. Amer. Math. Soc. 69(6) (1963) 766–770.
  * [25] W. Littman, $L^{p}-L^{q}$ estimates for singular integral operators arising from hyperbolic equations, Proc. Symp. Pure Math., 23 (1973) 479–481.
  * [26] E. Lorist and Z. Nieraeth, Sparse domination implies vector-valued sparse domination, Math. Z. 301 (2022) 1–35.
  * [27] D. M. Oberlin, Multilinear convolutions defined by measures on spheres, Trans. Amer. Math. Soc. 310(2) (1988) 821–835.
  * [28] E. A. Palsson, S. R. Sovine, Sparse bounds for maximal triangle and bilinear spherical averaging operators, arXiv:2110.08928
  * [29] L. Roncal, S. Shrivastava and K. Shuin, Bilinear spherical maximal functions of product type, J. Fourier Anal. Appl. 27(4) (2021) Article No. 73, 42 pp.
  * [30] A. Seeger and J. Wright, Problems on averages and lacunary maximal functions, Banach Cent. Publ. 95(1) (2011) 235–250.
  * [31] S. Shrivastava and K. Shuin, $L^{p}$ estimates for multilinear convolution operators defined with spherical measure, Bull. London Math. Soc. 53(4) (2021) 1045–1060.
  * [32] E. M. Stein, Maximal functions: Spherical means, Proc. Natl. Acad. Sci. USA, 73(7) (1976) 2174–2175.
  * [33] R. S. Strichartz, Convolutions with kernels having singularities on a sphere, Trans. Amer. Math. Soc. 148(2) (1970), 461–471.