On pointwise a.e. convergence of multilinear operators

Abstract

In this work, we obtain the pointwise almost everywhere convergence for two families of multilinear operators: (a) the doubly truncated homogeneous singular integral operators associated with $L^q$ functions on the sphere and (b) lacunary multiplier operators of limited smoothness. The a.e. convergence is deduced from the $L^2\times \cdots \times L^2\to L^{2/m}$ boundedness of the associated maximal multilinear operators.

1 Introduction and preliminaries

The pointwise a.e. convergence of sequences of operators is of paramount importance and has been widely studied in several areas of analysis, such as harmonic analysis, PDE, and ergodic theory. This area boasts challenging problems (indicatively see [5, 6, 12, 24]), and is intimately connected with the boundedness of the associated maximal operators; on this, see [27]. Moreover, techniques and tools employed to study a.e. convergence have led to important developments in harmonic analysis.

Multilinear harmonic analysis has made significant advances in recent years. The founders of this area are Coifman and Meyer [8], who realized the applicability of multilinear operators and introduced their study in analysis in the mid-1970s. Focusing on operators that commute with translations, a fundamental difference between the multilinear theory and the linear theory is the existence of a straightforward characterization of boundedness at an initial point, usually $L^2\to L^2$ . The lack of an easy characterization of boundedness at an initial point in the multilinear theory creates difficulties in their study. Criteria that get very close to characterization of boundedness have recently been obtained by the first two authors and Slavíková [19] and by Kato, Miyachi, and Tomita [25] in the bilinear case. These criteria were extended to the general m-linear case for $m\ge 2$ by the authors of this article in [18]. This reference also contains initial $L^2\times \cdots \times L^2\to L^{2/m}$ estimates for rough homogeneous multilinear singular integrals associated with $L^q$ functions on the sphere and multilinear multipliers of Hörmander type.

The purpose of this work is to obtain the pointwise a.e. convergence of doubly truncated multilinear homogeneous singular integrals and lacunary multilinear multipliers by establishing boundedness for their associated maximal operators.

We first introduce multilinear (singly) truncated singular integral operators. Let $\Omega $ be an integrable function, defined on the sphere $\mathbb {S}^{mn-1}$ , satisfying the mean value zero property

(1.1) $$ \begin{align} \int_{\mathbb{S}^{mn-1}}\Omega~ d\sigma_{mn-1}=0. \end{align} $$

Then we define

$$ \begin{align*} K(\vec{{y}}\,):=\frac{\Omega(\vec{{y}}\,')}{|\vec{{y}}\,|^{mn}}, \qquad \vec y \neq 0, \end{align*} $$

where $\vec {{y}}\,':=\vec {{y}}\,/|\vec {{y}}\,|\in \mathbb {S}^{mn-1}$ , and the corresponding truncated multilinear operator ${\mathcal {L}}_{\Omega }^{(\epsilon )}$ by

$$ \begin{align*}\mathcal L^{(\epsilon)}_{\Omega}\big(f_1,\dots,f_m\big)(x):=\int_{(\mathbb R^n)^m\setminus B(0,\epsilon)}{K(\vec{{y}}\,)\prod_{j=1}^{m}f_j(x-y_j)}~d\,\vec{{y}}\, \end{align*} $$

acting on Schwartz functions $f_1,\dots ,f_m$ on $\mathbb R^n$ , where $x\in \mathbb R^n$ , $\vec {{y}}\,:=(y_1,\dots ,y_m)\in (\mathbb R^n)^m$ , and $B(0,\epsilon )$ , is the ball centered at zero with radius $\epsilon>0$ in $(\mathbb R^n)^m$ . Moreover, by taking $\epsilon \searrow 0$ , we obtain the multilinear homogeneous Calderón–Zygmund singular integral operator

(1.2) $$ \begin{align} \mathcal L_{\Omega}\big(f_1,\dots,f_m\big)(x)&:=\lim_{\epsilon\searrow 0}\mathcal L^{(\epsilon)}_{\Omega}\big(f_1,\dots,f_m\big)(x) \\ &=p.v. \int_{(\mathbb R^n)^m}{K(\vec{{y}}\,)\prod_{j=1}^{m}f_j(x-y_j)}~d\,\vec{{y}}\,. \nonumber \end{align} $$

This is still well defined for any Schwartz functions $f_1, \dots ,f_m$ on $\mathbb R^n$ . In [18], we showed that if $\Omega $ lies in $ L^q (\mathbb {S}^{mn-1}) $ with $q>\frac {2m}{m+1}$ , then the multilinear singular integral operator ${\mathcal {L}}_{\Omega }$ admits a bounded extension from $L^2(\mathbb R^n)\times \cdots \times L^2(\mathbb R^n)$ to $L^{2/m}(\mathbb R^n)$ . In order words, given $f_j\in L^2(\mathbb R^n)$ , ${\mathcal {L}}_{\Omega }(f_1,\dots ,f_m)$ is well defined and is in $L^{2/m}(\mathbb R^n)$ .

We now define the doubly truncated multilinear operator ${\mathcal {L}}_{\Omega }^{(\epsilon ,\epsilon ^{-1})}$ by

$$\begin{align*}{\mathcal{L}}_{\Omega}^{(\epsilon , \epsilon^{-1})}\big(f_1,\dots , f_m\big) := {\mathcal{L}}_{\Omega}^{(\epsilon)}\big(f_1,\dots , f_m\big)- {\mathcal{L}}_{\Omega}^{(\epsilon^{-1} )}\big(f_1,\dots , f_m\big) \end{align*}$$

for Schwartz functions $f_j$ , $j=1,\dots ,m$ . We observe that if $\Omega \in L^q(\mathbb {S}^{mn-1})$ for $\frac {2m}{m+1}<q\le \infty $ , then

$$ \begin{align*}\lim_{\epsilon\to 0} {\mathcal{L}}_{\Omega}^{(\epsilon , \epsilon^{-1})}\big(f_1,\dots , f_m\big)=\lim_{\epsilon\to 0} {\mathcal{L}}_{\Omega}^{(\epsilon )}\big(f_1,\dots , f_m\big) \end{align*} $$

for $f_j$ in the Schwartz class.

We define, for fixed $0<\epsilon _0<1$ ,

$$\begin{align*}{\mathcal{L}}_{\Omega}^{*, \epsilon_0 }(\varphi_1,\dots , \varphi_m) := \sup_{\epsilon\ge \epsilon_0 } \big| {\mathcal{L}}_{\Omega}^{(\epsilon , \epsilon^{-1})}(\varphi_1,\dots , \varphi_m)\big| \end{align*}$$

and

(1.3) $$ \begin{align} {\mathcal{L}}_{\Omega}^{** }(\varphi_1,\dots , \varphi_m) := \sup_{\epsilon>0 } \big| {\mathcal{L}}_{\Omega}^{(\epsilon , \epsilon^{-1})}(\varphi_1,\dots , \varphi_m)\big| =\lim_{\epsilon_0\to 0}{\mathcal{L}}_{\Omega}^{*, \epsilon_0 }(\varphi_1,\dots , \varphi_m) \end{align} $$

for $\varphi _j$ in the Schwartz class. One main difficulty to study the boundedness of $\mathcal L^{**}_{\Omega }$ is to show that the doubly truncated operator is well defined pointwise a.e. for $f_j\in L^2(\mathbb {R}^n)$ . To overcome this difficulty, we need to utilize the boundedness of $\mathcal M_{\Omega }$ introduced in Section 3 (see Section 5 for the detailed proof).

Our first main result is as follows.

Theorem 1.1 Let $m\ge 2$ , $\frac {2m}{m+1}<q\le \infty $ , and $\Omega \in L^q (\mathbb S^{mn-1})$ satisfy (1.1). Then

(1.4) $$ \begin{align} \big\|{\mathcal{L}}_{\Omega}^{** }(f_1,\dots , f_m)\big\|_{L^{2/m}(\mathbb{R}^n) } \le C\Vert \Omega\Vert_{L^q(\mathbb{S}^{mn-1})} \prod_{j=1}^{\infty}\| f_j\|_{L^{2}(\mathbb{R}^n)} \end{align} $$

for $f_j\in L^2(\mathbb {R}^n)$ . Moreover, the doubly truncated singular integral ${\mathcal {L}}_{\Omega }^{(\epsilon ,\epsilon ^{-1})}(f_1,\dots , f_m)$ converges to ${\mathcal {L}}_{\Omega }(f_1,\dots ,f_m)$ pointwise a.e. as $\epsilon \to 0$ when $f_j\in L^2(\mathbb {R}^n)$ , $j=1,\dots , m$ . That is, the multilinear singular integral ${\mathcal {L}}_{\Omega }(f_1,\dots ,f_m)$ is well defined a.e. when $f_j\in L^2(\mathbb {R}^n)$ , $j=1,\dots , m$ .

In order to achieve this goal, we initially prove the following result, which provides the boundedness of the associated maximal singular integral operator:

$$ \begin{align*} {\mathcal{L}}_{\Omega}^*\big(f_1,\dots,f_m\big)(x):=\sup_{\epsilon>0}\Big| {\mathcal{L}}_{\Omega}^{(\epsilon)}\big(f_1,\dots,f_m\big)(x)\Big| \end{align*} $$

for Schwartz functions $f_j$ , $j=1,\dots ,m$ .

Theorem 1.2 Let $m\ge 2$ , $\frac {2m}{m+1}<q\le \infty $ , and $\Omega \in L^q (\mathbb S^{mn-1})$ satisfy (1.1). Then there exists a constant $C>0$ such that

(1.5) $$ \begin{align} \big\Vert {\mathcal{L}}_{\Omega}^*(f_1,\dots,f_m)\big\Vert_{L^{2/m}(\mathbb R^n)}\le C\Vert \Omega\Vert_{L^q(\mathbb{S}^{mn-1})}\prod_{j=1}^{m}\Vert f_j\Vert_{L^2(\mathbb R^n)} \end{align} $$

for Schwartz functions $f_1,\dots ,f_m$ on $\mathbb R^n$ .

This extends and improves a result obtained in [3] which treated the case $m=2$ and $q=\infty $ . Theorem 1.2 follows from Propositions 4.1 and 4.2, which are counterparts of Propositions 5 and 4 in [3], respectively. We improve the two propositions in the m-linear settings. Remark that the assumption $\Omega \in L^2(\mathbb {S}^{2n-1})$ in Proposition 5 and Theorem 2 in [3] should be $\Omega \in L^{\infty }(\mathbb {S}^{2n-1})$ . One of the main improvements is the $L^{p_1}\times \cdots \times L^{p_m}\to L^p$ estimate for $\mathfrak {T}_K^{**}$ in (4.10) with a bound $\Vert \Omega \Vert _{L^1(\mathbb {S}^{mn-1})}$ , while a simple m-linear extension of the arguments in [3] requires the bound $\Vert \Omega \Vert _{L^{\infty }(\mathbb {S}^{mn-1})}$ for the estimate, which originated simply from the kernel estimate

(1.6) $$ \begin{align} \big| K_{\tau}^{**}(\vec{{y}}\,)\big|\lesssim_N \Vert \Omega\Vert_{L^{\infty}(\mathbb{S}^{mn-1})}2^{mn\tau}\big(1+2^{-\tau}|\vec{{y}}\,| \big)^{-N}, \end{align} $$

where $\mathfrak {T}_K^{**}$ is defined in (4.6) and its kernel $K_{\tau }^{**}$ is in (4.7). For the improvement, we incorporate a delicate decomposition, as we are unable to use the kernel estimate (1.6) (see (4.8)). To obtain the results in Proposition 4.2, we suitably combine Littlewood–Paley techniques and wavelet decompositions to reduce the boundedness of ${\mathcal {L}}_{\Omega ,\mu }^{\sharp }$ to estimates for norms of maximal operators associated with lattice bumps with suitable decay. This is the essential contribution of this article in view of the fact that the bilinear argument in [3, Proposition 4] does not apply due to the complicated structure of general m-linear operators for $m\ge 3$ (see (4.14) for the exact formulation). This result is actually proved in terms of Plancherel-type inequalities, recently developed in [18] and stated in Proposition 2.1.

The tools used to establish Theorem 1.1 turn out to be useful in the study of pointwise convergence problems of several related operators. As an example, let us take multilinear multipliers with limited decay to demonstrate our idea.

For a smooth function $\sigma \in \mathscr {C}^{\infty }((\mathbb R^n)^m)$ and $\nu \in \mathbb {Z}$ , let

(1.7) $$ \begin{align} S_{\sigma}^{\nu}\big(f_1,\dots,f_m\big)(x):=\int_{(\mathbb R^n)^m}{\sigma(2^{\nu}\vec{{\xi}}\,)\bigg( \prod_{j=1}^{m}\widehat{f_j}(\xi_j)\bigg)e^{2\pi i\langle x,\sum_{j=1}^{m}\xi_j \rangle}} ~d\,\vec{{\xi}}\, \end{align} $$

for Schwartz functions $f_1,\dots ,f_m$ on $\mathbb R^n$ , where $\vec {{\xi }}\,:=(\xi _1,\dots ,\xi _m) \in (\mathbb R^n)^m$ .

We are interested in the pointwise convergence of $S^{\nu }_{\sigma }$ when $\nu \to -\infty $ . We pay particular attention to $\sigma $ satisfying the limited decay property (for some fixed a)

$$ \begin{align*}\big|\partial^{\beta}\sigma(\vec{{\xi}}\,) \big|\lesssim_{\beta}|\vec{{\xi}}\,|^{-a} \end{align*} $$

for sufficiently many $\beta $ . Examples of multipliers of this type include $\widehat \mu $ , the Fourier transform of the spherical measure $\mu $ (see [4, 7, 26] for the corresponding linear results).

The second contribution of this work is the following result.

Theorem 1.3 Let $m\ge 2$ and $a>\frac {(m-1)n}{2}$ . Let $\sigma \in \mathscr {C}^{\infty }((\mathbb R^n)^m)$ satisfy

(1.8) $$ \begin{align} \big|\partial^{\beta}\sigma(\vec{{\xi}}\,) \big|\lesssim_{\beta}|\vec{{\xi}}\,|^{-a} \end{align} $$

for all $|\beta |\le \big [ \frac {(m-1)n}{2} \big ]+1$ , where $ \left [ r\right ]$ denotes the integer part of r. Then, for $f_j$ in $L^2(\mathbb {R}^n)$ , $j=1,\dots , m$ , the functions $S_{\sigma }^{\nu }(f_1,\dots , f_m) $ converge to $\sigma (0) f_1 \cdots f_m$ pointwise a.e. as ${\nu \to -\infty }$ and to zero pointwise a.e. as $\nu \to \infty $ .

The precise definition of the action of the multilinear operator $S_{\sigma }^{\nu }$ on $L^2$ functions will be discussed after Theorem 1.4.

The a.e. convergence claimed in Theorem 1.3 is related to the boundedness of the associated m-(sub)linear lacunary maximal multiplier operator defined by

$$ \begin{align*} \mathscr{M}_{\sigma}\big(f_1,\dots,f_m\big) :=\sup_{\nu \in\mathbb{Z}}{\big|S_{\sigma}^{\nu}\big(f_1,\dots,f_m\big) \big|}. \end{align*} $$

$\mathscr {M}_{\sigma }$ is the so-called multilinear spherical maximal function when $\sigma =\widehat \mu $ , which was studied extensively recently by [1, 2, 10, 22, 23]. In particular, a bilinear version of the following theorem was previously obtained in [17].

Theorem 1.4 Let $m\ge 2$ and $a>\frac {(m-1)n}{2}$ . Let $\sigma \in \mathscr {C}^{\infty }((\mathbb R^n)^m)$ be as in Theorem 1.3. Then there exists a constant $C>0$ such that

$$ \begin{align*} \big\Vert \mathscr{M}_{\sigma}(f_1,\dots,f_m)\big\Vert_{L^{2/m}(\mathbb R^n)}\le C \prod_{j=1}^{m}\Vert f_j\Vert_{L^2(\mathbb R^n)} \end{align*} $$

for Schwartz functions $f_1,\dots ,f_m$ on $\mathbb R^n$ .

One of the main difficulties in dealing with general m-linear cases for $m\ge 3$ is that the target space $L^{2/m}$ is not a Banach space if $2/m<1$ . As a result, the condition $a>\frac {(m-1)n}{2}$ cannot be exploited by a simple adaptation of the bilinear argument in [17]. Additional combinatorial complexity arises from the multilinear extension, and in order to address these issues, we apply a more refined decomposition, recently introduced in [18], so that l-linear Plancherel-type estimates ( $1\le l\le m$ ) can be applied. These key estimates are stated in Proposition 2.1.

With the help of Theorem 1.4, we notice that the multilinear operator $S_{\sigma }^{\nu }$ is also well defined for $f_j\in L^2(\mathbb {R}^n)$ . Indeed, given $f_j$ in $L^2(\mathbb {R}^n)$ , we find a sequence of Schwartz functions $f_j^k$ that converge to $f_j$ in $L^2(\mathbb {R}^n)$ as $k\to \infty $ . Then Theorem 1.4 implies that the sequence

$$\begin{align*}\big\{ S_{\sigma}^{\nu}(f_1^k,\dots, f_m^k) \big\}_k \end{align*}$$

is a Cauchy sequence in $L^{2/m}$ and, thus, it has a unique limit in $L^{2/m}$ which we call $S_{\sigma }^{\nu }(f_1 ,\dots , f_m )$ . It is easy to verify that this limit does not depend on the choice of $f_j^k$ .

The paper is organized as follows. Section 2 is dedicated to preliminaries, introducing a wavelet decomposition that is one of the main ingredients to establish maximal inequalities in Theorems 1.2 and 1.4, and studying general properties of the decomposition. Another maximal inequality for rough singular integrals will be given in Section 3. We prove first Theorem 1.2 in Section 4 as it is necessary for the proof of Theorem 1.1 in Section 5. The proof of Theorems 1.4 and 1.3 will be given in turn in the last two sections.

2 Preliminary material

We adapt some notations and key estimates from [18]. For the sake of independent reading, we review the main tools and notation. We begin with certain orthonormal bases of $L^2$ due to Triebel [30], that will be of great use in our work. The idea is as follows. For any fixed $L\in \mathbb {N}$ , one can construct real-valued compactly supported functions $\psi _F, \psi _M$ in $\mathscr {C}^L(\mathbb {R})$ satisfying the following properties: ${\Vert \psi _F\Vert _{L^2(\mathbb {R})}=\Vert \psi _M\Vert _{L^2(\mathbb {R})}=1}$ , $\int _{\mathbb {R}}{x^{\alpha }\psi _M(x)}dx=0$ for all $0\le \alpha \le L$ , and moreover, if $\Psi _{\vec {{G}}}$ is a function on $\mathbb {R}^{mn}$ , defined by

$$ \begin{align*}\Psi_{\vec{{G}}}(\vec{{x}}\,):=\psi_{g_1}(x_1)\cdots \psi_{g_{mn}}(x_{mn})\end{align*} $$

for $\vec {{x}}\,:=(x_1,\dots ,x_{mn})\in \mathbb {R}^{mn}$ and $\vec {{G}}:=(g_1,\dots ,g_{mn})$ in the set

$$ \begin{align*}\mathcal{I}:=\big\{\vec{{G}}:=(g_1,\dots,g_{mn}):g_i\in\{F,M\} \big\},\end{align*} $$

then the family of functions

$$ \begin{align*} \bigcup_{\lambda\in\mathbb{N}_0}\bigcup_{\vec{{k}}\,\in \mathbb{Z}^{mn}}\big\{ 2^{\lambda{mn/2}}\Psi_{\vec{{G}}}(2^{\lambda}\vec{{x}}\,-\vec{{k}}\,):\vec{{G}}\in \mathcal{I}^{\lambda}\big\} \end{align*} $$

forms an orthonormal basis of $L^2(\mathbb {R}^{mn})$ , where $\mathcal {I}^0:=\mathcal {I}$ and for $\lambda \ge 1$ , we set ${\mathcal {I}^{\lambda }:=\mathcal {I}\setminus \{(F,\dots ,F)\}}$ .

We consistently use the notation $\vec {{\xi }}\,:=(\xi _1,\dots ,\xi _m) $ for elements of $ (\mathbb R^n)^m$ , $\vec {{G}}:=(G_1,\dots ,G_m)\in (\{F,M\}^n)^m$ , and $ \Psi _{\vec {{G}}}(\vec {{\xi }}\,)=\Psi _{G_1}(\xi _1)\cdots \Psi _{G_m}(\xi _m). $ For each ${\vec {{k}}:=(k_1,\dots ,k_m)\in (\mathbb Z^n)^m}$ and $\lambda \in \mathbb {N}_0$ , let

$$ \begin{align*}\Psi_{G_i,k_i}^{\lambda}(\xi_i):=2^{\lambda n/2}\Psi_{G_i}(2^{\lambda}\xi_i-k_i), \qquad 1\le i\le m, \end{align*} $$

and

$$ \begin{align*}\Psi_{\vec{{G}},\vec{{k}}\,}^{\lambda}(\vec{{\xi}}\,\,):=\Psi_{G_1,k_1}^{\lambda}(\xi_1)\cdots \Psi_{G_m,k_m}^{\lambda}(\xi_m). \end{align*} $$

We also assume that the support of $\psi _{g_i}$ is contained in $\{\xi \in \mathbb {R}: |\xi |\le C_0 \}$ for some $C_0>1$ , which implies that

$$ \begin{align*} \mathrm{Supp} (\Psi_{G_i,k_i}^{\lambda})\subset \big\{\xi_i\in\mathbb R^n: |2^{\lambda}\xi_i-k_i|\le C_0\sqrt{n}\big\}. \end{align*} $$

In other words, the support of $\Psi _{G_i,k_i}^{\lambda }$ is contained in the ball centered at $2^{-\lambda }k_i$ and radius $C_0\sqrt {n}2^{-\lambda }$ . Then we note that for a fixed $\lambda \in \mathbb {N}_0$ , elements of $\big \{\Psi _{\vec {{G}},\vec {{k}}\,}^{\lambda }\big \}_{\vec {{k}}\,}$ have (almost) disjoint compact supports.

It is also known in [29] that if L is sufficiently large, then every tempered distribution H on $\mathbb {R}^{mn}$ can be represented as

(2.1) $$ \begin{align} H(\vec{{x}}\,)=\sum_{\lambda\in\mathbb{N}_0}\sum_{\vec{{G}}\in\mathcal{I}^{\lambda}}\sum_{\vec{{k}}\,\in \mathbb{Z}^{mn}}b_{\vec{{G}},\vec{{k}}\,}^{\lambda}2^{\lambda mn/2}\Psi_{\vec{{G}}}(2^{\lambda} \vec{{x}}\, -\vec{{k}}\,), \end{align} $$

and for $1<q<\infty $ and $s\ge 0$ ,

$$ \begin{align*}\Big\Vert \Big(\sum_{\vec G,\ \vec k} \big|b_{\vec G,\vec k}^{\lambda} \Psi^{\lambda}_{\vec G,\vec k}\big|^2\Big)^{1/2} \Big\Vert_{L^q(\mathbb{R}^{mn})} \le C2^{-s\lambda}\|H\|_{L^q_s(\mathbb{R}^{mn})}, \end{align*} $$

where

$$ \begin{align*} b_{\vec{{G}},\vec{{k}}\,}^{\lambda}:=\int_{\mathbb{R}^{mn}}{H(\vec{{x}}\,)\Psi^{\lambda}_{\vec G,\vec k}(\vec{{x}}\,)}~d\vec{{x}}\, \end{align*} $$

and $L^q_s$ is the Sobolev space of functions H such that $(I-\Delta )^{s/2} H \in L^q(\mathbb R^{mn})$ . Moreover, it follows from the last estimate and from the (almost) disjoint support property of the $\Psi ^{\lambda }_{\vec G,\vec k}$ ’s that

(2.2) $$ \begin{align} \big\Vert \big\{b_{\vec{{G}},\vec{{k}}\,}^{\lambda}\big\}_{\vec{{k}}\,\in \mathbb{Z}^{mn}}\big\Vert_{\ell^{q}} \approx&\Big(2^{\lambda mn(1-q/2)}\int_{\mathbb{R}^{mn}}\Big( \sum_{\vec{{k}}\,} \big|b^{\lambda}_{\vec G,\vec k}\Psi^{\lambda}_{\vec G,\vec k}(\vec x)\big|^2\Big)^{q/2}~ d\vec{{x}}\,\Big)^{1/q}\notag \\ \lesssim& \; 2^{-\lambda (s-mn/q+mn/2)}\Vert H\Vert_{L^q_s(\mathbb{R}^{mn})}. \end{align} $$

Now we study an essential estimate in [18] which will play a significant role in the proof of both Theorems 1.2 and 1.4. We define the operator $L_{G_i,k_i}^{\lambda ,\gamma }$ by

(2.3) $$ \begin{align} L_{G_i,k_i}^{\lambda,\gamma}f:=\big( \Psi_{G_i,k_i}^{\lambda}(\cdot/2^{\gamma})\widehat{f}\kern2pt\big)^{\vee}, \qquad \gamma\in\mathbb{Z}. \end{align} $$

For $\mu \in \mathbb {Z}$ , let

(2.4) $$ \begin{align} \mathcal{U}^{\mu}:=\big\{ \vec{{k}}\,\in (\mathbb Z^n)^m: 2^{\mu-2}\le |\vec{{k}}\,|\le 2^{\mu+2},~ |k_1|\ge\cdots \ge |k_m| \big\} \end{align} $$

and split the set into m disjoint subsets $\mathcal {U}_l^{\mu }$ as below:

$$ \begin{align*} \mathcal{U}_1^{{\mu}}&:=\big\{\vec{{k}}\,\in \mathcal{U}^{{\mu}}:|k_1|\ge 2C_0\sqrt{n}>|k_2|\ge\cdots \ge |k_m| \big\}\\ \mathcal{U}_2^{{\mu}}&:=\big\{\vec{{k}}\,\in \mathcal{U}^{{\mu}}:|k_1|\ge |k_2|\ge 2C_0\sqrt{n}>|k_3|\ge\cdots \ge |k_m| \big\}\\ &\vdots\\ \mathcal{U}_m^{{\mu}}&:=\big\{\vec{{k}}\,\in \mathcal{U}^{{\mu}}:|k_1|\ge \cdots\ge |k_m|\ge 2C_0\sqrt{n} \big\}. \end{align*} $$

Then we have the following two observations that appear in [18].

• • For $\vec {{k}}\,\in \mathcal {U}_l^{\lambda +\mu }$ ,

  (2.5) $$ \begin{align} L_{G_j,k_j}^{\lambda,\gamma}f=L_{G_j,k_j}^{\lambda,\gamma}f^{\lambda,\gamma,{\mu}} \quad \text{ for }~ 1\le j\le l \end{align} $$
  due to the support of $\Psi _{G_j,k_j}^{\lambda }$ , where $\widehat {f^{\lambda ,\gamma ,\mu }}(\xi ):=\widehat {f}(\xi )\chi _{C_0\sqrt {n}2^{\gamma -\lambda }\le |\xi |\le 2^{\gamma +\mu +3}}$ .

• • For $\mu \ge 1$ and $\lambda \in \mathbb {N}_0$ ,

  (2.6) $$ \begin{align} \Big( \sum_{\gamma\in\mathbb{Z}}{\big\Vert f^{\lambda,\gamma,{\mu}}\big\Vert_{L^2}^2}\Big)^{1/2}\lesssim ({\mu}+\lambda)^{1/2}\Vert f\Vert_{L^2} \lesssim \mu^{1/2}(\lambda+1)^{1/2}\Vert f\Vert_{L^2} \end{align} $$
  where Plancherel’s identity is applied in the first inequality.

Proposition 2.1 [18, Proposition 2.4]

Let m be a positive integer with $m\ge 2$ and $0<q<\frac {2m}{m-1}$ . Fix $\lambda \in \mathbb {N}_0$ and $\vec {{G}}\in \mathcal {I}^{\lambda }$ . Suppose that $\{b_{\vec {{G}},\vec {{k}}\,}^{\lambda ,\gamma ,\mu }\}_{\vec {{G}}\in \mathcal {I}^{\lambda }, \gamma ,\mu \in \mathbb {Z}, \vec {{k}}\,\in (\mathbb Z^n)^m}$ is a sequence of complex numbers satisfying

$$ \begin{align*} \sup_{\gamma \in \mathbb{Z}}\big\Vert \{b_{\vec{{G}},\vec{{k}}\,}^{\lambda,\gamma,\mu}\}_{\vec{{k}}\,\in (\mathbb Z^n)^m} \big\Vert_{\ell^{\infty}}\le A_{\vec{{G}},\lambda,\mu} \end{align*} $$

and

$$ \begin{align*} \sup_{\gamma \in\mathbb{Z}}\big\Vert \{b_{\vec{{G}},\vec{{k}}\,}^{\lambda,\gamma,\mu}\}_{\vec{{k}}\,\in (\mathbb Z^n)^m} \big\Vert_{\ell^{q}}\le B_{\vec{{G}},\lambda,\mu,q}. \end{align*} $$

Then the following statements hold:

1. (1) For $1\le r\le 2$ , there exists a constant $C>0$ , independent of $,\vec {{G}}, \lambda , \mu $ , such that

   $$ \begin{align*} &\Big\Vert \bigg(\sum_{\gamma\in\mathbb{Z}}\Big| \sum_{\vec{{k}}\,\in \mathcal{U}_1^{\lambda+\mu}} b_{\vec{{G}},\vec{{k}}\,}^{\lambda,\gamma,\mu}L_{G_1,k_1}^{\lambda,\gamma}f_1^{\lambda,\gamma,\mu}\prod _{j=2}^{m}L_{G_j,k_j}^{\lambda,\gamma}f_j\Big|^r\bigg)^{1/r}\Big\Vert_{L^{2/m}}\\ &\le C A_{\vec{{G}},\lambda,\mu} 2^{\lambda mn/2}\Big(\sum_{\gamma\in\mathbb{Z}}{\Vert f_1^{\lambda,\gamma,\mu}\Vert_{L^2}^r} \Big)^{1/r} \prod_{j=2}^{m}\Vert f_{j}\Vert_{L^2} \end{align*} $$

   for Schwartz functions $f_1,\dots ,f_m$ on $\mathbb R^n$ .

2. (2) For $2\le l\le m$ , there exists a constant $C>0$ , independent of $\vec {{G}}, \lambda , \mu $ , such that

   $$ \begin{align*} &\Big\Vert \sum_{\gamma\in\mathbb{Z}} \Big| \sum_{\vec{{k}}\,\in \mathcal{U}_l^{\lambda+\mu}} b_{\vec{{G}},\vec{{k}}\,}^{\lambda,\gamma,\mu}\bigg( \prod_{j=1}^{l}L_{G_j,k_j}^{\lambda,\gamma}f_j^{\lambda,\gamma,\mu}\bigg)\bigg( \prod_{j=l+1}^{m}L_{G_j,k_{j}}^{\lambda,\gamma}f_{j}\bigg) \Big| \Big\Vert_{L^{2/m}}\\ &\le C A_{\vec{{G}},\lambda,\mu}^{1-\frac{(l-1)q}{2l}}B_{\vec{{G}},\lambda,\mu,q}^{\frac{(l-1)q}{2l}} 2^{\lambda mn/2}\bigg[ \prod_{j=1}^{l}\Big(\sum_{\gamma\in\mathbb{Z}}{\Vert f_j^{\lambda,\gamma,\mu}\Vert_{L^2}^2} \Big)^{1/2}\bigg] \bigg[\prod_{j=l+1}^{m}\Vert f_{j}\Vert_{L^2}\bigg] \end{align*} $$

   for Schwartz functions $f_1,\dots ,f_m$ on $\mathbb R^n$ , where $\prod _{m+1}^m$ is understood as the function 1.

In view of (2.5), (2.6), and Proposition 2.1, we actually obtain

(2.7) $$ \begin{align} &\Big\Vert \bigg( \sum_{\gamma\in\mathbb{Z}}\Big| \sum_{\vec{{k}}\,\in\mathcal{U}_{1}^{\lambda+\mu}}b_{\vec{{G}},\vec{{k}}\,}^{\lambda,\gamma,\mu} \prod_{j=1}^{m}L_{G_j,k_j}^{\lambda,\gamma}f_j\Big|^2\bigg)^{1/2}\Big\Vert_{L^{2/m}}\nonumber \\ &\lesssim A_{\vec{{G}},\lambda,\mu}\mu^{1/2}2^{\lambda mn/2}(\lambda+1)^{1/2}\prod_{j=1}^{m}\Vert f_j\Vert_{L^2}, \end{align} $$

and for $2\le l\le m$ ,

(2.8) $$ \begin{align} &\Big\Vert \sum_{\gamma\in\mathbb{Z}}\Big| \sum_{\vec{{k}}\,\in\mathcal{U}_{l}^{\lambda+\mu}}b_{\vec{{G}},\vec{{k}}\,}^{\lambda,\gamma,\mu}\prod_{j=1}^{m}L_{G_j,k_j}^{\lambda,\gamma}f_j\Big| \Big\Vert_{L^{2/m}}\nonumber\\ &\lesssim A_{\vec{{G}},\lambda,\mu}^{1-\frac{(l-1)q}{2l}}B_{\vec{{G}},\lambda,\mu,q}^{\frac{(l-1)q}{2l}} \mu^{l/2}2^{\lambda mn/2}(\lambda+1)^{l/2} \prod_{j=1}^{m}\Vert f_{j}\Vert_{L^2}. \end{align} $$

3 An auxiliary lemma

We have the following extension of Lemma 5 in [3].

Lemma 3.1 Let $1<q\le \infty $ and $\Omega \in L^q(\mathbb S^{mn-1})$ . Suppose $1<p_1,\dots , p_m<\infty $ and $1/m<p<\infty $ satisfies $1/p=1/p_1+\cdots +1/p_m$ and

(3.1) $$ \begin{align} \frac 1p< \frac{1}{q}+\frac{m}{q' }. \end{align} $$

Given $f_j\in L^{p_j}(\mathbb {R}^n)$ , there is a set of measure zero E such that for $x\in \mathbb R^n\setminus E$ ,

(3.2) $$ \begin{align} \idotsint\limits_{|\vec y\, |\le R} |\Omega (\vec{{y}}\,' ) | \prod_{j=1}^m \big| f_j(x-y_j) \big| ~ d\vec{{y}}\, <\infty, \end{align} $$

for all $R>0$ . Then, for $x\in \mathbb R^n\setminus E$ , the maximal operator

(3.3) $$ \begin{align} \mathcal M_{\Omega}(f_1,\dots, f_m)(x)= \sup_{R>0} \frac{1}{R^{mn} } \idotsint\limits_{|\vec y|\le R} |\Omega (\vec{{y}}\,' ) | \prod_{j=1}^m \big| f_j(x-y_j) \big| ~ d\vec{{y}}\, \end{align} $$

is well defined and maps $L^{p_1}(\mathbb R^n) \times \cdots \times L^{p_m}(\mathbb R^n) $ to $L^{p }(\mathbb R^n) $ with norm bounded by a constant multiple of $\|\Omega \|_{L^q(\mathbb {S}^{mn-1})}$ . Precisely, there is a constant $C>0$ such that

(3.4) $$ \begin{align} \|\mathcal M_{\Omega}(f_1,\dots, f_m)\|_{L^{p}}\le C\|\Omega\|_{L^q(\mathbb S^{mn-1})} \|f_1\|_{L^{p_1}}\cdots\|f_m\|_{L^{p_m}} \end{align} $$

for functions $f_j\in L^{p_j}(\mathbb {R}^n)$ , $1\le j\le m$ .

Proof Since $\Vert \Omega \Vert _{L^r(\mathbb {S}^{mn-1})}\lesssim \Vert \Omega \Vert _{L^{\infty }(\mathbb {S}^{mn-1})}$ for all $1<r<\infty $ and there exists $1<q<\infty $ such that $1/p<1/q+m/q'<m~ (=1/\infty +m/1)$ , we may assume that $1<q<\infty $ . Without loss of generality, we may also assume that $\|\Omega \|_{L^q(\mathbb {S}^{mn-1})}=1$ .

We split

$$\begin{align*}\Omega= \Omega_0+\sum_{l=1}^{\infty} \Omega_l , \end{align*}$$

where $\Omega _0=\Omega \chi _{|\Omega |\le 2}$ and $\Omega _l=\Omega \chi _{2^l<|\Omega |\le 2^{l+1} }$ for $l\ge 1$ . Then Hölder’s inequality and Chebyshev’s inequality give

$$\begin{align*}\|\Omega_l\|_{L^1} \le |\mathrm{Supp} \, \Omega_l|^{\frac{1}{q'}} \le \|\Omega\|_{L^q}^{\frac{q}{q'}} 2^{-l \frac{q}{q'}} =2^{-l \frac{q}{q'}}, \end{align*}$$

and obviously,

(3.5) $$ \begin{align} \|\Omega_l \|_{L^{\infty}} \le 2^{l+1}. \end{align} $$

We first claim that for $1<r,r_1,\dots ,r_m<\infty $ with $1/r=1/r_1+\cdots +1/r_m$ , we have

(3.6) $$ \begin{align} \big\|\mathcal M_{\Omega_l} (S_1,\dots, S_m) \big\|_{L^{r_1}\times \cdots \times L^{r_m}\to L^{r}} \lesssim 2^{-l \frac{q}{q'}} \prod_{j=1}^m\|S_j\|_{L^{r_j}(\mathbb{R}^n)} \end{align} $$

for simple functions $S_j$ . To verify this estimate, we choose indices $\mu _1,\dots ,\mu _m$ satisfying

$$ \begin{align*}1/\mu_1+\cdots+1/\mu_m=1\end{align*} $$

and

$$ \begin{align*}1<\mu_j<r_j \qquad \text{ for each }~ 1\le j\le m.\end{align*} $$

Then a direct computation using Hölder’s inequality yields

$$\begin{align*}\mathcal M_{\Omega_l}\big(S_1,\dots, S_m\big)(x) \le \int_{\mathbb S^{mn-1}} \big|\Omega_l (\vec \theta\, )\big| \prod_{j=1}^{m}\mathcal{M}_{\mu_j}^{\theta_j}S_j(x) ~ d\vec \theta\, , \end{align*}$$

where the directional maximal operator $\mathcal {M}_{\mu _j}^{\theta _j}$ is defined by

$$\begin{align*}\mathcal M_{\mu_j}^{\theta_j} g(x) := \sup_{R>0}\Big( \frac{1}{R }\int_0^R \big| g (x-t\theta_j) \big|^{\mu_j} ~dt\Big)^{1/\mu_j}. \end{align*}$$

It follows from this that

$$\begin{align*}\big\| \mathcal M_{\Omega_l}(S_1,\dots, S_m)\big\|_{L^{r}} \le \int_{\mathbb S^{mn-1}} |\Omega_l (\vec \theta\, ) | \prod_{j=1}^m \big\| \mathcal M_{\mu_j}^{\theta_j}S_j \big\|_{L^{r_j}} ~ d\vec \theta, \end{align*}$$

where Minkowski’s inequality and Hölder’s inequality are applied. Using the $L^{r_j}$ boundedness of $\mathcal M_{\mu _j}^{\theta _j} $ for $0<\mu _j<r_j$ with constants independent of $\theta _j$ (by the method of rotations), we obtain (3.6).

Then the case $p>1$ (for which $q>1$ implies the assumption (3.1)) in the assertion follows from summing the estimates (3.6) over $l\ge 0$ .

The other case $1/m<p\le 1$ can be proved by interpolation with the $L^1\times \cdots \times L^1\to L^{1/m,\infty }$ estimate. Let $\mathcal {M}$ be the Hardy–Littlewood maximal operator. Then, using (3.5), it is easy to verify the pointwise estimate

$$\begin{align*}\mathcal M_{\Omega_l}\big(f_1,\dots , f_m\big)(x) \le 2^{l+1} \prod_{j=1}^m \mathcal M f_j(x), \end{align*}$$

for $f_1,\dots , f_m$ in $L^1(\mathbb R^n)$ , and this yields that

(3.7) $$ \begin{align} \big\|\mathcal M_{\Omega_l} \big\|_{L^1\times \cdots \times L^1\to L^{1/m,\infty}} \lesssim 2^l, \end{align} $$

using Hölder’s inequality for weak-type spaces [14, p. 16] and the weak $(1,1)$ boundedness of $\mathcal {M}$ . Now we fix $0<p_1,\dots ,p_m<\infty $ and $1/m<p\le 1$ , and choose $r>1$ such that

$$ \begin{align*} \frac{1}{p}<\frac{1}{rq}+\frac{m}{q'} \,\,\,\bigg( \!\!<\frac{1}{q}+\frac{m}{q'} \bigg), \end{align*} $$

or, equivalently,

$$ \begin{align*}\frac{q(m-1/p)}{q'(m-1/r)}-\frac{1/p-1/r}{m-1/r}>0.\end{align*} $$

Interpolating between (3.7) and (3.6) with appropriate $(r_1,\dots , r_m)$ satisfying $1/r=1/r_1+\dots +1/r_m$ (using [15, Theorem 7.2.2]) yields

$$ \begin{align*}\|\mathcal M_{\Omega_l}(S_1,\dots, S_m)\|_{L^{p}}\le C2^{-l(\frac{q(m-1/p)}{q'(m-1/r)}-\frac{1/p-1/r}{m-1/r})} \|S_1\|_{L^{p_1}}\cdots\|S_m\|_{L^{p_m}} \end{align*} $$

for simple functions $S_j$ . The exponential decay in l obtained above together with the fact that $\|\cdot \|_{L^p}^p$ is a subadditive quantity for $0<p\le 1$ implies, for p and q satisfying (3.1),

(3.8) $$ \begin{align} \|\mathcal M_{\Omega}(S_1,\dots, S_m)\|_{L^{p}}\le C \|S_1\|_{L^{p_1}}\cdots\|S_m\|_{L^{p_m}} \end{align} $$

for simple functions $S_1,\dots , S_m$ .

Next, we extend $\mathcal M_{\Omega }(f_1,\dots , f_m)$ to functions $f_j\in L^{p_j}(\mathbb {R}^n).$ To achieve this goal, we choose nonnegative simple functions $S_j^k$ that increase pointwise to $|f_j|$ as $k\to \infty $ .

It follows from (3.8) that for any $R>0$ ,

$$\begin{align*}\bigg\{\int_{\mathbb R^{n}} \bigg[ \frac{1}{R^{mn}} \idotsint\limits_{|\vec y\, |\le R} |\Omega (\vec{{y}}\,' ) | \prod_{j=1}^m S_j^k(x-y_j) ~ d\vec{{y}}\, \bigg]^{p} dx \bigg\}^{\frac 1p}\le C\,\|f_1\|_{L^{p_1}}\cdots\|f_m\|_{L^{p_m}}, \end{align*}$$

and from this, we obtain

$$\begin{align*}\bigg\{\int_{\mathbb R^{n}} \bigg[ \frac{1}{R^{mn}} \idotsint\limits_{|\vec y\, |\le R} |\Omega (\vec{{y}}\,' ) | \prod_{j=1}^m |f_j^k(x-y_j)| ~ d\vec{{y}}\, \bigg]^{p} dx \bigg\}^{\frac 1p}\le C\, \|f_1\|_{L^{p_1}}\cdots\|f_m\|_{L^{p_m}} \end{align*}$$

via Lebesgue’s monotone convergence theorem. We conclude that for any $R\in \mathbb Z^+$ , there is a set of measure zero $E^R$ such that

(3.9) $$ \begin{align} \frac{1}{R^{mn}} \idotsint\limits_{|\vec y\, |\le R} |\Omega (\vec{{y}}\,' ) | \prod_{j=1}^m |f_j^k(x-y_j)| ~ d\vec{{y}}\, <\infty \end{align} $$

for all $x\in \mathbb R^n\setminus E^R$ . Setting $E= \cup _{R=1}^{\infty } E^R$ , we obtain (3.2) for $x\in \mathbb R^n\setminus E$ .

This allows us to define $\mathcal M_{\Omega }(f_1,\dots , f_m)(x) $ for $f_j\in L^{p_j}(\mathbb {R}^n)$ and $x\in \mathbb R^n\setminus E$ as the supremum of the expressions in (3.9). Now, for $x\in \mathbb R^n\setminus E$ ,

$$ \begin{align*} \mathcal M_{\Omega}(f_1,\dots, f_m)(x)= &\sup_{R>0}\frac1{R^{mn}}\idotsint\limits_{|\vec y\, |\le R} |\Omega (\vec{{y}}\,' ) | \prod_{j=1}^m \big| f_j(x-y_j) \big| ~ d\vec{{y}}\,\\ =&\sup_{R>0}\lim_{k\to\infty} \frac1{R^{mn}}\idotsint\limits_{|\vec y\, |\le R} |\Omega (\vec{{y}}\,' ) | \prod_{j=1}^m S^k_j(x-y_j) ~ d\vec{{y}}\,\\ \le&\liminf_{k\to\infty} \mathcal M_{\Omega}(S^k_1,\dots, S^k_m)(x). \end{align*} $$

As $\mathcal M_{\Omega }(S^k_1,\dots , S^k_m)(x)$ is increasing in k, we obtain from (3.8) by Fatou’s lemma that

$$ \begin{align*}\| \mathcal M_{\Omega}(f_1,\dots, f_m)\|_{L^p(\mathbb{R}^n)} \lesssim \liminf_{k\to\infty} \|\mathcal M_{\Omega}(S^k_1,\dots, S^k_m)\|_{L^p(\mathbb{R}^n)} \lesssim \prod_{j=1}^m\|f_j\|_{L^{p_j}(\mathbb{R}^n)}. \end{align*} $$

In particular, this shows that for $f_j\in L^{p_j}(\mathbb {R}^n)$ , there is a set $E'$ of measure zero such that

$$\begin{align*}\sup_{R>0} \frac{1}{R^{mn} } \idotsint\limits_{|\vec y|\le R} |\Omega (\vec{{y}}\,' ) | \prod_{j=1}^m \big| f_j(x-y_j) \big| ~ d\vec{{y}}\,<\infty \end{align*}$$

for all. $x\in \mathbb {R}^n\setminus E'$ 3.9

4 Proof of Theorem 1.2

Let $\frac {2m}{m+1}<q<2$ and $\Omega $ in $L^q(\mathbb S^{mn-1})$ . We use a dyadic decomposition introduced by Duoandikoetxea and Rubio de Francia [13]. We choose a Schwartz function $\Phi ^{(m)}$ on $(\mathbb R^n)^m$ such that its Fourier transform $\widehat {\Phi ^{(m)}}$ is supported in the annulus $\{\vec {{\xi }}\,\in (\mathbb R^n)^m: 1/2\le |\vec {{\xi }}\,|\le 2\}$ and satisfies $\sum _{j\in \mathbb {Z}}\widehat {\Phi ^{(m)}_j}(\vec {{\xi }}\,)=1$ for $\vec {{\xi }}\,\not = \vec {\boldsymbol {0}}$ where $\widehat {\Phi _j^{(m)}}(\vec {{\xi }}\,):=\widehat {\Phi ^{(m)}}(\vec {{\xi }}\,/2^j)$ . For $\gamma \in \mathbb {Z}$ , let

$$ \begin{align*}K^{\gamma}(\vec{{y}}\,):=\widehat{\Phi^{(m)}}(2^{\gamma}\vec{{y}}\,)K(\vec{{y}}\,), \quad \vec{{y}}\,\in (\mathbb R^n)^m,\end{align*} $$

and then we observe that $K^{\gamma }(\vec {{y}}\,)=2^{\gamma mn} K^0(2^{\gamma } \vec {{y}}\,)$ . For $\mu \in \mathbb {Z}$ , we define

(4.1) $$ \begin{align} K_{\mu}^{\gamma}(\vec{{y}}\,):=\Phi_{{\mu}+\gamma}^{(m)}\ast K^{\gamma}(\vec{{y}}\,)=2^{\gamma mn}[\Phi_{{\mu}}^{(m)}\ast K^{0}](2^{\gamma} \vec{{y}}\,). \end{align} $$

It follows from this definition that

$$ \begin{align*}\widehat{K^{\gamma}_{\mu}}(\vec{{\xi}}\,)= \widehat{\Phi^{(m)}}(2^{-(\mu+\gamma)}\vec{{\xi}}\,)\widehat{K^0}(2^{-\gamma}\vec{{\xi}}\,)=\widehat{K^0_{\mu}}(2^{-\gamma} \vec{{\xi}}\,), \end{align*} $$

which implies that $\widehat {K^{\gamma }_{\mu }}$ is bounded uniformly in $\gamma $ , while they have almost disjoint supports, so it is natural to add them together as follows:

$$ \begin{align*}K_{\mu}(\vec{{y}}\,):=\sum_{\gamma\in \mathbb{Z}}{K_{\mu}^{\gamma}(\vec{{y}}\,)}.\end{align*} $$

4.1 Reduction

We introduce the maximal operator

$$ \begin{align*} {\mathcal{L}}_{\Omega}^{\sharp}\big(f_1,\dots,f_m\big)(x) := \sup_{\tau\in \mathbb Z} \bigg| \sum_{\gamma<\tau} \int_{(\mathbb R^n)^m }{K^{\gamma}(\vec{{y}}\,)\prod_{j=1}^{m}f_j(x-y_j)}~d\,\vec{{y}}\, \bigg| \end{align*} $$

for $x\in \mathbb R^n$ . Then we claim that

(4.2) $$ \begin{align} {\mathcal{L}}_{\Omega}^*\big(f_1,\dots,f_m\big) \le \mathcal{M}_{\Omega}\big(f_1,\dots,f_m\big)(x)+ {\mathcal{L}}_{\Omega}^{\sharp}\big(f_1,\dots,f_m\big). \end{align} $$

To prove (4.2), we introduce the notation

$$ \begin{align*}K^{(\epsilon)} ( \vec y\,) := K ( \vec y\,) \chi_{|\vec y\,| \ge \epsilon}, \qquad \widetilde{K}^{(\epsilon)} ( \vec y\,) := K ( \vec y\,) \big( 1- \widehat{\Theta^{(m)}} (\vec y/\epsilon) \big),\end{align*} $$

setting $\widehat {\Theta ^{(m)}}(\vec {{y}}\,):=1-\sum _{\gamma \in \mathbb {N}}{\widehat {\Phi ^{(m)}}(\vec {{y}}\,/2^{\gamma })}$ so that

$$\begin{align*}\mathrm{Supp}(\widehat{\Theta^{(m)}})\subset \{\vec{{y}}\,\in (\mathbb R^n)^m:|\vec{{y}}\,|\le 2\} \end{align*}$$

and $\widehat {\Theta ^{(m)}}(\vec {{y}}\,)=1$ for $|\vec {{y}}\,|\le 1$ .

Given $\epsilon>0$ , choose $\rho \in \mathbb Z$ such that $2^{\rho }\le \epsilon <2^{\rho +1} $ . Then we write

(4.3) $$ \begin{align} &\bigg| \int_{(\mathbb R^n)^m\setminus B(0,\epsilon)}{K(\vec{{y}}\,)\prod_{j=1}^{m}f_j(x-y_j)}~d\,\vec{{y}}\, \bigg| \notag\\ &\qquad\qquad \le \bigg| \int_{(\mathbb R^n)^m }{\big( K^{(\epsilon)}(\vec{{y}}\,) - \widetilde{K}^{(2^{\rho})}(\vec{{y}}\,) \big) \prod_{j=1}^{m}f_j(x-y_j)}~d\,\vec{{y}}\, \bigg| \end{align} $$

(4.4) $$ \begin{align} &\qquad \qquad\qquad \qquad+ \bigg| \int_{(\mathbb R^n)^m} \widetilde{K}^{(2^{\rho})}(\vec{{y}}\,) \prod_{j=1}^{m}f_j(x-y_j) ~d\,\vec{{y}}\,\bigg|. \end{align} $$

Term (4.4) is clearly less than

$$ \begin{align*} \bigg|\sum_{\gamma\in\mathbb{Z}: \gamma<-\rho}\int_{(\mathbb R^n)^m}K^{\gamma}(\vec{{y}}\,) \prod_{j=1}^{m}f_j(x-y_j) ~d\,\vec{{y}}\, \bigg|\le {\mathcal{L}}_{\Omega}^{\sharp}\big(f_1,\dots,f_m\big)(x), \end{align*} $$

while (4.3) is controlled by $ \mathcal {M}_{\Omega }\big (f_1,\dots ,f_m\big )(x) $ as

$$ \begin{align*} \big|K^{(\epsilon)}(\vec{{y}}\,) - \widetilde K^{(2^{\rho})}(\vec{{y}}\,) \big|\lesssim |K(\vec{{y}}\,)| \chi_{|\vec{{y}}\,|\approx 2^{\rho}}\lesssim \frac{|\Omega(\vec{{y}}\,')|}{2^{\rho mn}}\chi_{|\vec{{y}}\,|\lesssim 2^{\rho}}. \end{align*} $$

Thus, (4.2) follows after taking the supremum over all $\epsilon>0$ .

Since the boundedness of $\mathcal {M}_{\Omega }$ follows from Lemma 3.1 with the fact that $q>\frac {2m}{m+1}$ implies $\frac {m}{2} <\frac {1}{q}+\frac {m}{q'}$ , matters reduce to the boundedness of ${\mathcal {L}}_{\Omega }^{\sharp }$ .

For each $\gamma \in \mathbb {Z}$ , let

$$ \begin{align*}K_{\mu}:=\sum_{\gamma\in\mathbb{Z}}K_{\mu}^{\gamma}.\end{align*} $$

In the study of multilinear rough singular integral operators ${\mathcal {L}}_{\Omega }$ in [18] whose kernel is $\sum _{\gamma \in \mathbb {Z}}K^{\gamma }=\sum _{\mu \in \mathbb {Z}}\sum _{\gamma \in \mathbb {Z}}K_{\mu }^{\gamma }=\sum _{\mu \in \mathbb {Z}}K_{\mu }$ , the part where $\mu $ is less than a constant is relatively simple because the Fourier transform of $K_{\mu }$ satisfies the estimate

(4.5) $$ \begin{align} \big|\partial^{\alpha}\widehat{K_{\mu}}(\vec{{\xi}}\,)\big|\lesssim \Vert \Omega\Vert_{L^q(\mathbb{S}^{mn-1})}|\vec{{\xi}}\,|^{-|\alpha|}Q(\mu),\qquad 1<q\le \infty, \end{align} $$

for all multi-indices $\alpha $ and $\vec {{\xi }}\,\in \mathbb {R}^{mn}\setminus \{0\}$ , where $Q(\mu )=2^{(mn-\delta ')\mu }$ if $\mu \ge 0$ and $Q(\mu )=2^{\mu (1-\delta ')}$ if $\mu <0$ for some $0<\delta '<1/q'$ , which is the condition of the Coifman–Meyer multiplier theorem ([9], [15, Theorem 7.5.3]) with constant $\Vert \Omega \Vert _{L^q(\mathbb S^{mn-1})}Q(\mu )$ . The remaining case when $\mu $ is large enough was handled by using product-type wavelet decompositions. We expect that a similar strategy would work in handling ${\mathcal {L}}_{\Omega }^{\sharp }$ .

To argue strictly, we write

$$ \begin{align*} {\mathcal{L}}_{\Omega}^{\sharp}\big(f_1,\dots,f_m\big) \le \widetilde {\mathcal{L}}_{\Omega}^{\sharp}\big(f_1,\dots,f_m\big) + \sum_{\mu\in\mathbb{Z}:2^{\mu-10}>C_0\sqrt{mn}} {\mathcal{L}}_{\Omega,\mu}^{\sharp}\big(f_1,\dots,f_m\big) , \end{align*} $$

where we set

$$ \begin{align*}\widetilde {\mathcal{L}}_{\Omega}^{\sharp}\big(f_1,\dots,f_m\big) (x) := \sup_{\tau \in \mathbb Z} \bigg| \int_{(\mathbb R^n)^m }{ \sum_{ \gamma<\tau} \sum_{\mu\in\mathbb{Z}:2^{\mu-10}\le C_0\sqrt{mn}} K^{\gamma}_{\mu}(\vec{{y}}\,) \prod_{j=1}^{m}f_j(x-y_j)}~d\,\vec{{y}}\, \bigg| \end{align*} $$

and

$$ \begin{align*}{\mathcal{L}}_{\Omega,\mu}^{\sharp}\big(f_1,\dots,f_m\big)(x) := \sup_{\tau\in \mathbb Z} \bigg| \sum_{\gamma<\tau} \int_{(\mathbb R^n)^m }{ K^{\gamma}_{\mu}(\vec{{y}}\,) \prod_{j=1}^{m}f_j(x-y_j)}~d\,\vec{{y}}\, \bigg|. \end{align*} $$

Then Theorem 1.2 follows from the following two propositions.

Proposition 4.1 Let $1<p_1,\dots ,p_m\le \infty $ and $1/p=1/p_1+\cdots +1/p_m$ . Suppose that $1<q<\infty $ and $\Omega \in L^q(\mathbb {S}^{mn-1})$ with $\int _{\mathbb {S}^{mn-1}}\Omega d\sigma =0$ . Then there exists a constant $C>0$ such that

$$ \begin{align*} \big\Vert \widetilde{\mathcal{L}}_{\Omega}^{\sharp}(f_1,\dots,f_m)\big\Vert_{L^p}\le C\Vert \Omega\Vert_{L^q(\mathbb{S}^{mn-1})}\prod_{j=1}^{m}\Vert f_j\Vert_{L^{p_j}} \end{align*} $$

for Schwartz functions $f_1,\dots ,f_m$ on $\mathbb R^n$ .

Proposition 4.2 Let $\frac {2m}{m+1}<q\le \infty $ and $\Omega \in L^q(\mathbb {S}^{mn-1})$ with $\int _{\mathbb {S}^{mn-1}}\Omega d\sigma =0$ . Suppose that $\mu \in \mathbb {Z}$ satisfies $2^{\mu -10}>C_0\sqrt {mn}$ . Then there exist $C, \epsilon _0>0$ such that

$$ \begin{align*} \big\| {\mathcal{L}}_{\Omega,\mu}^{\sharp}(f_1,\dots,f_m)\big\|_{L^{2/m} } \lesssim 2^{-\epsilon_0 \mu} \| \Omega\|_{L^q(\mathbb{S}^{mn-1})}\prod_{j=1}^{m}\Vert f_j\Vert_{L^2} \end{align*} $$

for Schwartz functions $f_1,\dots ,f_m$ on $\mathbb R^n$ .

4.2 Proof of Proposition 4.1

We decompose $\widetilde {\mathcal {L}}_{\Omega }^{\sharp }$ further so that the Coifman–Meyer multiplier theorem is involved: setting

$$ \begin{align*}\widetilde{K}(\vec{{y}}\,):=\sum_{\mu\in\mathbb{Z}: 2^{\mu-10}\le C_0\sqrt{mn}}K_{\mu}(\vec{{y}}\,)=\sum_{\mu\in\mathbb{Z}: 2^{\mu-10}\le C_0\sqrt{mn}}\;\sum_{\gamma\in\mathbb{Z}}K_{\mu}^{\gamma}(\vec{{y}}\,),\end{align*} $$

$\widetilde {\mathcal {L}}^{\sharp }_{\Omega }\big (f_1,\dots ,f_m\big )(x)$ is controlled by the sum of

$$ \begin{align*} T_{\widetilde{K}}^*\big(f_1,\dots,f_m\big)(x):= \sup_{\tau\in\mathbb{Z}}\Big|\int_{|y|> 2^{-\tau}}\widetilde{K}(\vec{{y}}\,)\prod_{j=1}^{m}f_j(x-y_j)~d\,\vec{{y}}\, \Big| \end{align*} $$

and

(4.6) $$ \begin{align} \mathfrak{T}_{K}^{**}\big(f_1,\dots,f_m\big)(x):= \sup_{\tau\in\mathbb{Z}}\Big| \int_{(\mathbb R^n)^m} {K}^{**}_{\tau}(\vec{{y}}\,) \prod_{j=1}^{m}f_j(x-y_j) ~ d\,\vec{{y}}\, \Big|, \end{align} $$

where

(4.7) $$ \begin{align} {K}^{**}_{\tau}(\vec{{y}}\,):=\Big(\sum_{\mu\in\mathbb{Z}: 2^{\mu-10}\le C_0\sqrt{mn}}\; \sum_{\gamma<\tau}K_{\mu}^{\gamma}(\vec{{y}}\,)\Big) -\widetilde{K}(\vec{{y}}\,)\chi_{|\vec{{y}}\,|> 2^{-\tau}}. \end{align} $$

To obtain the boundedness of $T_{\widetilde {K}}^*$ , we claim that $\widetilde {K}$ is an m-linear Calderón–Zygmund kernel with constant $C\Vert \Omega \Vert _{L^q(\mathbb {S}^{mn-1})}$ for $1<q< \infty $ . Indeed, it follows from (4.5) that

$$ \begin{align*} \big| \partial^{\alpha} \widehat{\widetilde{K}}(\vec{{\xi}}\,)\big|\le \sum_{\mu\in\mathbb{Z}: 2^{\mu-10}\le C_0\sqrt{mn}}\big|\partial^{\alpha}\widehat{K_{\mu}}(\vec{{\xi}}\,)\big|\lesssim \Vert \Omega\Vert_{L^q(\mathbb{S}^{mn-1})}|\vec{{\xi}}\,|^{-|\alpha|} \end{align*} $$

as the sum of $Q(\mu )$ over $\mu $ satisfying $2^{\mu -10}\le C_0\sqrt {mn}$ converges. Then $\widetilde {K}$ satisfies the size and smoothness conditions for m-linear Calderón–Zygmund kernel with constant $C\Vert \Omega \Vert _{L^q(\mathbb {S}^{mn-1})}$ , as mentioned in the proof of [21, Proposition 6]. Since $\widetilde {K}$ is a Calderón–Zygmund kernel, Cotlar’s inequality in [20, Theorem 1] yields that $T_{\widetilde {K}}^*$ is bounded on the full range of exponents with constant $C\Vert \Omega \Vert _{L^{q}(\mathbb {S}^{mn-1})}$ .

To handle the boundedness of the operator $\mathfrak {T}_K^{**}$ , we observe that the kernel ${K}_{\tau }^{**}$ can be written as

(4.8) $$ \begin{align} {K}_{\tau}^{**}(\vec{{y}}\,)=\sum_{\mu\in\mathbb{Z}:2^{\mu-10}\le C_0\sqrt{mn} }\; \Big( \sum_{\gamma<\tau}K_{\mu}^{\gamma}(\vec{{y}}\,)\chi_{|\vec{{y}}\,|\le 2^{-\tau}}- \sum_{\gamma\ge \tau}K_{\mu}^{\gamma}(\vec{{y}}\,)\chi_{|\vec{{y}}\,|>2^{-\tau}}\Big), \end{align} $$

and thus

$$ \begin{align*} \mathfrak{T}_{K}^{**}\big(f_1,\dots,f_m\big)(x) \le \sup_{\tau\in\mathbb{Z}}\; \sum_{\mu\in\mathbb{Z}:2^{\mu-10}\le C_0\sqrt{mn} } \mathcal{I}_{\mu,\tau}(x)+\mathcal{J}_{\mu,\tau}(x), \end{align*} $$

where

$$ \begin{align*} \mathcal{I}_{\mu,\tau}(x):= \sum_{\gamma<\tau}\Big|\int_{|\vec{{y}}\,|<2^{-\tau}} K_{\mu}^{\gamma}(\vec{{y}}\,) \prod_{j=1}^{m}f_j(x-y_j) \;d\,\vec{{y}}\, \Big|, \end{align*} $$

$$ \begin{align*} \mathcal{J}_{\mu,\tau}(x):= \sum_{\gamma \ge \tau}\Big|\int_{|\vec{{y}}\,|\ge 2^{-\tau}} K_{\mu}^{\gamma}(\vec{{y}}\,) \prod_{j=1}^{m}f_j(x-y_j) \;d\,\vec{{y}}\, \Big|. \end{align*} $$

We claim that there exists $\epsilon>0$ such that

(4.9) $$ \begin{align} \mathcal{I}_{\mu,\tau}+ \mathcal{J}_{\mu,\tau}\lesssim_{C_0,m,n} 2^{\epsilon \mu}\Vert \Omega\Vert_{L^1(\mathbb{S}^{mn-1})}\prod_{j=1}^{m}\mathcal{M}f_j ~\text{ uniformly in ~} \tau\in\mathbb{Z} \end{align} $$

for $\mu $ satisfying $2^{\mu -10}\le C_0\sqrt {mn}$ , where we recall $\mathcal M$ is the Hardy–Littlewood maximal operator. Then, using Hölder’s inequality and the boundedness of $\mathcal {M}$ , we obtain

(4.10) $$ \begin{align} \big\Vert \mathfrak{T}_{K}^{**}\big(f_1,\dots,f_m\big) \big\Vert_{L^p}\lesssim \Vert \Omega\Vert_{L^1(\mathbb{S}^{mn-1})} \Big\Vert \prod_{j=1}^{m}\mathcal{M}f_j \Big\Vert_{L^p}\lesssim \Vert \Omega\Vert_{L^1(\mathbb{S}^{mn-1})}\prod_{j=1}^{m}\Vert f_j\Vert_{L^{p_j}} \end{align} $$

for $1<p_1,\dots ,p_m\le \infty $ and $0<p\le \infty $ satisfying $1/p=1/p_1+\dots +1/p_m$ as $\sum _{\mu : 2^{\mu -10}\le C_0\sqrt {mn}}2^{\epsilon \mu }$ converges. Therefore, let us prove (4.9).

Using (4.1), we have

$$ \begin{align*} \mathcal{I}_{\mu,\tau}(x)&\lesssim \sum_{\gamma<\tau}\int_{|\vec{{y}}\,|<2^{-\tau}}\int_{|\vec{{z}}\,|\approx 1}2^{\gamma mn}2^{\mu mn}|\Omega(\vec{{z}}\,')|d\vec{{z}}\, \prod_{j=1}^{m}|f_j(x-y_j)|~d\,\vec{{y}}\,\\ &\lesssim 2^{\mu mn}\Vert \Omega\Vert_{L^1(\mathbb{S}^{mn-1})}\frac{1}{2^{-\tau mn}}\int_{|\vec{{y}}\,|<2^{-\tau}} \prod_{j=1}^{m}|f_j(x-y_j)| ~d\,\vec{{y}}\,\\ &\lesssim 2^{ \mu mn}\Vert \Omega\Vert_{L^1(\mathbb{S}^{mn-1})}\prod_{j=1}^{m}\mathcal{M}f_j(x), \end{align*} $$

as desired.

In addition,

$$ \begin{align*} \mathcal{J}_{\mu,\tau}(x)\le \sum_{\gamma\ge \tau}\int_{|\vec{{y}}\,|\ge 2^{-\tau}}2^{\gamma mn}\Big|\int_{|\vec{{z}}\,|\approx 1}{\Phi_{\mu}(2^{\gamma}\vec{{y}}\,-\vec{{z}}\,)\Omega(\vec{{z}}\,')}d\,\vec{{z}}\, \Big|\prod_{j=1}^{m}|f_j(x-y_j)| ~ d\,\vec{{y}}\,. \end{align*} $$

Since $\Omega $ has vanishing mean, we have

$$ \begin{align*} &\Big|\int_{|\vec{{z}}\,|\approx 1}{\Phi_{\mu}(2^{\gamma}\vec{{y}}\,-\vec{{z}}\,)\,\Omega(\vec{{z}}\,')}\,d\,\vec{{z}}\, \Big| \\ &\lesssim 2^{\mu (mn+1)}\int_{|\vec{{z}}\,|\approx 1}\int_0^1\big| \nabla \Phi (2^{\mu+\gamma}\vec{{y}}\,-2^{\mu}t\vec{{z}}\,)\big|\;dt\; |\Omega(\vec{{z}}\,')|~ d\, \vec{{z}}\,. \end{align*} $$

Now we choose a constant M such that $mn<M<mn+1$ and see that

$$ \begin{align*} \big| \nabla \Phi (2^{\mu+\gamma}\vec{{y}}\,-2^{\mu}t\vec{{z}}\,)\big|&\lesssim_M \frac{1}{(1+|2^{\mu+\gamma}\vec{{y}}\,-2^{\mu}t\vec{{z}}\,|)^M}\\ &\lesssim_{C_0,m,n,M} \frac{1}{(1+2^{\mu+\gamma}|\vec{{y}}\,|)^M}\le \frac{1}{2^{M(\mu+\gamma)}}\frac{1}{|\vec{{y}}\,|^{M}}, \end{align*} $$

as $|\vec {{z}}\,|\approx 1$ , $0<t<1$ , and $2^{\mu -10}\le C_0\sqrt {mn}$ . This yields that

$$ \begin{align*} \mathcal{J}_{\mu,\tau}(x)&\lesssim 2^{\mu(mn+1-M)}\Vert \Omega\Vert_{L^1(\mathbb{S}^{mn-1})}\Big(\sum_{\gamma\ge \tau}2^{-\gamma(M- mn)}\Big)\\ &\qquad \qquad \qquad \qquad \times \int_{|\vec{{y}}\,|\ge 2^{-\tau}}{\frac{1}{|\vec{{y}}\,|^M}\prod_{j=1}^{m}|f_j(x-y_j)|}~ d\, \vec{{y}}\,. \end{align*} $$

Since $M>mn$ , the sum over $\gamma \ge \tau $ converges to $2^{-\tau (M-mn)}$ and the integral over $|\vec {{y}}\,|\ge 2^{-\tau }$ is estimated by

$$ \begin{align*} &\sum_{l=0}^{\infty}\int_{2^{-\tau+l}\le |\vec{{y}}\,|<2^{-\tau+l+1}}{\frac{1}{|\vec{{y}}\,|^M}\prod_{j=1}^{m}|f_j(x-y_j)|}~ d\,\vec{{y}}\,\\ &\lesssim 2^{\tau(M-mn)}\sum_{l=0}^{\infty}2^{-l(M-mn)} \bigg(\frac{1}{2^{(-\tau+l+1)mn}}\int_{|\vec{{y}}\,|\le 2^{-\tau+l+1}}\prod_{j=1}^{m}|f_j(x-y_j)| ~ d\,\vec{{y}}\,\bigg)\\ &\lesssim 2^{\tau(M-mn)}\prod_{j=1}^{m}\mathcal{M}f_j(x). \end{align*} $$

Finally, we have

$$ \begin{align*} \mathcal{J}_{\mu,\tau} \lesssim 2^{\mu(mn+1-M)}\Vert \Omega\Vert_{L^1(\mathbb{S}^{mn-1})}\prod_{j=1}^{m}\mathcal{M}f_j , \end{align*} $$

which completes the proof of (4.9).

4.3 Proof of Proposition 4.2

The proof is based on the wavelet decomposition and the recent developments in [18]. Recalling that $\widehat {K_{\mu }^0}\in L^{q'}$ , we apply the wavelet decomposition (2.1) to write

$$ \begin{align*} \widehat{K_{\mu}^0}(\vec{{\xi}}\,)=\sum_{\lambda\in\mathbb{N}_0}\sum_{\vec{{G}}\in\mathcal{I}^{\lambda}}\sum_{\vec{{k}}\,\in (\mathbb Z^n)^m}b_{\vec{{G}},\vec{{k}}\,}^{\lambda,\mu}\Psi_{G_1,k_1}^{\lambda}(\xi_1)\cdots \Psi_{G_m,k_m}^{\lambda}(\xi_m), \end{align*} $$

where

$$ \begin{align*} b_{\vec{{G}},\vec{{k}}\,}^{\lambda,\mu}:=\int_{(\mathbb R^n)^m}{\widehat{K_{\mu}^0}(\vec{{\xi}}\,)\Psi_{\vec{{G}},\vec{{k}}\,}^{\lambda}(\vec{{\xi}}\,)}~ d\,\vec{{\xi}}\,. \end{align*} $$

It is known in [18] that for any $0<\delta <1/q'$ ,

(4.11) $$ \begin{align} \big\Vert \{b_{\vec{{G}},\vec{{k}}\,}^{\lambda,\mu}\}_{\vec{{k}}\,}\big\Vert_{\ell^{\infty}}\lesssim 2^{-\delta {\mu}}2^{-\lambda (M+1+mn)} \Vert \Omega\Vert_{L^q(\mathbb{S}^{mn-1})}, \end{align} $$

where M is the number of vanishing moments of $\Psi _{\vec {{G}}}$ . Moreover, it follows from the inequality (2.2), the Hausdorff–Young inequality, and Young’s inequality that

(4.12) $$ \begin{align} \big\Vert \{b_{\vec{{G}},\vec{{k}}\,}^{\lambda,\mu}\}_{\vec{{k}}\,}\big\Vert_{\ell^{q'}}&\lesssim 2^{-\lambda mn (1/2-1/q')}\Vert \widehat{K_{\mu}^0}\Vert_{L^{q'}}\lesssim 2^{-\lambda mn(1/q-1/2)}\Vert \Omega \Vert_{L^q(\mathbb{S}^{mn-1})}. \end{align} $$

Now we may assume that $2^{\lambda +\mu -2}\le |\vec k|\le 2^{\lambda +\mu +2}$ due to the compact supports of $\widehat {K_{\mu }^0}$ and $\Psi _{\vec {{G}},\vec {{k}}\,}^{\lambda }$ . In addition, by symmetry, it suffices to focus on the case $|k_1|\ge \cdots \ge |k_m|$ . Since $\widehat {K_{\mu }^{\gamma }}(\vec {{\xi }}\,)=\widehat {K_{\mu }^0}(\vec {{\xi }}\,/2^{\gamma })$ , the boundedness of $ {\mathcal {L}}_{\Omega ,\mu }^{\sharp }$ is reduced to the inequality

(4.13) $$ \begin{align} &\bigg\Vert \sup_{\tau\in\mathbb{Z}}\Big| \sum_{\lambda\in\mathbb{N}_0}\sum_{\vec{{G}}\in\mathcal{I}^{\lambda}} \sum_{\gamma\in\mathbb{Z}: \gamma<\tau}\sum_{\vec{{k}}\,\in\mathcal{U}^{\lambda+\mu}} b_{\vec{{G}},\vec{{k}}\,}^{\lambda,\mu}\prod_{j=1}^{m} L_{G_j,k_j}^{\lambda,\gamma}f_j\Big|\bigg\Vert_{L^{2/m}}\nonumber\\ &\lesssim 2^{-\epsilon_0 \mu}\Vert \Omega\Vert_{L^q(\mathbb{S}^{mn-1})}\prod_{j=1}^{m}\Vert f_j\Vert_{L^2}, \end{align} $$

where the operators $L_{G_j,k_j}^{\lambda ,\gamma }$ and the set $\mathcal {U}^{\lambda +\mu }$ are defined as in (2.5) and (2.4). We split $\mathcal {U}^{\lambda +{\mu }}$ into m disjoint subsets $\mathcal {U}_l^{\lambda +{\mu }}$ ( $1\le l\le m$ ) as before such that for $k\in \mathcal {U}^{\lambda +\mu }_l$ , we have

$$ \begin{align*}|k_1|\ge\cdots \ge |k_l|\ge 2C_0\sqrt n\ge |k_{l+1}|\ge\cdots\ge |k_m|.\end{align*} $$

Then the left-hand side of (4.13) is estimated by

$$ \begin{align*} \bigg( \sum_{l=1}^{m}\sum_{\lambda\in\mathbb{N}_0}\sum_{\vec{{G}}\in\mathcal{I}^{\lambda}} \bigg\Vert \sup_{\tau\in\mathbb{Z}}\Big| \sum_{\gamma\in\mathbb{Z}:\gamma<\tau} \mathcal{T}_{\vec{{G}},l}^{\lambda,\gamma,\mu}(f_1,\dots,f_m) \Big| \bigg\Vert_{L^{2/m}}^{2/m}\bigg)^{m/2}, \end{align*} $$

where $\mathcal {T}_{\vec {{G}},l}^{\lambda ,\gamma ,\mu }$ is defined by

$$ \begin{align*} \mathcal{T}_{\vec{{G}},l}^{\lambda,\gamma,\mu}\big(f_1,\dots,f_m\big):=\sum_{\vec{{k}}\,\in\mathcal{U}_l^{\lambda+{\mu}}}b_{\vec{{G}},\vec{{k}}\,}^{\lambda,\mu} \bigg(\prod_{j=1}^{m}L_{G_j,k_j}^{\lambda,\gamma}f_j \bigg). \end{align*} $$

We claim that for each $1\le l\le m$ , there exists $\epsilon _0, M_0>0$ such that

(4.14) $$ \begin{align} \begin{aligned} \bigg\Vert \sup_{\tau\in\mathbb{Z}}\Big| &\sum_{\gamma\in\mathbb{Z}:\gamma<\tau} \mathcal{T}_{\vec{{G}},l}^{\lambda,\gamma,\mu}(f_1,\dots,f_m) \Big| \bigg\Vert_{L^{2/m}} \\ &\lesssim 2^{-\epsilon_0 \mu_0}2^{-\lambda M_0}\Vert \Omega\Vert_{L^q(\mathbb{S}^{mn-1})}\prod_{j=1}^{m}\Vert f_j\Vert_{L^2}, \end{aligned} \end{align} $$

which concludes (4.13). Therefore, it remains to prove (4.14).

4.3.1 Proof of (4.14)

When $2\le l\le m$ , we apply (2.8) with $2<q'<\frac {2m}{m-1}$ , along with (4.11), and (4.12) to obtain

$$ \begin{align*} &\bigg\Vert \sup_{\tau\in\mathbb{Z}}\Big| \sum_{\gamma\in\mathbb{Z}:\gamma<\tau}\mathcal{T}_{\vec{{G}},l}^{\lambda,\gamma,\mu}\big(f_1,\dots,f_m\big) \bigg\Vert_{L^{2/m}}\le \bigg\Vert \sum_{\gamma\in\mathbb{Z}} \big| \mathcal{T}_{\vec{{G}},l}^{\lambda,\gamma,\mu}\big(f_1,\dots,f_m\big) \big| \bigg\Vert_{L^{2/m}} \\ &\lesssim \big\Vert \big\{ b_{\vec{{G}},\vec{{k}}\,}^{\lambda,\mu}\big\}_{\vec{{k}}\,}\big\Vert_{\ell^{\infty}}^{1-\frac{(m-1)q'}{2m}} \big\Vert \big\{ b_{\vec{{G}},\vec{{k}}\,}^{\lambda,\mu}\big\}_{\vec{{k}}\,}\big\Vert_{\ell^{q'}}^{\frac{(m-1)q'}{2m}} 2^{\lambda mn/2}(\lambda+1)^{l/2}{\mu}^{l/2}\prod_{j=1}^{m}\Vert f_j\Vert_{L^2}\\ &\lesssim \Vert \Omega\Vert_{L^q(\mathbb{S}^{mn-1})}2^{-\delta {\mu}(1-\frac{(m-1)q'}{2m})}{\mu}^{m/2}2^{-\lambda C_{M,m,n,q}}(\lambda+1)^{m/2}\prod_{j=1}^{m}\Vert f_j\Vert_{L^2} , \end{align*} $$

where

$$ \begin{align*} C_{M,m,n,q}:=(M+1+mn)\bigg(1-\frac{(m-1)q'}{2m}\bigg)+mn(1/q-1/2)\frac{(m-1)q'}{2m}-\frac{mn}{2}. \end{align*} $$

Here, we used the fact that $\frac {l-1}{2l}\le \frac {m-1}{2m}$ for $l\le m$ . Then (4.14) follows from choosing M sufficiently large so that $C_{M,m,n,q}>0$ since $1-\frac {(m-1)q'}{2m}>0$ .

Now let us prove (4.14) for $l=1$ . In this case, we first see the estimate

(4.15) $$ \begin{align} \Big\Vert \Big( \sum_{\gamma\in\mathbb{Z}}\big| \mathcal{T}_{\vec{{G}},1}^{\lambda,\gamma,\mu}(f_1,\dots,f_m) \big|^2\Big)^{1/2}\Big\Vert_{L^{2/m}} \lesssim 2^{-\epsilon_0\mu} 2^{-M_0\lambda} \Vert \Omega\Vert_{L^q(\mathbb{S}^{mn-1})}\prod_{j=1}^{m}\Vert f_j\Vert_{L^2} \end{align} $$

for some $\epsilon _0, M_0>0$ , which can be proved, as in [18, Section 6], by using (2.7) and (4.11).

Choose a Schwartz function $\Gamma $ on $\mathbb R^n$ whose Fourier transform is supported in the ball $ \{\xi \in \mathbb R^n: |\xi |\le 2\}$ and is equal to $1$ for $|\xi |\le 1$ , and define $\Gamma _k:=2^{kn}\Gamma (2^k\cdot )$ so that $\mathrm {Supp}(\widehat {\Gamma _k})\subset \{\xi \in \mathbb R^n: |\xi |\le 2^{k+1}\}$ and $\widehat {\Gamma _k}(\xi )=1$ for $|\xi |\le 2^k$ .

Since the Fourier transform of $\mathcal {T}_{\vec {{G}},1}^{\lambda ,\gamma ,\mu }(f_1,\dots ,f_m)$ is supported in the set $\big \{ \xi \in \mathbb R^n: 2^{\gamma +\mu -5}\le |\xi |\le 2^{\gamma +\mu +4} \big \}$ , we can write

$$ \begin{align*} \sum_{\gamma\in\mathbb{Z}: \gamma<\tau}\mathcal{T}_{\vec{{G}},1}^{\lambda,\gamma,\mu}(f_1,\dots,f_m)=\Gamma_{\tau+\mu+3}\ast \Big( \sum_{\gamma\in\mathbb{Z}: \gamma<\tau}\mathcal{T}_{\vec{{G}},1}^{\lambda,\gamma,\mu}(f_1,\dots,f_m)\Big) \end{align*} $$

and then split the right-hand side into

$$ \begin{align*} & \Gamma_{\tau+\mu+3}\ast \Big( \sum_{\gamma\in\mathbb{Z}}\mathcal{T}_{\vec{{G}},1}^{\lambda,\gamma,\mu}(f_1,\dots,f_m)\Big)-\Gamma_{\tau+\mu+3}\ast \Big( \sum_{ \gamma\in\mathbb{Z} :\gamma\ge \tau}\mathcal{T}_{\vec{{G}},1}^{\lambda,\gamma,\mu}(f_1,\dots,f_m)\Big). \end{align*} $$

Due to the Fourier support conditions of $\Gamma _{\tau +\mu +3}$ and $\mathcal {T}_{\vec {{G}},1}^{\lambda ,\gamma ,\mu }(f_1,\dots ,f_m)$ , the sum in the second term can be actually taken over $\tau \le \gamma \le \tau +9$ . Therefore, the left-hand side of (4.14) is controlled by the sum of

(4.16) $$ \begin{align} I:=\bigg\Vert \sup_{\nu\in\mathbb{Z}}\Big| \Gamma_{\nu}\ast \Big(\sum_{\gamma\in\mathbb{Z}}{ \mathcal{T}_{\vec{{G}},1}^{\lambda,\gamma,\mu}(f_1,\dots,f_m) } \Big)\Big| \bigg\Vert_{L^{2/m}} \end{align} $$

and

(4.17) $$ \begin{align} II:=\sum_{\gamma=0}^{9}\Big\Vert \sup_{\tau\in\mathbb{Z}}\big| \Gamma_{\tau+\mu+3}\ast T_{\vec{{G}},1}^{\lambda,\tau+\gamma,\mu}(f_1,\dots,f_m)\big|\Big\Vert_{L^{2/m}}. \end{align} $$

First of all, when $0\le \gamma \le 9$ , the Fourier supports of both $\Gamma _{\tau +\mu +3}$ and $T_{\vec {{G}},1}^{\lambda ,\tau +\gamma ,\mu }(f_1,\dots ,f_m)$ are $\{\xi \in \mathbb R^n: |\xi |\sim 2^{\tau +\mu }\}$ . This implies that for any $0<r<1$ ,

$$ \begin{align*} &\big| \Gamma_{\tau+\mu+3}\ast T_{\vec{{G}},1}^{\lambda,\tau+\gamma,\mu}(f_1,\dots,f_m)(x)\big|\\ &\lesssim 2^{(\tau+\mu)(n/r-n)}\Big( \int_{\mathbb R^n} \big| \Gamma_{\tau+\mu+3}(x-y)\big|^r \big| T_{\vec{{G}},1}^{\lambda,\tau+\gamma,\mu}(f_1,\dots,f_m)(y)\big|^r ~dy\Big)^{1/r}\\ &\lesssim \Big(\mathcal{M}\big(|T_{\vec{{G}},1}^{\lambda,\tau+\gamma,\mu}(f_1,\dots,f_m)|^r\big)(x)\Big)^{1/r}, \end{align*} $$

where the Nikolskii inequality (see [28, Proposition 1.3.2]) is applied in the first inequality. Setting $0<r<2/m$ , and using the maximal inequality for $\mathcal {M}$ and the embedding $\ell ^2\hookrightarrow \ell ^{\infty }$ we obtain

(4.18) $$ \begin{align} II&\lesssim \big\Vert \sup_{\tau\in\mathbb{Z}}\big| T_{\vec{{G}},1}^{\lambda,\tau,\mu}(f_1,\dots,f_m) \big| \big\Vert_{L^{2/m}} \end{align} $$

$$\begin{align*} &\le \Big\Vert \Big( \sum_{\gamma\in\mathbb{Z}}\big| \mathcal{T}_{\vec{{G}},1}^{\lambda,\gamma,\mu}(f_1,\dots,f_m) \big|^2\Big)^{1/2}\Big\Vert_{L^{2/m}}. \end{align*}$$

Then the $L^{2/m}$ norm is bounded by the right-hand side of (4.14), thanks to (4.15). This completes the estimate for $II$ defined in (4.17), and we turn our attention to I defined in (4.16).

In the sequel, we will make use of the following inequality: if $\widehat {g_{\gamma }}$ is supported on $\{\xi \in \mathbb {R}^n : C^{-1} 2^{\gamma +\mu }\leq |\xi |\leq C2^{\gamma +\mu }\}$ for some $C>1$ and $\mu \in \mathbb {Z}$ , then

(4.19) $$ \begin{align} \Big\Vert \Big\{ \Phi^{(1)}_j\!\ast\! \Big(\sum_{\gamma\in\mathbb{Z}}{g_{\gamma}}\!\Big)\!\Big\}_{\! j\in\mathbb{Z}}\Big\Vert_{L^p(\ell^q)}\lesssim_{C} \big\Vert \big\{ g_j\big\}_{j\in\mathbb{Z}}\big\Vert_{L^p(\ell^q)} \quad \text{uniformly in }~\mu \end{align} $$

for $0<p<\infty $ . The proof of (4.19) is elementary and standard, so it is omitted here (see [16, equation (13)] and [31, Theorem 3.6] for related arguments).

To obtain the bound of I, we note that

$$ \begin{align*} I \approx \Big\Vert \sum_{\gamma\in\mathbb{Z}}{ \mathcal{T}_{\vec{{G}},1}^{\lambda,\gamma,\mu}(f_1,\dots,f_m) }\Big\Vert_{H^{2/m}}, \end{align*} $$

where $H^{2/m}$ is the Hardy space. We refer to [15, Corollary 2.1.8] for the above estimate. Then, using the Littlewood–Paley theory for Hardy space (see, for instance, [15, Theorem 2.2.9]) and (4.19), there exists a unique polynomial $Q^{\lambda ,\mu ,\vec {{G}}}(x)$ such that

(4.20) $$ \begin{align} \Big\Vert \sum_{\gamma\in\mathbb{Z}}{ \mathcal{T}_{\vec{{G}},1}^{\lambda,\gamma,\mu}(f_1,\dots,f_m) } -Q^{\lambda,\mu,\vec{{G}}} \Big\Vert_{H^{2/m}}\notag &\lesssim \Big\Vert \Big( \sum_{\gamma\in\mathbb{Z}}\big| \mathcal{T}_{\vec{{G}},1}^{\lambda,\gamma,\mu}(f_1,\dots,f_m) \big|^2\Big)^{1/2}\Big\Vert_{L^{2/m}}\nonumber\\ & \lesssim 2^{-\epsilon_0\mu} 2^{-M_0\lambda} \Vert \Omega\Vert_{L^q(\mathbb{S}^{mn-1})}\prod_{j=1}^{m}\Vert f_j\Vert_{L^2}, \end{align} $$

where (4.15) is applied. Furthermore,

$$ \begin{align*} &\Big\Vert \sum_{\gamma\in\mathbb{Z}}{ \mathcal{T}_{\vec{{G}},1}^{\lambda,\gamma,\mu}(f_1,\dots,f_m) } \Big\Vert_{H^{2/m}}\\ &\approx \bigg\Vert \sup_{\nu\in\mathbb{Z}}\Big|\Gamma_{\nu}\ast\Big( \sum_{\gamma\in\mathbb{Z}}{ \mathcal{T}_{\vec{{G}},1}^{\lambda,\gamma,\mu}(f_1,\dots,f_m) } \Big)\Big|\bigg\Vert_{L^{2/m}}\\ &=\bigg\Vert \sup_{\nu\in\mathbb{Z}}\Big|\Gamma_{\nu}\ast\Big( \sum_{\gamma\in\mathbb{Z}: \gamma\le \nu-\mu+5}{ \mathcal{T}_{\vec{{G}},1}^{\lambda,\gamma,\mu}(f_1,\dots,f_m) } \Big)\Big|\bigg\Vert_{L^{2/m}}\\ &\lesssim \bigg\Vert \sup_{\nu\in\mathbb{Z}} \Big| \sum_{\gamma\in\mathbb{Z}: \gamma\le \nu-\mu+5}{ \mathcal{T}_{\vec{{G}},1}^{\lambda,\gamma,\mu}(f_1,\dots,f_m) }\Big|\bigg\Vert_{L^{2/m}}\\ &\le \Big\Vert \sum_{\gamma\in\mathbb{Z}}{ \big| \mathcal{T}_{\vec{{G}},1}^{\lambda,\gamma,\mu}(f_1,\dots,f_m) \big| }\Big\Vert_{L^{2/m}}, \end{align*} $$

where the argument that led to (4.18) is applied in the first inequality. As we have discussed in [18, Section 6.1], this quantity is finite for all Schwartz functions $f_1,\dots ,f_m$ . Accordingly, we have

$$ \begin{align*}\sum_{\gamma\in\mathbb{Z}}{ \mathcal{T}_{\vec{{G}},1}^{\lambda,\gamma,\mu}(f_1,\dots,f_m) } -Q^{\lambda,\mu,\vec{{G}}}\in H^{2/m}\end{align*} $$

and

$$ \begin{align*}\sum_{\gamma\in\mathbb{Z}}{ \mathcal{T}_{\vec{{G}},1}^{\lambda,\gamma,\mu}(f_1,\dots,f_m) } \in H^{2/m},\end{align*} $$

and thus $Q^{\lambda ,\mu ,\vec {{G}}}=0$ . Now it follows from (4.20) that

$$ \begin{align*} I \lesssim 2^{-\epsilon_0\mu} 2^{-M_0\lambda} \Vert \Omega\Vert_{L^q(\mathbb{S}^{mn-1})}\prod_{j=1}^{m}\Vert f_j\Vert_{L^2}, \end{align*} $$

as expected. This completes the proof of (4.14).

5 Proof of Theorem 1.1

We now prove Theorem 1.1 by making use of Theorem 1.2. Recall that in Theorem 1.2 we established the boundedness only for Schwartz functions. An important obstacle we address is the pointwise definition of the maximal operator $\mathcal L^{**}$ (see (1.3)) for general $L^2$ functions. This definition can be given via an abstract extension (see [11]), but this is not as useful for our purposes. We provide below a concrete approach that preserves the pointwise bounds provided by the supremum.

5.1 A variant of Theorem 1.2 for general $L^2$ functions

We note that when $f_j\in L^{2}(\mathbb {R}^n)$ , by Lemma 3.1, there exists a set $E^{\Omega }_{f_1,\dots , f_m}$ of measure $0$ such that

$$ \begin{align*}\mathcal M_{\Omega} \big(f_1,\dots , f_m\big)(x)<\infty \end{align*} $$

when $x\notin E^{\Omega }_{f_1,\dots , f_m}$ . Therefore, for $x\notin E^{\Omega }_{f_1,\dots , f_m}$ , we have

(5.1) $$ \begin{align} \int_{\epsilon_0 \le |\vec{{y}}\,| \le \epsilon_0^{-1}} \frac{|\Omega (\vec{{y}}\,')|}{|\vec{{y}}\,|^{mn}}\, \prod_{j=1}^{m}\big|f_j(x-y_j)\big|\; d\vec{{y}}\, \le C \frac{1 }{\epsilon_0^{2mn}} \mathcal M_{\Omega} \big(f_1,\dots , f_m\big)(x), \end{align} $$

and thus ${\mathcal {L}}_{\Omega }^{ ( \epsilon , \epsilon ^{-1}) }(f_1,\dots , f_m)(x)$ and ${\mathcal {L}}_{\Omega }^{*, \epsilon _0 }(f_1,\dots , f_m)(x)$ are well defined, for $f_j$ in $L^2(\mathbb {R}^n)$ and $\Omega $ in $L^q(\mathbb {S}^{mn-1})$ . Moreover,

$$\begin{align*}{\mathcal{L}}_{\Omega}^{(\epsilon , \epsilon^{-1})}(f_1, \dots , f_m)(x)\le {\mathcal{L}}_{\Omega}^{*,\epsilon_0}(f_1, \dots , f_m)(x) \end{align*}$$

pointwise for $x\in \mathbb R^n\setminus E_{f_1,\dots , f_m}^{\Omega }$ .

For given $f_j\in L^2(\mathbb {R}^n)$ , we pick sequences of Schwartz functions $f_j^k$ converging to $f_j$ in $L^2$ as $k\to \infty $ , by density. Using the identity

(5.2) $$ \begin{align} a_1a_2\cdots a_m - b_1b_2\cdots b_m= \sum_{j=1}^m b_1 \cdots b_{j-1} (a_j-b_j) a_{j+1}\cdots a_m \end{align} $$

(with the obvious modification when $j=1$ or $j=m$ ), the inequality

$$\begin{align*}{\mathcal{L}}_{\Omega}^{*,\epsilon_0 }\big(f_1,\dots , f_m\big)\le 2{\mathcal{L}}_{\Omega}^{*}(f_1^k,\dots , f_m^k) +\sum_{j=1}^m {\mathcal{L}}_{\Omega}^{*,\epsilon_0 }(f_1^k,\dots f_{j-1}^k, f_j-f_j^k, f_{j+1}\dots , f_m ) \end{align*}$$

is valid pointwise on the complement of the set

(5.3) $$ \begin{align} E^{\Omega}:=E_{f_1,\dots, f_m}^{\Omega} \cup \left( \bigcup_{k=1}^{\infty} E_{f_1^k,\dots, f_m^k}^{\Omega}\right) \cup \bigg( \bigcup_{j=1}^m\bigcup_{k=1}^{\infty} E_{f_1^k,\dots f_{j-1}^k, f_j-f_j^k, f_{j+1}\dots , f_m }^{\Omega} \bigg), \end{align} $$

which is a set of measure zero. Now we take the $L^{2/m} $ quasi-norm on both sides and apply Theorem 1.3 for Schwartz functions and the estimate (5.1), combined with Lemma 3.1, for the terms in the sum. Taking $k\to \infty $ , we obtain

$$ \begin{align*} \big\|{\mathcal{L}}_{\Omega}^{*,\epsilon_0 }(f_1,\dots , f_m)\big\|_{L^{2/m}(\mathbb{R}^n) } \le C\Vert \Omega\Vert_{L^q(\mathbb{S}^{mn-1})} \prod_{j=1}^{\infty}\| f_j\|_{L^{2}(\mathbb{R}^n)}. \end{align*} $$

Finally, letting $\epsilon _0 \to 0$ and using the monotone convergence theorem, we conclude

(5.4) $$ \begin{align} \big\|{\mathcal{L}}_{\Omega}^{** }(f_1,\dots , f_m)\big\|_{L^{2/m}(\mathbb{R}^n) } \le C\Vert \Omega\Vert_{L^q(\mathbb{S}^{mn-1})} \prod_{j=1}^{\infty}\| f_j\|_{L^{2}(\mathbb{R}^n)} \end{align} $$

for $f_j\in L^2(\mathbb {R}^n)$ .

5.2 Proof of Theorem 1.1

Let $f_1,\dots ,f_m$ be given $L^2$ functions and pick sequences $\{f_j^k\}$ of Schwartz functions such that $f_j^k$ converges to $f_j$ in $L^2(\mathbb {R}^n)$ as $k\to \infty $ . Recall that ${\mathcal {L}}_{\Omega }(f_1 ,\dots , f_m )$ is defined as the $L^{2/m}$ limit of ${\mathcal {L}}_{\Omega }(f_1^k ,\dots , f_m^k )$ as $k\to \infty $ . Then there exists a subsequence $\{k_l\}$ of $\{k\}$ such that ${\mathcal {L}}_{\Omega }(f_1^{k_l} ,\dots , f_m^{k_l} ) \to {\mathcal {L}}_{\Omega }(f_1 ,\dots , f_m )$ pointwise on $\mathbb R^n\setminus E$ , for some set E of measure zero. Let us denote the subsequence $\{k_l\}$ still by $\{k\}$ for notational convenience. Then, on $\mathbb R^n\setminus (E\cup E^{\Omega })$ , where $E^{\Omega }$ is as in (5.3), we have

$$ \begin{align*} \big|{\mathcal{L}}_{\Omega}^{(\epsilon,\epsilon^{-1})}&(f_1,\dots, f_m) -{\mathcal{L}}_{\Omega}(f_1 ,\dots, f_m )\big|\\ &\le \big|{\mathcal{L}}_{\Omega}^{(\epsilon,\epsilon^{-1})}(f_1,\dots, f_m) -{\mathcal{L}}_{\Omega}^{(\epsilon,\epsilon^{-1})}(f_1^k ,\dots, f_m^k )\big|\\ & \quad + \big| {\mathcal{L}}_{\Omega}^{(\epsilon,\epsilon^{-1})}(f_1^k ,\dots, f_m^k ) - {\mathcal{L}}_{\Omega} (f_1^k ,\dots, f_m^k ) \big|\\ & \quad \quad + \big| {\mathcal{L}}_{\Omega} (f_1^k ,\dots, f_m^k ) -{\mathcal{L}}_{\Omega}(f_1 ,\dots, f_m ) \big|. \end{align*} $$

We first take the $\limsup _{\epsilon \to 0}$ on both sides and then the middle term on the right vanishes. Then we apply $\liminf _{k\to \infty }$ so that the last term vanishes. Consequently, we have

$$ \begin{align*} \limsup_{\epsilon \to 0}|&{\mathcal{L}}_{\Omega}^{(\epsilon,\epsilon^{-1})}(f_1,\dots, f_m) -{\mathcal{L}}_{\Omega}(f_1 ,\dots, f_m )|\\ &\le \liminf_{k\to \infty} \sum_{j=1}^m {\mathcal{L}}_{\Omega}^{**} (f_1,\dots, f_{j-1} , f_j-f_j^k, f_{j+1}^k, \dots , f_m^k) \end{align*} $$

on $\mathbb R^n\setminus \big (E\cup E^{\Omega }\big )$ , where the identity (5.2) is applied. It follows that

$$ \begin{align*} & \Big|\big\{x:\,\,\limsup_{\epsilon \to 0}|{\mathcal{L}}_{\Omega}^{(\epsilon,\epsilon^{-1})}(f_1,\dots, f_m) -{\mathcal{L}}_{\Omega}(f_1 ,\dots, f_m )|>\lambda\big\}\Big| \\ &\le \Big|\big\{x:\,\, \liminf_{k\to \infty} \sum_{j=1}^m {\mathcal{L}}_{\Omega}^{**} (f_1,\dots, f_{j-1} , f_j-f_j^k, f_{j+1}^k, \dots , f_m^k)>\lambda\big\}\Big| \\ &\le \lambda^{-\frac 2m} \sum_{j=1}^m \Big\| \liminf_{k\to \infty} \sum_{j=1}^m {\mathcal{L}}_{\Omega}^{**} (f_1,\dots, f_{j-1} , f_j-f_j^k, f_{j+1}^k, \dots , f_m^k)\big\|_{L^{\frac 2m}}^{\frac 2m} \end{align*} $$

by Chebyshev’s inequality. But this last expression tends to zero as $k\to \infty $ in view of Fatou’s lemma and (5.4). We conclude that

$$ \begin{align*}\limsup_{\epsilon \to 0}|{\mathcal{L}}_{\Omega}^{(\epsilon,\epsilon^{-1})}(f_1,\dots, f_m) -{\mathcal{L}}_{\Omega}(f_1 ,\dots, f_m )| \end{align*} $$

equals zero a.e. and this finishes the proof.

6 Proof of Theorem 1.4

Let $\mu _0$ be the smallest integer satisfying $2^{{\mu }_0-3}>C_0\sqrt {mn}$ and

$$ \begin{align*}\widehat{\Theta^{(m)}_{{\mu}_0-1}}(\vec{{\xi}}\,):=1-\sum_{{\mu}={\mu}_0}^{\infty}{\widehat{\Phi_{\mu}^{(m)}}(\vec{{\xi}}\,)}.\end{align*} $$

Clearly,

$$ \begin{align*} \widehat{\Theta^{(m)}_{{\mu}_0-1}}(\vec{{\xi}}\,)+\sum_{{\mu}={\mu}_0}^{\infty}{\widehat{\Phi_{\mu}^{(m)}}(\vec{{\xi}}\,)}=1, \end{align*} $$

and thus we can write

$$ \begin{align*} \sigma(\vec{{\xi}}\,)=\widehat{\Theta_{{\mu}_0-1}^{(m)}}(\vec{{\xi}}\,)\sigma(\vec{{\xi}}\,)+\sum_{{\mu}={\mu}_0}^{\infty}{\widehat{\Phi_{\mu}^{(m)}}(\vec{{\xi}}\,)\sigma(\vec{{\xi}}\,)}=:\sigma_{{\mu}_0-1}(\vec{{\xi}}\,)+\sum_{{\mu}={\mu}_0}^{\infty}{\sigma_{\mu}(\vec{{\xi}}\,)}. \end{align*} $$

Note that $\sigma _{{\mu }_0-1}$ is a compactly supported smooth function and thus the corresponding maximal multiplier operator $\mathscr {M}_{\sigma _{{\mu }_0-1}}$ , defined by

$$ \begin{align*} &\mathscr{M}_{\sigma_{{\mu}_0-1}}\big(f_1,\dots,f_m\big)(x)\\ &:=\sup_{\nu \in\mathbb{Z}}\Big|\int_{(\mathbb R^n)^m}{\sigma_{{\mu}_0-1}(2^{\nu} \vec{{\xi}}\,)\bigg( \prod_{j=1}^{m}\widehat{f_j}(\xi_j)\bigg) e^{2\pi i\langle x,\sum_{j=1}^{m}\xi_j \rangle}}d\vec{{\xi}}\, \Big|, \end{align*} $$

is bounded by a constant multiple of $ \mathcal {M}f_1(x)\cdots \mathcal {M}f_m(x), $ where $\mathcal {M}$ is the Hardy–Littlewood maximal operator on $\mathbb R^n$ as before. Using Hölder’s inequality and the $L^2$ -boundedness of $\mathcal {M}$ , we can prove

$$ \begin{align*} \big\Vert \mathscr{M}_{\sigma_{{\mu}_0-1}}(f_1,\dots,f_m) \big\Vert_{L^{2/m}}\lesssim \prod_{j=1}^{m}\Vert f_j\Vert_{L^2}. \end{align*} $$

It remains to show that

(6.1) $$ \begin{align} \Big\Vert \sum_{{\mu}={\mu}_0}^{\infty}\mathscr{M}_{\sigma_{\mu}}(f_1,\dots,f_m)\Big\Vert_{L^{2/m}}\lesssim \prod_{j=1}^{m}\Vert f_j\Vert_{L^2}. \end{align} $$

Using the decomposition (2.1), write

(6.2) $$ \begin{align} \sigma_{\mu}(\vec{{\xi}}\,)=\sum_{\lambda\in\mathbb{N}_0}\sum_{\vec{{G}}\in\mathcal{I}^{\lambda}}\sum_{\vec{{k}}\,\in (\mathbb Z^n)^m}{b_{\vec{{G}},\vec{{k}}\,}^{\lambda,\mu}\Psi_{G_1,k_1}^{\lambda}(\xi_1)\cdots\Psi_{G_m,k_m}^{\lambda}(\xi_m)}, \end{align} $$

where

$$ \begin{align*} b_{\vec{{G}},\vec{{k}}\,}^{\lambda,\mu}:=\int_{(\mathbb R^n)^m}{\sigma_{\mu}(\vec{{\xi}}\,)\Psi_{\vec{{G}},\vec{{k}}\,}^{\lambda}(\vec{{\xi}}\,)}d\vec{{\xi}}\,. \end{align*} $$

Let $M:=\Big [ \frac {(m-1)n}{2}\Big ]+1$ and choose $1<q<\frac {2m}{m-1}$ such that

(6.3) $$ \begin{align} \frac{(m-1)n}{2}<\frac{mn}{q}<\min{(a,M)}. \end{align} $$

In view of (2.2), we have

(6.4) $$ \begin{align} \big\Vert \{b_{\vec{{G}},\vec{{k}}\,}^{\lambda,\mu}\}_{\vec{{k}}\,\in (\mathbb Z^n)^m}\big\Vert_{\ell^q} &\lesssim 2^{-\lambda (M-mn/q+mn/2)}\Vert \sigma_{\mu}\Vert_{L_M^q((\mathbb R^n)^m)}\nonumber\\ & \lesssim 2^{-\lambda (M-mn/q+mn/2)} 2^{-{\mu}(a-mn/q)}, \end{align} $$

where the assumption (1.8) is applied in the last inequality.

We observe that if ${\mu }\ge {\mu }_0$ , then $b_{\vec {{G}},\vec {{k}}\,}^{\lambda ,\mu }$ vanishes unless $2^{\lambda +{\mu }-2}\le |\vec {{k}}\,|\le 2^{\lambda +{\mu }+2}$ due to the compact supports of $\sigma _{\mu }$ and $\Psi _{\vec {{G}},\vec {{k}}\,}^{\lambda }$ , which allows us to replace the sum over $\vec {{k}}\,\in (\mathbb Z^n)^m$ in (6.2) by the sum over $2^{\lambda +{\mu }-1}\le |\vec {{k}}\,|\le 2^{\lambda +{\mu }+1}$ . Moreover, we may consider only the case $|k_1|\ge \cdots \ge |k_m|$ as in the previous section. Therefore, in the rest of the section, we assume

$$ \begin{align*} \sigma_{\mu}(\vec{{\xi}}\,)&=\sum_{\lambda\in\mathbb{N}_0}\sum_{\vec{{G}}\in \mathcal{I}^{\lambda}}\sum_{\vec{{k}}\,\in \mathcal{U}^{\lambda+{\mu}}}{ b_{\vec{{G}},\vec{{k}}\,}^{\lambda,\mu}\Psi_{G_1,k_1}^{\lambda}(\xi_1)\cdots\Psi_{G_m,k_m}^{\lambda}(\xi_m) }\\ &=\sum_{l=1}^{m}\sum_{\lambda\in\mathbb{N}_0}\sum_{\vec{{G}}\in \mathcal{I}^{\lambda}}\sum_{\vec{{k}}\,\in \mathcal{U}_l^{\lambda+{\mu}}}{ b_{\vec{{G}},\vec{{k}}\,}^{\lambda,\mu}\Psi_{G_1,k_1}^{\lambda}(\xi_1)\cdots\Psi_{G_m,k_m}^{\lambda}(\xi_m) }\\ &=:\sum_{l=1}^{m}\sum_{\lambda\in\mathbb{N}_0}\sum_{\vec{{G}}\in\mathcal{I}^{\lambda}}\sigma_{\mu,l}^{\lambda,\vec{{G}}}(\vec{{\xi}}\,), \end{align*} $$

where the sets $\mathcal {U}^{\lambda +\mu }$ and $\mathcal {U}_l^{\lambda +\mu }$ are defined as before. Then the left-hand side of (6.1) can be controlled by

(6.5) $$ \begin{align} \bigg(\sum_{l=1}^{m}\sum_{{\mu}={\mu}_0}^{\infty}\sum_{\lambda\in\mathbb{N}_0}\sum_{\vec{{G}}\in\mathcal{I}^{\lambda}}\big\Vert \mathscr{M}_{\sigma_{{\mu},l}^{\lambda,\vec{{G}}}}(f_1,\dots,f_m)\big\Vert_{L^{2/m}}^{2/m} \bigg)^{m/2}. \end{align} $$

Now we claim that

(6.6) $$ \begin{align} &\big\Vert \mathscr{M}_{\sigma_{{\mu},l}^{\lambda,\vec{{G}}}}(f_1,\dots,f_m)\big\Vert_{L^{2/m}}\nonumber\\ &\lesssim 2^{-\lambda (M-mn/q)}(\lambda+1)^{l/2}2^{-{\mu}(a-mn/q)}{\mu}^{l/2}\prod_{j=1}^{m}\Vert f_j\Vert_{L^2}. \end{align} $$

Then (6.5) is less than a constant multiple of $\prod _{j=1}^{m}\Vert f_j\Vert _{L^2}$ as desired, due to the choice of q in (6.3).

In order to prove (6.6), we use the estimates (2.7) and (2.8). We first rewrite

$$ \begin{align*} \mathscr{M}_{\sigma_{{\mu},l}^{\lambda,\vec{{G}}}}\big(f_1,\dots,f_m\big)(x)=\sup_{\gamma\in\mathbb{Z}}\left| \sum_{\vec{{k}}\,\in \mathcal{U}_l^{\lambda+{\mu}}}{b_{\vec{{G}},\vec{{k}}\,}^{\lambda,\mu} \bigg( \prod_{j=1}^{m}L_{G_j,k_j}^{\lambda,\gamma}f_i(x)\bigg) }\right|, \end{align*} $$

where $L_{G,k}^{\lambda ,\gamma }$ is defined as in (2.3).

When $l=1$ , applying the embeddings $\ell ^2\hookrightarrow \ell ^{\infty }$ , $\ell ^q\hookrightarrow \ell ^{\infty }$ , and (2.7), the left-hand side of (6.6) is less than

$$ \begin{align*} &\bigg\Vert \bigg(\sum_{\gamma\in\mathbb{Z}} \Big| \sum_{\vec{{k}}\,\in\mathcal{U}_{1}^{\lambda+{\mu}}}{b_{\vec{{G}},\vec{{k}}\,}^{\lambda,\mu} \prod_{j=1}^{m}{L_{G_j,k_j}^{\lambda,\gamma}f_j} } \Big|^2 \bigg)^{1/2}\bigg\Vert_{L^{2/m}}\\ &\lesssim \big\Vert \{ b_{\vec{{G}},\vec{{k}}\,}^{\lambda,\mu} \}_{\vec{{k}}\,\in (\mathbb Z^n)^m}\big\Vert_{\ell^{q}} \mu^{1/2}2^{\lambda mn/2}(\lambda+1)^{1/2}\prod_{j=1}^{m}\Vert f_j\Vert_{L^2}\\ &\lesssim 2^{-\lambda(M-mn/q)}(\lambda+1)^{1/2} 2^{-{\mu}(a-mn/q)} {\mu}^{1/2}\prod_{j=1}^{m}\Vert f_j\Vert_{L^2}, \end{align*} $$

where (6.4) is applied in the last inequality.

For the case $2\le l\le m$ , we can bound the left-hand side of (6.6) by

$$ \begin{align*} &\bigg\Vert \sum_{\gamma\in\mathbb{Z}}\Big| \sum_{\vec{{k}}\,\in\mathcal{U}_{l}^{\lambda+{\mu}}} b_{\vec{{G}},\vec{{k}}\,}^{\lambda,\mu} \prod_{j=11}^{m} L_{G_j,k_j}^{\lambda,\gamma}f_j \Big| \bigg\Vert_{L^{2/m}}\\ &\lesssim \big\Vert \{b_{\vec{{G}},\vec{{k}}\,}^{\lambda,\mu}\}_{\vec{{k}}\,\in (\mathbb Z^n)^m}\big\Vert_{\ell^{q}} {\mu}^{l/2}2^{\lambda mn/2}(\lambda+1)^{l/2}\prod_{j=1}^{m}\Vert f_j\Vert_{L^2}. \end{align*} $$

Here, we used the inequality (2.8) and the embedding $\ell ^q\hookrightarrow \ell ^{\infty }$ . Then the preceding expression is estimated by

$$ \begin{align*} 2^{-\lambda (M-mn/q)}(\lambda+1)^{l/2}2^{-{\mu} (a-mn/q)}{\mu}^{l/2}\prod_{j=1}^{m}\Vert f_j\Vert_{L^2} \end{align*} $$

in view of (6.4). This completes the proof of (6.6).

7 Proof of Theorem 1.3

We now prove Theorem 1.3 taking Theorem 1.4 for granted.

First of all, it is easy to see that if $f_j$ are Schwartz functions on $\mathbb R^n$ , then

(7.1) $$ \begin{align} \lim_{\nu\to -\infty}S_{\sigma}^{\nu}(f_1,\dots,f_m)(x)=\sigma(0)f_1(x)\cdots f_m(x) \end{align} $$

and

$$ \begin{align*} \lim_{\nu\to \infty}S_{\sigma}^{\nu}(f_1,\dots,f_m)(x)=0, \end{align*} $$

using the Lebesgue dominated convergence theorem and the property that

$$ \begin{align*}\lim_{\nu\to\infty}\sigma(2^{\nu}\vec{\xi}\, )=0. \end{align*} $$

7.1 Extension of Theorem 1.4 to general $f_j\in L^2(\mathbb {R}^n)$

Let $f_1,\dots ,f_m$ be given $L^2$ functions on $\mathbb {R}^n$ . As $S_{\sigma }^{\nu }(f_1 ,\dots , f_m ) $ is finite a.e. for each $\nu \in \mathbb {Z}$ , there exists a set $E_{\nu }$ of measure zero so that $\big |S_{\sigma }^{\nu }(f_1 ,\dots , f_m ) \big |<\infty $ on the complement of $E_{\nu }$ . Since $\nu $ ranges over a countable set $\mathbb {Z}$ , the measure of $E:=\bigcup _{\nu \in \mathbb {Z}}E_{\nu }$ is clearly zero and thus we can define

$$\begin{align*}\mathscr{M}_{\sigma}(f_1,\dots , f_m)= \sup_{\nu\in\mathbb{Z}} | S_{\sigma}^{\nu}(f_1 ,\dots, f_m )| \end{align*}$$

on $E^c$ . That is, $\mathscr {M}_{\sigma }(f_1,\dots , f_m)$ is well defined pointwise a.e. Moreover, it controls every $\big | S_{\sigma }^{\nu }(f_1 ,\dots , f_m )\big |$ pointwise, whenever the latter is finite.

We now extend Theorem 1.5 to $f_j\in L^2(\mathbb R^n)$ using the above definition. Without loss of generality, we only consider the case when $\nu \to -\infty $ as the case $\nu \to \infty $ follows similarly. As every sequence that converges in $L^{2/m}$ has a subsequence that converges a.e., there are a set of measure zero $E_{f_1,\dots , f_m}^1 $ and a subsequence

$$\begin{align*}k_1^1<k_2^1<k_3^1<\cdots <k_{\ell}^1<\cdots \end{align*}$$

of the sequence of k’s such that

$$\begin{align*}S_{\sigma}^{-1}(f_1^{k_{\ell}^1},\dots, f_m^{k_{\ell}^1}) (x) \to S_{\sigma}^{-1}(f_1,\dots, f_m)(x) \end{align*}$$

for all $x\in \mathbb R^n\setminus E_{f_1,\dots , f_m}^1$ . Next, there are a set of measure zero $E_{f_1,\dots , f_m}^2$ and a subsequence

$$\begin{align*}k_1^2<k_2^2<k_3^2<\cdots <k_{\ell}^2<\cdots \end{align*}$$

of

$$\begin{align*}k_1^1<k_2^1<k_3^1<\cdots <k_{\ell}^1<\cdots \end{align*}$$

such that

$$\begin{align*}S_{\sigma}^{-2}(f_1^{k_{\ell}^2},\dots, f_m^{k_{\ell}^2}) (x) \to S_{\sigma}^{-2}(f_1,\dots, f_m)(x) \end{align*}$$

for all $x\in \mathbb R^n \setminus (E_{f_1,\dots , f_m}^2\cup E_{f_1,\dots , f_m}^2)$ . There are a set of measure zero $E_{f_1,\dots , f_m}^3$ and a subsequence

$$\begin{align*}k_1^3<k_2^3<k_3^3<\cdots <k_{\ell}^3<\cdots \end{align*}$$

of

$$\begin{align*}k_1^2<k_2^2<k_3^2<\cdots <k_{\ell}^2<\cdots \end{align*}$$

such that

$$\begin{align*}S_{\sigma}^{-3}(f_1^{k_{\ell}^3},\dots, f_m^{k_{\ell}^3}) (x) \to S_{\sigma}^{-3}(f_1,\dots, f_m)(x) \end{align*}$$

for all $x\in \mathbb R^n\setminus (E_{f_1,\dots , f_m}^1 \cup E_{f_1,\dots , f_m}^2 \cup E_{f_1,\dots , f_m}^3)$ . Iterating this process, we can take a diagonal sequence

$$\begin{align*}k_1^1<k_2^2<k_3^3<\cdots <k_{\ell}^{\ell}<\cdots, \end{align*}$$

which is a subsequence of all subsequences, for which $f_j^{k_{\ell }^{\ell }}(x)\to f_j(x)$ for all $1\le j\le m$ and

$$\begin{align*}S_{\sigma}^{\nu}(f_1^{k_{\ell}^{\ell}},\dots, f_m^{k_{\ell}^{\ell}}) (x) \to S_{\sigma}^{\nu}(f_1,\dots, f_m)(x), \end{align*}$$

as $\ell \to \infty $ for $x\in \mathbb R^n\setminus \bigcup _{\rho =1}^{\infty } E_{f_1,\dots , f_m}^{\rho }$ and all $\nu =-1,-2,\dots $ .

Now, on the set $ \mathbb R^n\setminus \bigcup _{\rho =1}^{\infty } E_{f_1,\dots , f_m}^{\rho }$ , we have

$$\begin{align*}\big|S_{\sigma}^{\nu}(f_1,\dots, f_m) \big|= \lim_{\ell\to \infty} \big|S_{\sigma}^{\nu}(f_1^{k_{\ell}^{\ell}},\dots, f_m^{k_{\ell}^{\ell}}) \big|= \liminf_{\ell\to \infty} \big|S_{\sigma}^{\nu}(f_1^{k_{\ell}^{\ell}},\dots, f_m^{k_{\ell}^{\ell}}) \big|, \end{align*}$$

for every $\nu =-1,-2,\dots $ , and thus

$$\begin{align*}\big|S_{\sigma}^{\nu}(f_1,\dots, f_m) \big|\le \liminf_{\ell\to \infty} \mathscr{M}_{\sigma}(f_1^{k_{\ell}^{\ell}},\dots, f_m^{k_{\ell}^{\ell}}). \end{align*}$$

This deduces

$$\begin{align*}\mathscr{M}_{\sigma}(f_1,\dots, f_m) |\le \liminf_{\ell\to \infty} \mathscr{M}_{\sigma}(f_1^{k_{\ell}^{\ell}},\dots, f_m^{k_{\ell}^{\ell}}) \end{align*}$$

on $ \mathbb R^n\setminus \bigcup _{\rho =1}^{\infty } E_{f_1,\dots , f_m}^{\rho }$ . Taking the $L^{2/m} $ quasi-norm on the both sides and using Fatou’s lemma and Theorem 1.4 for Schwartz functions, we finally obtain

(7.2) $$ \begin{align} \big\Vert \mathscr{M}_{\sigma}(f_1,\dots, f_m) \big\Vert_{L^{2/m}(\mathbb{R}^n)}\lesssim \prod_{j=1}^{m}\Vert f_j\Vert_{L^2(\mathbb R^n)} \end{align} $$

for $f_j\in L^2(\mathbb R^n)$ .

7.2 Proof of Theorem 1.3

Let $f_j$ , $j=1,\dots ,m$ , be functions in $L^2(\mathbb R^n)$ and $\{f_j^{k_{\ell }^{\ell }}\}_{\ell }$ be sequences that appeared above so that on $(\bigcup _{\rho =1}^{\infty }E_{f_1,\dots ,f_m}^{\rho })^c$ ,

(7.3) $$ \begin{align} \lim_{\ell\to \infty} S_{\sigma}^{\nu}(f_1^{k_{\ell}^{\ell}},\dots, f_m^{k_{\ell}^{\ell}}) = S_{\sigma}^{\nu}(f_1,\dots, f_m) \end{align} $$

for each $\nu =-1,-2,\dots $ .

On $(\bigcup _{\rho =1}^{\infty }E_{f_1,\dots ,f_m}^{\rho })^c$ , we write

$$ \begin{align*} &\big| S_{\sigma}^{\nu}(f_1,\dots, f_m)(x) - \sigma(0)f_1(x)\cdots f_m(x) \big|\\ &\le \big| S_{\sigma}^{\nu}(f_1,\dots, f_m) - S_{\sigma}^{\nu}(f_1^{k_{\ell}^{\ell}},\dots, f_m^{k_{\ell}^{\ell}}) \big|\\ &\quad + \big| S_{\sigma}^{\nu}(f_1^{k_{\ell}^{\ell}},\dots, f_m^{k_{\ell}^{\ell}})- \sigma(0) f_1^{k_{\ell}^{\ell}} \cdots f_m^{k_{\ell}^{\ell}} \big|\\ &\quad \quad + \big| \sigma(0) f_1^{k_{\ell}^{\ell}} \cdots f_m^{k_{\ell}^{\ell}}-\sigma(0) f_1\cdots f_m \big|. \end{align*} $$

We first take $\limsup _{\nu \to -\infty }$ and use (7.1) to make the middle term on the right vanish, and then apply $\liminf _{\ell \to \infty }$ to handle the last term on the right which will vanish as well. As a result, we obtain

$$ \begin{align*} &\limsup_{\nu \to -\infty}\big| S_{\sigma}^{\nu}(f_1,\dots, f_m) - \sigma(0) f_1\cdots f_m \big|\\ & \le \liminf_{\ell \to \infty} \sup_{\nu<0}\big| S_{\sigma}^{\nu}(f_1,\dots, f_m) - S_{\sigma}^{\nu}(f_1^{k_{\ell}^{\ell}},\dots, f_m^{k_{\ell}^{\ell}}) \big|. \end{align*} $$

Using the identity (5.2), we control the preceding expression pointwise by

$$\begin{align*}\liminf_{\ell \to \infty} \sum_{i=1}^m \mathscr M_{\sigma} (f_1^{k_{\ell}^{\ell}},\dots, f_{i-1}^{k_{\ell}^{\ell}} ,f_i^{k_{\ell}^{\ell}} -f_i, f_{i+1},\dots, f_m ) \end{align*}$$

on the complement of the set

$$\begin{align*}\bigcup_{\rho=1}^{\infty} \bigg[E_{f_1,\dots, f_m}^{\rho} \cup \bigg( \bigcup_{i=1}^m \bigcup_{\ell=1}^{\infty} E_{f_1^{k_{\ell}^{\ell}},\dots, f_{i-1}^{k_{\ell}^{\ell}} ,f_i^{k_{\ell}^{\ell}} -f_i, f_{i+1},\dots, f_m}^{\rho} \bigg)\bigg], \end{align*}$$

which has full measure. Since

$$\begin{align*}\Big\| \liminf_{\ell \to \infty}\sum_{i=1}^m \mathscr M_{\sigma} (f_1^{k_{\ell}^{\ell}},\dots, f_{i-1}^{k_{\ell}^{\ell}} ,f_i^{k_{\ell}^{\ell}} -f_i, f_{i+1},\dots, f_m ) \Big\|_{L^{2/m}(\mathbb R^n)} =0 \end{align*}$$

in view of Fatou’s lemma and (7.2), we finally obtain

$$\begin{align*}\limsup_{\nu \to -\infty}\big| S_{\sigma}^{\nu}(f_1,\dots, f_m) - \sigma(0) f_1\cdots f_m \big|=0 \end{align*}$$

for almost all points in $ \mathbb R^n $ , which proves one part of the claimed a.e. convergences.

8 Concluding remarks

As of this writing, we are uncertain how to extend Theorem 1.4 in the nonlacunary case. A new ingredient may be necessary to accomplish this.

We have addressed the boundedness of several multilinear and maximal multilinear operators at the initial point $L^2\times \cdots \times L^2\to L^{2/m}$ . Our future investigation related to this project has two main directions: (a) to extend this initial estimate to many other operators, such as the general maximal multipliers considered in [17, 26] and (b) to obtain $L^{p_1}\times \cdots \times L^{p_m}\to L^{p}$ bounds for all of these operators in the largest possible range of exponents possible. Additionally, one could consider the study of related endpoint estimates. We hope to achieve this goal in future publications.