Equivalence of (quasi-)norms on a vector-valued function space and its applications to multilinear operators

TL;DR

This paper establishes (quasi-)norm equivalences on vector-valued function spaces, extending to Triebel-Lizorkin spaces, and applies these results to improve multilinear multiplier theorems and boundedness of bilinear pseudo-differential operators.

Contribution

It introduces new (quasi-)norm equivalences on vector-valued function spaces and extends these to Triebel-Lizorkin spaces, enhancing existing multilinear operator theorems.

Findings

Extended (quasi-)norm equivalence to Triebel-Lizorkin spaces.

Improved multilinear Hormander's multiplier theorem.

Enhanced boundedness results for bilinear pseudo-differential operators.

Abstract

In this paper we present (quasi-)norm equivalence on a vector-valued function space $L^p_A(l^q)$ and extend the equivalence to $p=\infty$ and $0<q<\infty$ in the scale of Triebel-Lizorkin space, motivated by Fraizer-Jawerth. By applying the results, we improve the multilinear Hormander's multiplier theorem of Tomita, that of Grafakos-Si, and the boundedness results for bilinear pseudo-differential operators, given by Koezuka-Tomita.

Equations

$$L_{A}^{p}(l^{q}):=\big{\{}\{f_{k}\}_{k\in\mathbb{Z}}\subset S^{\prime}:f_{k}\in\mathcal{E}(A2^{k}),\big{\|}\{f_{k}\}_{k\in\mathbb{Z}}\big{\|}_{L^{p}(l^{q})}<\infty\big{\}}.$$

$$\mathfrak{M}_{\sigma,2^{k}}^{t}f(x):=2^{kd/t}\Big{\|}\frac{f_{k}(x-\cdot)}{(1+2^{k}|\cdot|)^{\sigma}}\Big{\|}_{L^{t}},$$

$$\displaystyle\sup_{P\in\mathcal{D},l(P)\leq 2^{-\mu}}{\Big{(}\frac{1}{|P|}\int_{P}{\sum_{k=-\log_{2}{l(P)}}^{\infty}{|f_{k}(x)|^{q}}}dx\Big{)}^{1/q}}$$

$$\displaystyle\mathrel{\phantom{=}}\approx\Big{\|}\Big{\{}\sum_{Q\in\mathcal{D}_{k}}{\Big{(}\inf_{y\in Q}{\mathfrak{M}_{\sigma,2^{k}}^{t}f_{k}(y)}\Big{)}\chi_{S_{Q}}}\Big{\}}_{k\geq\mu}\Big{\|}_{L^{\infty}(l^{q})},\quad\text{uniformly in }~{}\mu.$$

$$\big{\|}\{f_{k}\}_{k\in\mathbb{Z}}\big{\|}_{L^{p}(l^{q})}\approx\Big{\|}\Big{\{}\sum_{Q\in\mathcal{D}_{k}}{\Big{(}\inf_{y\in Q}{\mathfrak{M}_{\sigma,2^{k}}^{t}f_{k}(y)}\Big{)}\chi_{S_{Q}}}\Big{\}}_{k\in\mathbb{Z}}\Big{\|}_{L^{p}(l^{q})},\quad\{f_{k}\}_{k\in\mathbb{Z}}\in L_{A}^{p}(l^{q})$$

$$\|f\|_{L^{p}}\approx\big{\|}\big{\{}\phi_{k}\ast f\big{\}}_{k\in\mathbb{Z}}\big{\|}_{L^{p}(l^{2})}$$

$$\Big{\|}\Big{\{}\sum_{Q\in\mathcal{D}_{k}}{\Big{(}\inf_{y\in Q}{\mathfrak{M}_{\sigma,2^{k}}^{t}\big{(}\phi_{k}\ast f\big{)}(y)}\Big{)}\chi_{S_{Q}}}\Big{\}}_{k\in\mathbb{Z}}\Big{\|}_{L^{p}(l^{2})}$$

$$\|f\|_{BMO}\approx\sup_{P\in\mathcal{D}}{\Big{(}\frac{1}{|P|}\int_{P}{\sum_{k=-\log_{2}{l(P)}}^{\infty}{\big{|}\phi_{k}\ast f(x)\big{|}^{2}}}dx\Big{)}^{1/2}}.$$

$$\|f\|_{BMO}\approx\Big{\|}\Big{\{}\sum_{Q\in\mathcal{D}_{k}}{\Big{(}\inf_{y\in Q}{\mathfrak{M}_{\sigma,2^{k}}^{t}\big{(}\phi_{k}\ast f\big{)}(y)}\Big{)}\chi_{S_{Q}}}\Big{\}}_{k\in\mathbb{Z}}\Big{\|}_{L^{\infty}(l^{2})}.$$

$$\big{|}\langle f,g\rangle\big{|}\lesssim\|f\|_{BMO}\|g\|_{H^{1}}.$$

$$T_{m}\big{(}f_{1},\dots,f_{n}\big{)}(x):={\int_{(\mathbb{R}^{d})^{n}}{m(\vec{\boldsymbol{\xi}})\Big{(}\prod_{j=1}^{n}\widehat{f_{j}}(\xi_{j})\Big{)}e^{2\pi i\langle x,\sum_{j=1}^{n}{\xi_{j}}\rangle}}d\vec{\boldsymbol{\xi}}}$$

$$\mathcal{L}_{s}^{r,\vartheta^{(n)}}[m]:=\sup_{l\in\mathbb{Z}}{\big{\|}m(2^{l}\cdot_{1},\dots,2^{l}\cdot_{n})\vartheta^{(n)}\big{\|}_{L_{s}^{r}((\mathbb{R}^{d})^{n})}}.$$

$$\big{\|}T_{m}\big{(}f_{1},\dots,f_{n}\big{)}\big{\|}_{L^{p}}\leq C\mathcal{L}_{s}^{2,\vartheta^{(n)}}[m]\prod_{j=1}^{n}\|f_{j}\|_{L^{p_{j}}}.$$

$$\big{\|}T_{m}\big{(}f_{1},\dots,f_{n}\big{)}\big{\|}_{L^{p}}\lesssim\mathcal{L}_{s}^{r,\vartheta^{(n)}}[m]\prod_{j=1}^{n}\|f_{j}\|_{L^{p_{j}}}.$$

$$X^{p}:=\Big{\{}\begin{array}[]{ll}H^{p}&\text{if}\quad p<\infty\\ BMO&\text{if}\quad p=\infty\end{array}.$$

$$\frac{1}{p}=\frac{1}{p_{i,1}}+\cdots +\frac{1}{p_{i,n}}\text{for }1\le i\le n.$$

$$\big{\|}T_{m}\big{(}f_{1},\dots,f_{n}\big{)}\big{\|}_{L^{p}}\lesssim\mathcal{L}_{s}^{2,\vartheta^{(n)}}[m]\sum_{i=1}^{n}{\Big{(}\|f_{i}\|_{X^{p_{i,i}}}\prod_{1\leq j\leq n,j\not=i}{\|f_{j}\|_{L^{p_{i,j}}}}\Big{)}}.$$

$$\big{\|}T_{m}\big{(}f_{1},\dots,f_{n}\big{)}\big{\|}_{L^{p}}\lesssim\mathcal{L}_{s}^{u,\vartheta^{(n)}}[m]\sum_{i=1}^{n}{\Big{(}\|f_{i}\|_{X^{p_{i,i}}}\prod_{1\leq j\leq n,j\not=i}{\|f_{j}\|_{H^{p_{i,j}}}}\Big{)}}$$

$$\big{|}\partial_{\vec{\boldsymbol{\xi}}}^{{\boldsymbol{\alpha}}}\partial_{x}^{\beta}a(x,\vec{\boldsymbol{\xi}})\big{|}\leq C\Big{(}1+\sum_{j=1}^{n}|\xi_{j}|\Big{)}^{m-|{\boldsymbol{\alpha}}|+|\beta|}$$

$$T_{[a]}\big{(}f_{1},\dots,f_{n}\big{)}(x):=\int_{(\mathbb{R}^{d})^{n}}{a(x,\vec{\boldsymbol{\xi}})\prod_{j=1}^{n}\widehat{f_{j}}(\xi_{j})e^{2\pi i\langle x,\sum_{j=1}^{n}\xi_{j}\rangle}}d\vec{\boldsymbol{\xi}}$$

$$\big{\|}T_{[a]}(f_{1},f_{2})\big{\|}_{L_{s}^{p}}\lesssim\|f_{1}\|_{L_{s}^{p_{1}}}\|f_{2}\|_{L^{p_{2}}}+\|f_{1}\|_{L^{p_{1}}}\|f_{2}\|_{L_{s}^{p_{2}}}$$

$$\|a\|_{{\boldsymbol{M}}_{n}\mathcal{S}_{1,1,N}^{m}}:=\max{\Big{[}\sup\Big{(}1+\sum_{j=1}^{n}|\xi_{j}|\Big{)}^{-m+|{\boldsymbol{\alpha}}|-|\beta|}\big{|}\partial_{\vec{\boldsymbol{\xi}}}^{{\boldsymbol{\alpha}}}\partial_{x}^{\beta}a(x,\vec{\boldsymbol{\xi}})\big{|}\Big{]}}$$

$${\tau }_p:=d/\min (1,p)-d,{\tau }_{p,q}:=d/\min (1,p,q)-d.$$

$$s>\Big{\{}\begin{array}[]{ll}\tau_{p,q}&\text{if}\quad q<\infty\\ \tau_{p,\infty}+d&\text{if}\quad q=\infty\end{array},$$

$$\big{\|}T_{[a]}\big{(}f_{1},\dots,f_{n}\big{)}\big{\|}_{F_{p}^{s,q}}\lesssim\|a\|_{{\boldsymbol{M}}_{n}\mathcal{S}_{1,1,N}^{m}}\sum_{i=1}^{n}{\Big{(}\|f_{i}\|_{F_{p_{i,i}}^{s+m,q}}\prod_{1\leq j\leq n,j\not=i}{\|f_{j}\|_{h^{p_{i,j}}}}\Big{)}}$$

$$Y_{s}^{p}:=\Big{\{}\begin{array}[]{ll}h_{s}^{p}&\text{if}\quad p<\infty\\ bmo_{s}&\text{if}\quad p=\infty\end{array}.$$

$$\big{\|}T_{[a]}\big{(}f_{1},\dots,f_{n}\big{)}\big{\|}_{Y_{s}^{p}}\lesssim\|a\|_{{\boldsymbol{M}}_{n}\mathcal{S}_{1,1,N}^{m}}\sum_{i=1}^{n}{\Big{(}\|f_{i}\|_{Y^{p_{i,i}}_{s+m}}\prod_{1\leq j\leq n,j\not=i}{\|f_{j}\|_{h^{p_{i,j}}}}\Big{)}}$$

$$\big{\|}J^{s}(fg)\big{\|}_{L^{p}}\lesssim\big{\|}J^{s}f\big{\|}_{L^{p_{1}}}\|g\|_{L^{p_{2}}}+\|f\|_{L^{\widetilde{p}_{1}}}\big{\|}J^{s}g\big{\|}_{L^{\widetilde{p}_{2}}}$$

$$\big{\|}J^{s}(fg)\big{\|}_{h^{p}}\lesssim\|J^{s}f\|_{h^{p_{1}}}\|g\|_{h^{p_{2}}}+\|f\|_{h^{\widetilde{p}_{1}}}\|J^{s}g\|_{h^{\widetilde{p}_{2}}}$$

$$\big{\|}J^{s}(fg)\big{\|}_{Y^{p}}\lesssim\|J^{s}f\|_{Y^{p_{1}}}\|g\|_{h^{p_{2}}}+\|f\|_{h^{\widetilde{p}_{1}}}\|J^{s}g\|_{Y^{\widetilde{p}_{2}}}$$

Full text

The author is supported in part by NRF grant 2019R1F1A1044075

1. Introduction

Let $T$ be multilinear operator, defined on $n$-fold products of $S(\mathbb{R}^{d})$, taking values in the space of tempered distributions. One of main problems in multilinear operator theory is $L^{p_{1}}\times\cdots\times L^{p_{n}}\to L^{r}$ boundedness estimates for $T$ when $1/r=1/p_{1}+\dots+1/p_{n}$, and this problem has been actively studied until recently. For example, the multilinear Calderón-Zygmund theory has been developed by Grafakos-Torres [22] while particular examples in the theory have been already studied by Coifman and Meyer [6, 7, 8, 9]. The boundedness of bilinear Hilbert transform was obtained by Lacey and Thiele [25, 26], and the multilinear versions of Hörmander multiplier theorem are investigated by Tomita [36], Grafakos-Si [21], Grafakos-Miyach-Tomita [18], Miyachi-Tomita [28], Grafakos-Nguyen [20], and Grafakos-Miyachi-Nguyen-Tomita [19]. The boundedness of multilinear pseudo-differential operators was investigated by Bényi-Torres [2], Miyachi-Tomita [29], Rodrǵuez-López-Staubach [35], Michalowski-Rule-Staubach [27], Naibo [31], and Koezuka-Tomita [24].

Hölder’s inequality $L^{p_{1}}\cdots L^{p_{n}}\subset L^{r}$, $1/r=1/p_{1}+\dots+1/p_{n}$, is primarily required to handle such multilinear operators, but the inequality seems to be insufficient to derive $BMO$ bound when $p_{j}=\infty$. In turn, the above results mostly treat finite $p_{j}$’s, and occasionally extend to $L^{\infty}$ rather than $BMO$ when $p_{j}=\infty$.

The aim of this paper is twofold. The first one is to introduce (quasi-)norm equivalence on a vector-valued function space, from which $\|f\|_{BMO}$ can be expressed as $L^{\infty}(l^{2})$ norm of a variant of $f$. The equivalence will enable us to still utilize Hölder’s inequality to obtain some boundedness results involving $BMO$-type function spaces. The second one is to study how the equivalence can be applied to generalize previous boundedness results for multilinear operators to $BMO$-type function spaces. We will actually extend and improve the multilinear version of Hörmander’s multiplier theorems of Tomita [36] and Grafakos-Si [21], and the boundedness result of multilinear pseudo-differential operators of Koezuka-Tomita [24].

1.1. Equivalence of (quasi-)norms on a vector-valued function space

For $r>0$ let $\mathcal{E}(r)$ denote the space of all distributions whose Fourier transforms are supported in \big{\{}\xi\in\mathbb{R}^{d}:|\xi|\leq 2r\big{\}}. Let $A>0$. For $0<p<\infty$ and $0<q\leq\infty$ or for $p=q=\infty$ we define

[TABLE]

Then it is known in [37] that $L_{A}^{p}(l^{q})$ is a quasi-Banach space (Banach space if $p,q\geq 1$) with a (quasi-)norm $\|\cdot\|_{L^{p}(l^{q})}$. We will study some (quasi-)norm equivalence on $L_{A}^{p}(l^{q})$ and one of main results is an extension of the norm equivalence to the case $p=\infty$ and $0<q<\infty$ in the scale of Triebel-Lizorkin space.

Let $\mathcal{D}$ denote the set of all dyadic cubes in $\mathbb{R}^{d}$, and for each $k\in\mathbb{Z}$ let $\mathcal{D}_{k}$ be the subset of $\mathcal{D}$ consisting of the cubes with side length $2^{-k}$. For $k\in\mathbb{Z}$, $\sigma>0$, and $0<t\leq\infty$ let

[TABLE]

which is a generalization of the Peetre’s maximal function $\mathfrak{M}_{\sigma,2^{k}}f(x):=\mathfrak{M}_{\sigma,2^{k}}^{\infty}f(x)$. We refer to Section 2 for properties of the operator $\mathfrak{M}_{\sigma,2^{k}}^{t}$.

Theorem 1.1**.**

Let $0<q<\infty$, $\sigma>d/t>d/q$, $0<\gamma<1$, and $\mu\in\mathbb{Z}$. Suppose $A>0$ and $f_{k}\in\mathcal{E}(A2^{k})$ for each $k\in\mathbb{Z}$. For $Q\in\mathcal{D}$ there exists a proper measurable subset $S_{Q}$ of $Q$, depending on $\gamma,q,\sigma,t,\{f_{k}\}_{k\in\mathbb{Z}}$, such that $|S_{Q}|>(1-\gamma)|Q|$ and

[TABLE]

We note that the constant in (1.1) is independent of $\{f_{k}\}_{k\in\mathbb{Z}}$, just depending on $\gamma$. The equivalence in Theorem 1.1 can be compared with the estimate in Lemma 3.1 that for $0<p<\infty$ or $p=q=\infty$

[TABLE]

if $\sigma>d/t>d/\min{(p,q)}$. Note that for $1<p<\infty$, according to Littlewood-Paley theory,

[TABLE]

and, using (1.3), this is also comparable to

[TABLE]

where $\{\phi_{k}\}_{k\in\mathbb{Z}}$ is a homogeneous Littlewood-Paley partition of unity, defined in Section 2. On the other hand, using a deep connection between $BMO$ and Carleson measure,

[TABLE]

The main value of Theorem 1.1 is that $\|\cdot\|_{BMO}$ can be expressed in the form $\|\cdot\|_{L^{\infty}(l^{2})}$ as an extension of (1.4) to $p=\infty$.

Corollary 1.2**.**

Let $\sigma>d/t>d/2$ and $0<\gamma<1$. For $Q\in\mathcal{D}$ there exists a proper measurable subset $S_{Q}$ of $Q$, depending on $\gamma,\sigma,t,f$, such that $|S_{Q}|>(1-\gamma)|Q|$ and

[TABLE]

A simple application of Corollary 1.2 is the inequality

[TABLE]

This provides one direction of the duality between $H^{1}$ and $BMO$, which was first announced in [10] and proved in [5, 11]. It can be also proved in a different way, using Corollary 1.2 and Hölder’s inequality. The proof will be given in Appendix A.

1.2. Hörmander multiplier theorem for multilinear operators

For simplicity we use the notation $\vec{\boldsymbol{\xi}}:=(\xi_{1},\dots,\xi_{n})$. For m\in L^{\infty}\big{(}(\mathbb{R}^{d})^{n}\big{)} the $n$-linear multiplier operator $T_{m}$ is defined by

[TABLE]

for $f_{j}\in S(\mathbb{R}^{d})$. Let $\vartheta^{(n)}\in S((\mathbb{R}^{d})^{n})$ have the properties that $0\leq\vartheta^{(n)}\leq 1$, $\vartheta^{(n)}=1$ for $2^{-1}\leq|\vec{\boldsymbol{\xi}}|\leq 2$, and Supp(\vartheta^{(n)})\subset\big{\{}\vec{\boldsymbol{\xi}}\in(\mathbb{R}^{d})^{n}:2^{-2}\leq|\vec{\boldsymbol{\xi}}|\leq 2^{2}\big{\}}. Define

[TABLE]

We recall the multilinear multiplier theorem of Tomita [36].

Theorem A**.**

Suppose $1<p,p_{1},\dots,p_{n}<\infty$ and $1/p=1/p_{1}+\cdots+1/p_{n}$. If $m\in L^{\infty}((\mathbb{R}^{d})^{n})$ satisfies $\mathcal{L}_{s}^{2,\vartheta^{(n)}}[m]<\infty$ for $s>nd/2$, then there exists a constant $C>0$ so that

[TABLE]

Another boundedness result was obtained by Grafakos-Si [21]

Theorem B**.**

Let $0<p<\infty$ and $1/p=1/p_{1}+\cdots+1/p_{n}$. Suppose $1<r\leq 2$ and $m$ satisfies $\mathcal{L}_{s}^{r,\vartheta^{(n)}}[m]<\infty$ for $s>nd/r$. Then there exists a number $\delta>0$, satisfying $0<\delta\leq r-1$, such that

[TABLE]

whenever $r-\delta<p_{j}<\infty$ for $1\leq j\leq n$.

Note that Theorem B takes into account a broader range of $p$ by giving stronger assumptions on $s$, while, under the same assumption $s>nd/2$ (when $r=2$), the estimate in Theorem B is a partial result of Theorem A. We also refer to [13, 14, 18, 19, 28] for further results.

We will generalize Theorem A and B. Let

[TABLE]

Theorem 1.3**.**

Let $1<p<\infty$ and $1<p_{i,j}\leq\infty$, $1\leq i,j\leq n$, satisfy

[TABLE]

Suppose $m$ satisfies $\mathcal{L}_{s}^{2,\vartheta^{(n)}}[m]<\infty$ for $s>nd/2$. Then

[TABLE]

Theorem 1.4**.**

Let $0<p<\infty$ and $0<p_{i,j}\leq\infty$, $1\leq i,j\leq n$, satisfy (1.6). Suppose $1<u\leq 2$, $0<r\leq 2$, and $m$ satisfies $\mathcal{L}_{s}^{u,\vartheta^{(n)}}[m]<\infty$ for $s>nd/r$. Then there exists a number $\delta>0$, satisfying $0<\delta\leq r$, such that

[TABLE]

whenever $r-\delta<p_{i,j}\leq\infty$ for $1\leq i,j\leq n$.

We remark that, under the same hypothesis $s>nd/r$, the condition $\mathcal{L}_{s}^{r,\vartheta^{(n)}}[m]<\infty$ in Theorem B is improved to $\mathcal{L}_{s}^{u,\vartheta^{(n)}}[m]<\infty$ for any $1<u\leq 2$ in Theorem 1.4. Due to the independence of $r$ in $\mathcal{L}_{s}^{u,\vartheta^{(n)}}[m]<\infty$, one has better freedom in the range $0<r\leq 2$ and $r-\delta<p_{i,j}\leq\infty$.

1.3. Multilinear pseudo-differential operators of type $(1,1)$

The $n$-linear Hörmander symbol class ${\boldsymbol{M}}_{n}\mathcal{S}_{1,1}^{m}$ consists of all a\in C^{\infty}\big{(}(\mathbb{R}^{d})^{n+1}\big{)} having the property that for all multi-indices $\alpha_{1}$,$\dots$,$\alpha_{n}$,$\beta$ there exists a constant $C=C_{{\boldsymbol{\alpha}},\beta}$ such that

[TABLE]

where ${\boldsymbol{\alpha}}:=(\alpha_{1},\dots,\alpha_{n})$ and $|{\boldsymbol{\alpha}}|:=|\alpha_{1}|+\dots+|\alpha_{n}|$. The corresponding $n$-linear pseudo-differential operator $T_{[a]}$ is defined by

[TABLE]

for $f_{1},\dots,f_{n}\in S(\mathbb{R}^{d})$. Denote by $Op{\boldsymbol{M}}_{n}\mathcal{S}_{1,1}^{m}$ the class of $n$-linear pseudo-differential operators with symbols in ${\boldsymbol{M}}_{n}\mathcal{S}_{1,1}^{m}$. Bilinear pseudo-differential operators(n=2) in $Op{\boldsymbol{M}}_{2}\mathcal{S}_{1,1}^{0}$ have bilinear Calderón-Zygmund kernels, but in general they are not bilinear Calderón-Zygmund operators. In particular, they do not always give rise to a mapping $L^{p_{1}}\times L^{p_{2}}\to L^{p}$ for $1<p,p_{1},p_{2}\leq\infty$ with $1/p=1/p_{1}+1/p_{2}$.

The boundedness properties of operators in $Op{\boldsymbol{M}}_{2}\mathcal{S}_{1,1}^{0}$ have been studied by Bényi-Torres [2], and Bényi-Nahmod-Torres [1] in the scale of Lebesgue-Sobolev spaces. To be specific, Bényi-Torres [2] proved that if $a\in{\boldsymbol{M}}_{2}\mathcal{S}_{1,1}^{0}$, then

[TABLE]

for $1<p_{1},p_{2},p<\infty$, $1/p_{1}+1/p_{2}=1/p$, and $s>0$. Moreover, this result was generalized to $a\in{\boldsymbol{M}}_{2}\mathcal{S}_{1,1}^{m}$, $m\in\mathbb{R}$, by Bényi-Nahmod-Torres [1]. Naibo [31] investigated bilinear pseudo-differential operators on Triebel-Lizorkin spaces and Koezuka-Tomita [24] slightly developed the result of Naibo. These works can be readily extended to multilinear operators. For $a\in{\boldsymbol{M}}_{n}\mathcal{S}_{1,1}^{m}$ and $N\in\mathbb{N}_{0}$ we define

[TABLE]

where the supremum is taken over $(x,\vec{\boldsymbol{\xi}})\in(\mathbb{R}^{d})^{n+1}$ and the maximum is taken over $|{\alpha}_{1}|,\dots,|{\alpha}_{n}|,|\beta|\leq N$. For $0<p,q\leq\infty$ let

[TABLE]

Theorem C**.**

[24, 31] Let $0<p<\infty$, $0<q\leq\infty$, $m\in\mathbb{R}$, and $a\in{\boldsymbol{M}}_{n}\mathcal{S}_{1,1}^{m}$. Let $0<p_{i,j}<\infty$, $1\leq i,j\leq n$, satisfy (1.6). If

[TABLE]

then there exists a positive integer $N$ such that

[TABLE]

for $f_{1},\dots,f_{n}\in S(\mathbb{R}^{d})$. Moreover, the inequality also holds for $p_{i,j}=\infty$, $i\not=j$.

We refer the reader to Section 2 for notations and definitions of some function spaces. Recall that $h^{p}=L^{p}$ for $1<p\leq\infty$ and $F_{p}^{s,2}=h^{p}_{s}$ for $0<p<\infty$.

Note that the condition (1.7) in Theorem C is due to the multiplier theorem of Triebel [37]. Recently, the author [34] has improved the result of Triebel, sharpening the condition on $s$, and extending the multiplier theorem to $p=\infty$ in the scale of Triebel-Lizorkin space. Using this result and Theorem 1.1 we will extend Theorem C to the full range $0<p,p_{i,j}\leq\infty$ with the weaker condition $s>\tau_{p,q}$, instead of (1.7).

Theorem 1.5**.**

Suppose $0<p,q\leq\infty$, $m\in\mathbb{R}$, and $a\in{\boldsymbol{M}}_{n}\mathcal{S}_{1,1}^{m}$. Let $0<p_{i,j}\leq\infty$, $1\leq i,j\leq n$, satisfy (1.6). If $s>\tau_{p,q}$, then there exists a positive integer $N$ such that

[TABLE]

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

As a corollary, from $h_{s}^{p}=F_{p}^{s,2}$ and $bmo_{s}=F_{\infty}^{s,2}$, the following estimates hold. Let

[TABLE]

Corollary 1.6**.**

Suppose $0<p\leq\infty$, $m\in\mathbb{R}$, and $a\in{\boldsymbol{M}}_{n}\mathcal{S}_{1,1}^{m}$. Let $\{p_{i,j}\}_{1\leq i,j\leq n}$ satisfy $0<p_{i,j}\leq\infty$ and (1.6). If $s>\tau_{p}$, then there exist positive integers $N>0$ such that

[TABLE]

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

Generalization of Kato-Ponce inequality

The classical Kato-Ponce commutator estimate [23] plays a key role in the wellposedness theory of Navier-Stokes and Euler equations in Sobolev spaces. The commutator estimate has been recast later on into the following fractional Leibniz rule, so called Kato-Ponce inequality. Let $J^{s}:=(1-\Delta)^{s/2}$ be the (inhomogeneous) fractional Laplacian operator. Then

[TABLE]

where $1/p=1/p_{1}+1/p_{2}=1/\widetilde{p}_{1}+1/\widetilde{p}_{2}$, $1<p<\infty$, and $1<p_{1},p_{2},\widetilde{p}_{1},\widetilde{p}_{2}\leq\infty$. Grafakos-Oh [17] and Muscalu-Schlag [30] extended the inequality (1.8) to the wider range $1/2<p<\infty$ under the assumption that $s>\tau_{p}$ or $s\in 2\mathbb{N}$. Recently, Naibo-Thomson [32] extend it to (weighted) local Hardy space for $0<p,p_{1},p_{2},\widetilde{p}_{1},\widetilde{p}_{2}<\infty$.

Theorem D**.**

Let $0<p,p_{1},p_{2},\widetilde{p}_{1},\widetilde{p}_{2}<\infty$ satisfy $1/p=1/p_{1}+1/p_{2}=1/\widetilde{p}_{1}+1/\widetilde{p}_{2}$. Suppose $s>\tau_{p}$. Then for $f,g\in S(\mathbb{R}^{d})$ one has

[TABLE]

Additionally, the case $p=\infty$ was settled by Bourgain-Li [3] and $BMO$ estimates for homogeneous Laplacian operators $D^{s}:=(-\Delta)^{s/2}$ was established by Brummer-Naibo [4].

As a consequence of Corollary 1.6 in the case $a\equiv 1$, one obtains the following extension of Kato-Ponce inequality, which includes an endpoint case of $bmo$ type.

Corollary 1.7**.**

Let $0<p,p_{1},p_{2},\widetilde{p}_{1},\widetilde{p}_{2}\leq\infty$ satisfy $1/p=1/p_{1}+1/p_{2}=1/\widetilde{p}_{1}+1/\widetilde{p}_{2}$. Suppose $s>\tau_{p}$. Then for $f,g\in S(\mathbb{R}^{d})$ one has

[TABLE]

where

[TABLE]

The main ingredient in the proof of Theorem 1.1 is the maximal inequalities for $\mathfrak{M}_{\sigma,2^{k}}^{t}$, which are stated in Lemma 2.5. Then for $0<q<\infty$ one obtains that for any proper measurable subset $S_{Q}$ of $Q$ the left hand side of (1.1) is comparable to

[TABLE]

Thus, in order to prove Theorem 1.1 one needs to show that there exists a subset $S_{Q}$ of $Q$ such that

[TABLE]

and one direction is clear because the essential supremum of a function dominates the supremum of averages. For the other direction, we take advantage of ”$\gamma$-median” and its nice properties. The proofs of Theorem 1.3, 1.4, and 1.5 are based on the Littlewood-Paley decomposition, breaking down operator $T$ in the form $T=\sum_{i=1}^{n}{T_{i}}$. Then we establish

[TABLE]

where $X$, $Y_{i}$’s, and $Z_{j}$’s are suitable spaces which appear in Theorem 1.3, 1.4, and 1.5. The improvement of the condition $\mathcal{L}_{s}^{u,\vartheta^{(n)}}[m]<\infty$ in Theorem 1.4 is provided by using Nikolskii’s inequality. Theorem 1.4 (with $r=u=2$) implies the case $2\leq p_{1},\dots,p_{n}\leq\infty$ in Theorem 1.3, and thus we first present the proof of Theorem 1.4. Then the technique of transposes for multilinear operators in [36] completes the proof of Theorem 1.3.

The paper is organized as follows. Some preliminary results are given in Section 2. In Section 3 we discuss several (quasi-)norm equivalence and prove Theorem 1.1. In Section 4 we prove Theorem 1.3 and 1.4. Section 5 is devoted to the proof of Theorem 1.5.

We make some convention on notation. Let $\mathbb{N}$ and $\mathbb{Z}$ be the collections of all natural numbers and all integers, respectively, and $\mathbb{N}_{0}:=\mathbb{N}\cup\{0\}$. We will use the symbol $A\lesssim B$ to indicate that $A\leq CB$ for some constant $C>0$, possibly different at each occurrence, and $A\approx B$ if $A\lesssim B$ and $B\lesssim A$ simultaneously.

2. Preliminaries

2.1. Function spaces

Let $\Phi_{0}\in S$ satisfy Supp(\widehat{\Phi_{0}})\subset\big{\{}\xi\in\mathbb{R}^{d}:|\xi|\leq 1\big{\}} and $\widehat{\Phi_{0}}(\xi)=1$ for $|\xi|\leq 1/2$. Define $\phi:=\Phi_{0}-2^{-d}\Phi(2^{-1}\cdot)$ and $\phi_{k}:=2^{kd}\phi(2^{k}\cdot)$. Then $\{\Phi_{0}\}\cup\{\phi_{k}\}_{k\in\mathbb{N}}$ and $\{\phi_{k}\}_{k\in\mathbb{Z}}$ form inhomogeneous and homogeneous Littlewood-Paley partition of unity, respectively. Note that Supp(\widehat{\phi_{k}})\subset\big{\{}\xi\in\mathbb{R}^{d}:2^{k-2}\leq|\xi|\leq 2^{k}\big{\}} and

[TABLE]

[TABLE]

For $0<p,q\leq\infty$ and $s\in\mathbb{R}$, inhomogeneous Triebel-Lizorkin space $F_{p}^{s,q}$ is the collection of all $f\in S^{\prime}$ such that

[TABLE]

[TABLE]

where the supremum is taken over all dyadic cubes whose side length is less than $1$. Similarly, homogeneous Triebel-Lizorkin space $\dot{F}_{p}^{s,q}$ is defined to be the collection of all $f\in S^{\prime}/\mathcal{P}$ (tempered distribution modulo polynomials) such that

[TABLE]

[TABLE]

Then these spaces provide a general framework that unifies classical function spaces.

[TABLE]

Recall that for $0<p\leq\infty$

[TABLE]

where $\Phi_{k}:=2^{kd}\Phi_{0}(2^{k}\cdot)$, and the space $bmo$ is a localized version of $BMO$ defined as the set of locally integrable functions $f$ satisfying

[TABLE]

where $f_{Q}$ is the average of $f$ over a cube $Q$. Moreover, for $s\in\mathbb{R}$

[TABLE]

[TABLE]

where $\widehat{D^{s}f}(\xi):=|\xi|^{s}\widehat{f}(\xi)$ and $\widehat{J^{s}f}(\xi):=(1+|\xi|^{2})^{s/2}\widehat{f}(\xi)$ are the fractional Laplacian operators as before. It is known that $(H^{1})^{*}=BMO$, $(h^{1})^{*}=bmo$, and $h^{p}=H^{p}=L^{p}$ for $1<p\leq\infty$. See [12, 15, 37] for more details.

2.2. Maximal inequalities

Let $\mathcal{M}$ be the Hardy-Littlewood maximal operator, defined by

[TABLE]

where the supremum is taken over all cubes containing $x$, and for $0<r<\infty$ let \mathcal{M}_{r}f:=\big{(}\mathcal{M}(|f|^{r})\big{)}^{1/r}. Then Fefferman-Stein’s vector-valued maximal inequality in [11] says that for $0<p<\infty$, $0<q\leq\infty$, and $0<r<\min{(p,q)}$ one has

[TABLE]

Clearly, (2.2) also holds when $p=q=\infty$.

We now introduce a variant of Hardy-Littlewood maximal function. For $\epsilon\geq 0$, $r>0$, and $k\in\mathbb{Z}$, let $\mathcal{M}_{r}^{k,\epsilon}$be defined by

[TABLE]

Note that $\mathcal{M}_{r}^{k,\epsilon}f(x)$ is decreasing function of $\epsilon$, and $\mathcal{M}_{r}^{k,0}f(x)\approx\mathcal{M}_{r}f(x)$. Then the following maximal inequality holds for the case $p=\infty$ and $0<q<\infty$.

Lemma 2.1**.**

[33]** Let $0<r<q<\infty$, $\epsilon>0$, and $\mu\in\mathbb{Z}$. Suppose $A>0$ and $f_{k}\in\mathcal{E}(A2^{k})$ for each $k\in\mathbb{Z}$. Then one has

[TABLE]

Here, the implicit constant of the inequality is independent of $\mu$.

We now continue with some properties of the operator $\mathfrak{M}_{\sigma,2^{k}}^{t}$, defined in (1.1).

Lemma 2.2**.**

Let $\sigma>0$, $0<t\leq s\leq\infty$, and $k\in\mathbb{Z}$. Suppose $A>0$ and $f\in\mathcal{E}(A2^{k})$. Then

[TABLE]

Lemma 2.3**.**

Let $\sigma>0$, $0<t\leq s\leq\infty$, and $k\in\mathbb{Z}$. Then

[TABLE]

Lemma 2.4**.**

Let $0<t<\infty$, $\sigma>d/t$, $0<\epsilon<\sigma-d/t$, and $k\in\mathbb{Z}$. Suppose $A>0$ and $f\in\mathcal{E}(A2^{k})$. Then

[TABLE]

The proofs of Lemma 2.2, 2.3, and 2.4 will be given in Appendix B.

Elementary considerations reveal that for $\sigma>0$ and $Q\in\mathcal{D}_{k}$

[TABLE]

and then it follows from Lemma 2.3 that for $0<t\leq\infty$

[TABLE]

if $f\in\mathcal{E}(A2^{k})$ for some $A>0$.

In addition, Lemma 2.4, (2.2), and Lemma 2.1 lead immediately to the following maximal inequalities.

Lemma 2.5**.**

Let $0<p,q\leq\infty$ and $\sigma>d/t>d/\min{(p,q)}$. Suppose $A>0$ and $f_{k}\in\mathcal{E}(A2^{k})$ for each $k\in\mathbb{Z}$.

1. (1)

For $0<p<\infty$ or $p=q=\infty$

[TABLE] 2. (2)

For $p=\infty$, $0<q<\infty$, and $\mu\in\mathbb{Z}$

[TABLE]

where the constant in the inequality is independent of $\mu$.

If $\min{(p,q)}\leq t\leq\infty$ and $\sigma>d/\min{(p,q)}$ then we may choose $0<t_{0}<\min{(p,q)}$ so that $\sigma>d/t_{0}>d/\min{(p,q)}$ and

[TABLE]

for $f_{k}\in\mathcal{E}(A2^{k})$. Then as a consequence of Lemma 2.5 one obtains the following lemma.

Lemma 2.6**.**

Let $0<p,q\leq\infty$, $\min{(p,q)}\leq t\leq\infty$, and $\sigma>d/\min{(p,q)}$. Suppose $A>0$ and $f_{k}\in\mathcal{E}(A2^{k})$ for each $k\in\mathbb{Z}$.

1. (1)

For $0<p<\infty$ or $p=q=\infty$

[TABLE] 2. (2)

For $p=\infty$, $0<q<\infty$, and $\mu\in\mathbb{Z}$

[TABLE]

where the constant in the inequality is independent of $\mu$.

2.3. Multiplier theorem for $L^{p}_{A}(l^{q})$

The next lemma states a vector-valued version of Hörmander’s multiplier theorem. It was partially proved by Triebel [37, 1.6.3, 2.4.9] and was completed by the author [34] recently.

Lemma E**.**

Let $0<p,q\leq\infty$ and $\mu\in\mathbb{Z}$. Suppose $f_{k}\in\mathcal{E}(A2^{k})$ for each $k\in\mathbb{Z}$, and $\{m_{k}\}_{k\in\mathbb{Z}}$ satisfies

[TABLE]

1. (1)

For $0<p<\infty$ or $p=q=\infty$,

[TABLE] 2. (2)

For $p=\infty$ and $0<q<\infty$

[TABLE]

uniformly in $\mu$.

Observe that for $j\geq 0$

[TABLE]

and the following result can be verified with the use of a change of variables.

Lemma 2.7**.**

Let $0<p,q\leq\infty$, $j\geq 0$, and $\mu\in\mathbb{Z}$. Suppose $f_{k}\in\mathcal{E}(A2^{k})$ for each $k\in\mathbb{Z}$, and $\{m_{k}\}_{k\in\mathbb{Z}}$ satisfies

[TABLE]

1. (1)

For $0<p<\infty$ or $p=q=\infty$,

[TABLE]

uniformly in $j$. 2. (2)

For $p=\infty$ and $0<q<\infty$

[TABLE]

uniformly in $\mu$ and $j$.

Proof.

(1) By using (2.5) and Lemma E (1), one has

[TABLE]

uniformly in $j$, since $f_{k+j}(2^{j}\cdot)\in\mathcal{E}(A2^{k})$.

(2) Similarly, (2.5) and Lemma E (2) yield that

[TABLE]

uniformly in $j$.

∎

3. Equivalence of (quasi-)norms by using $\mathfrak{M}^{t}_{\sigma,2^{k}}$

Let $f_{k}\in\mathcal{E}(A2^{k})$ for some $A>0$ and ${\boldsymbol{f}}:=\{f_{k}\}_{k\in\mathbb{Z}}$. For convenience in notation we will occasionally write

[TABLE]

It follows from (2.3) and Lemma 2.2 that for $\sigma>0$ and $0<t\leq\infty$

[TABLE]

Then Lemma 2.5 (1) gives the (quasi-)norm equivalence

[TABLE]

for $\sigma>d/t>d/\min{(p,q)}$ if $0<p<\infty$ or $p=q=\infty$. Similarly, if $0<q<\infty$ and $\mu\in\mathbb{Z}$, then

[TABLE]

for $\sigma>d/t>d/q$.

Lemma 3.1**.**

Let $0<p,q\leq\infty$, $\sigma>d/t>d/\min{(p,q)}$, $0<\gamma<1$, and $\mu\in\mathbb{Z}$. For each $Q\in\mathcal{D}$ let $S_{Q}$ be a measurable subset of $Q$ with $|S_{Q}|>(1-\gamma)|Q|$. Suppose $A>0$ and $f_{k}\in\mathcal{E}(A2^{k})$ for each $k\in\mathbb{Z}$.

1. (1)

For $0<p<\infty$ or $p=q=\infty$

[TABLE] 2. (2)

For $p=\infty$ and $0<q<\infty$

[TABLE]

Note that the constants in the estimates are independent of $S_{Q}$ as long as $|S_{Q}|>(1-\gamma)|Q|$.

Proof of Lemma 3.1.

The second assertion follows immediately from (3.3) and the condition $|S_{Q}|>(1-\gamma)|Q|$. Thus we only pursue the first one. Assume $0<p<\infty$ or $p=q=\infty$. Since $\chi_{Q}\geq\chi_{S_{Q}}$ one direction is obvious due to (3.1). We will base the converse on the pointwise estimate that for $0<r<\infty$

[TABLE]

which is due to the observation that for $x\in Q$

[TABLE]

Choose $r<p,q$ and then apply (3.1) and (3.4) to obtain

[TABLE]

where the maximal inequality (2.2) is applied in the third inequality (with a different countable index set $\mathcal{D}$). ∎

3.1. Proof of Theorem 1.1

One direction follows immediately from Lemma 3.1 (2). Therefore, we need to prove that there exists a measurable subset $S_{Q}$ such that $|S_{Q}|>(1-\gamma)|Q|$ and

[TABLE]

To choose such a subset $S_{Q}$ we set up notation and terminology. For $0<q\leq\infty$ and $P\in\mathcal{D}$ we define

[TABLE]

Recall that the nonincreasing rearrangement $f^{*}$ of a non-negative measurable function $f$ is given by

[TABLE]

and satisfies

[TABLE]

For $P\in\mathcal{D}$, $0<\gamma<1$, and a non-negative measurable function $f$, the “$\gamma$-median of $f$ over $P$” is defined as

[TABLE]

We consider the $\gamma$-median of $G_{P}^{q}(\mathbf{f})$ over $P$ and the supremums of the quantity over $P\in\mathcal{D}$, $l(P)\leq 2^{-\mu}$. That is,

[TABLE]

[TABLE]

Observe that

[TABLE]

and by (3.6) one has

[TABLE]

Moreover,

[TABLE]

Now for each $P\in\mathcal{D}$ we define

[TABLE]

Then (3.1) yields that

[TABLE]

and (3.1) can be deduced in the following proposition.

Proposition 3.2**.**

Let $0<q<\infty$, $\sigma>d/t>d/q$, $0<\gamma<1$, and $\mu\in\mathbb{Z}$. Suppose $A>0$ and $f_{k}\in\mathcal{E}(A2^{k})$ for each $k\in\mathbb{Z}$. Then

[TABLE]

uniformly in $\mu$.

**Remark **.

For $0<q<\infty$

[TABLE]

while for $0<p<\infty$ or $p=q=\infty$

[TABLE]

which is due to Lemma 3.1 (1).

Proof of Proposition 3.2.

Assume $0<q<\infty$, $\sigma>d/t>d/q$, and $\mu\in\mathbb{Z}$. Our claim is

[TABLE]

To verify (Claim 1) let $\nu\geq\mu$ and fix $P\in\mathcal{D}_{\nu}$ (i.e. $l(P)=2^{-\nu}\leq 2^{-\mu}$). Suppose $x\in P$. Then it suffices to show that

[TABLE]

due to (3.8). Suppose that the left hand side of (3.10) is a nonzero number. Then there exists the “maximal” dyadic cube $P_{0}(x)\subset P$ such that $x\in S_{P_{0}(x)}^{\gamma,q}(\mathfrak{M}_{\sigma}^{t}(\mathbf{f}))$, and thus

[TABLE]

where the second inequality follows from (3.8). The maximality of $P_{0}(x)$ yields that the left hand side of (3.10) is

[TABLE]

where the last one follows from (3.11). This proves (3.10).

We now prove (Claim 2) for “any $0<\gamma<\infty$”. Fix $\nu\geq\mu$ and let us assume

[TABLE]

Then, using Chebyshev’s inequality, (3.2), and (3.12), there exists a constant $C_{A,q,t,\sigma}>0$ such that for $R\in\mathcal{D}_{\nu}$

[TABLE]

This yields that

[TABLE]

So far, we have proved that for any $R\in\mathcal{D}_{\nu}$,

[TABLE]

which is equivalent to

[TABLE]

We complete the proof by taking the supremum over $R\in\mathcal{D},l(R)=2^{-\nu}\leq 2^{-\mu}$. ∎

We end this section by pointing out that the replacement of Lemma 2.5 in the above arguments by Lemma 2.6 provides the following corollaries.

Corollary 3.3**.**

Let $0<p,q\leq\infty$, $\min{(p,q)}\leq t\leq\infty$, $\sigma>d/\min{(p,q)}$, $0<\gamma<1$, and $\mu\in\mathbb{Z}$. For each $Q\in\mathcal{D}$ let $S_{Q}$ be a measurable subset of $Q$ with $|S_{Q}|>(1-\gamma)|Q|$. Suppose $A>0$ and $f_{k}\in\mathcal{E}(A2^{k})$ for each $k\in\mathbb{Z}$.

1. (1)

For $0<p<\infty$ or $p=q=\infty$

[TABLE] 2. (2)

For $p=\infty$ and $0<q<\infty$

[TABLE]

Corollary 3.4**.**

Let $0<q<\infty$, $q\leq t\leq\infty$, $\sigma>d/q$, $0<\gamma<1$, and $\mu\in\mathbb{Z}$. Suppose $A>0$ and $f_{k}\in\mathcal{E}(A2^{k})$ for each $k\in\mathbb{Z}$. For $Q\in\mathcal{D}$ there exists a proper measurable subset $S_{Q}$ of $Q$, depending on $\gamma,q,\sigma,t,\{f_{k}\}_{k\in\mathbb{Z}}$, such that $|S_{Q}|>(1-\gamma)|Q|$ and

[TABLE]

4. Proof of Theorem 1.3 and 1.4

We will first prove Theorem 1.4 and then turn to the proof of Theorem 1.3. Before proving the theorem we set up some notation. Write $\vec{\boldsymbol{f}}:=(f_{1},\dots,f_{n})$, $\vec{\boldsymbol{\xi}}:=(\xi_{1},\dots,\xi_{n})$, $\vec{\boldsymbol{v}}:=(v_{1},\dots,v_{n})$, $d\vec{\boldsymbol{\xi}}:=d\xi_{1}\cdots d\xi_{n}$, and $d\vec{\boldsymbol{v}}:=dv_{1}\cdots dv_{n}$.

4.1. Proof of Theorem 1.4

Choose $0<t<r$ such that $s>nd/t>nd/r(\geq nd/2)$ and let $\delta=r-t>0$. Suppose $p_{1},\dots,p_{n}>t=r-\delta$. Then

[TABLE]

Let $\widetilde{\vartheta^{(n)}}$ be a cutoff function on $(\mathbb{R}^{d})^{n}$ such that $0\leq\widetilde{\vartheta^{(n)}}\leq 1$, $\widetilde{\vartheta^{(n)}}(\vec{\boldsymbol{\xi}})=1$ for $2^{-2}n^{-1/2}\leq|\vec{\boldsymbol{\xi}}|\leq 2n^{1/2}$, and Supp(\widetilde{\vartheta^{(n)}})\subset\big{\{}\vec{\boldsymbol{\xi}}\in(\mathbb{R}^{d})^{n}:2^{-3}n^{-1/2}\leq|\vec{\boldsymbol{\xi}}|\leq 2^{2}n^{1/2}\big{\}}. Then using Calderón’s reproducing formula, Littlewood-Paley partition of unity $\{\phi_{k}\}_{k\in\mathbb{Z}}$, and triangle inequality, we first see that

[TABLE]

Thus it suffices to prove the estimate that

[TABLE]

We use a notation $\mathcal{L}_{s}^{u}[m]:=\mathcal{L}_{s}^{u,\widetilde{\vartheta^{(n)}}}[m]$.

By using Littlewood-Paley partition of unity $\{\phi_{k}\}_{k\in\mathbb{Z}}$, $m(\vec{\boldsymbol{\xi}})$ can be decomposed as

[TABLE]

Then (4.2) is a consequence of the following estimates that

[TABLE]

for each $1\leq i\leq n$. We only concern ourselves with the case $i=1$ by setting $p_{j}:=p_{1,j}$ for $1\leq j\leq n$, and use symmetry for other cases. Suppose $1/p=1/p_{1}+\dots+1/p_{n}$.

We write

[TABLE]

since $\widetilde{\vartheta^{(n)}}(\vec{\boldsymbol{\xi}}/2^{k})=1$ for $2^{k-2}\leq|\xi_{1}|\leq 2^{k}$ and $|\xi_{j}|\leq 2^{k}$ for $2\leq j\leq n$. Let

[TABLE]

Then we note that

[TABLE]

and

[TABLE]

We further decompose $m^{(1)}$ as

[TABLE]

where

[TABLE]

[TABLE]

We refer to $T_{m_{low}^{(1)}}$ as the low frequency part, and $T_{m_{high}^{(1)}}$ as the high frequency part of $T_{m^{(1)}}$ (due to the Fourier supports of $T_{m_{low}^{(1)}}\vec{\boldsymbol{f}}$ and $T_{m_{high}^{(1)}}\vec{\boldsymbol{f}}$).

4.1.1. Low frequency part

To obtain the estimates for the operator $T_{m^{(1)}_{low}}$, we observe that

[TABLE]

where $(g)_{l}:=\phi_{l}\ast g$ for $g\in S$ and $l\in\mathbb{Z}$. It suffices to treat only the sum over $k_{3},\dots,k_{n}\leq k_{2}$ and $k-3-\lfloor\log_{2}{n}\rfloor\leq k_{2}\leq k$, and we will actually prove that

[TABLE]

We define $\Phi_{l}:=2^{ld}\Phi_{0}(2^{l}\cdot)$ for $l\in\mathbb{Z}$ as before, and then observe that for any $g\in S$

[TABLE]

Then

[TABLE]

where $(f_{j})^{k_{2}}:=\Phi_{k_{2}}\ast f_{j}$. Since the second sum is a finite sum over $k_{2}$ near $k$, we may only consider the case $k_{2}=k$ and thus our claim is

[TABLE]

To prove (4.4) let $0<\epsilon<\min{(1,t)}$ such that $1/\epsilon=1-1/u+1/t$, which implies $u^{\prime}=\frac{1}{1/\epsilon-1/t}$ where $1/u+1/u^{\prime}=1$. Then using Nikolskii’s inequality and Hölder’s inequality with $u^{\prime}/\epsilon>1$ one has

[TABLE]

By using Hausdorff Young’s inequality with $u^{\prime}\geq 2$ and (4.3),

[TABLE]

and

[TABLE]

Therefore

[TABLE]

because $1/\epsilon-1+1/u-1/t=0$.

Let

[TABLE]

Then it follows from (3.9) that $S_{Q}^{(1)}$ and $S_{Q}^{(2)}$ are measurable subsets of $Q$ such that $|S_{Q}^{(1)}|,|S_{Q}^{(2)}|\geq\frac{3}{4}|Q|$. We observe that $|S_{Q}^{(1)}\cap S_{Q}^{(2)}|\geq\frac{1}{2}|Q|$ and thus, for any $\tau>0$

[TABLE]

using the argument in (3.4). Clearly, the constant in the inequality is independent of $Q$.

Now we choose $\tau<\min{(1,p)}$, and apply (4.1.1), (2.4), (4.6), and (2.2) to obtain

[TABLE]

By using Hölder’s inequality, the $L^{p}$ norm is dominated by a constant times

[TABLE]

where the inequality follows from Lemma 3.1 (1), Proposition 3.2, Lemma 2.5 (1), and (2.1) with (4.1). Since $\|f_{2}\|_{X^{p_{2}}}\lesssim\|f_{2}\|_{H^{p_{2}}}$, one finally obtains (4.4).

4.1.2. High frequency part

The proof for the high frequency part relies on the fact that if $\widehat{g_{k}}$ is supported on $\{\xi:C^{-1}2^{k}\leq|\xi|\leq C2^{k}\}$ for $C>1$ then

[TABLE]

for $h\in\mathbb{N}$. The proof of (4.7) is elementary and standard, so it will not pursued here. Just use the estimate $|\phi_{k}\ast g_{l}(x)|\lesssim\mathfrak{M}_{\sigma,2^{l}}g_{l}(x)$ for $k-h\leq l\leq k+h$ and apply Lemma 2.5 (1).

We note that

[TABLE]

where $(f_{j})^{k,n}:=\Phi_{k-4-\lfloor\log_{2}{n}\rfloor}\ast f_{j}$.

Observe that the Fourier transform of T_{m_{k}}\big{(}(f_{1})_{k},(f_{2})^{k,n},\dots,(f_{n})^{k,n}\big{)} is supported in \big{\{}\xi\in\mathbb{R}^{d}:2^{k-3}\leq|\xi|\leq 2^{k+2}\big{\}} and thus (4.7) yields that

[TABLE]

Using the argument that led to (4.1.1), one has

[TABLE]

Fix $0<\gamma<1$. For $Q\in\mathcal{D}_{k}$ let S_{Q}:=S_{Q}^{\gamma,2}\big{(}\mathfrak{M}_{s/n,2^{k}}^{t}(f_{1})_{k}\big{)} as before and proceed the similar arguments to obtain that

[TABLE]

4.2. Proof of Theorem 1.3

As in the proof of Theorem 1.4, it suffices to deal with $T_{m^{(1)}}$. Suppose $1<p<\infty$ and $1<p_{j}\leq\infty$ for $1\leq j\leq n$. Then we will prove

[TABLE]

for each $1\leq i\leq n$. First of all, it follows, from Theorem 1.4 with $r=u=2$, that (4.8) holds for $2\leq p_{j}\leq\infty$.

Now assume $1<p\leq\min{(p_{1},\dots,p_{n})}<2$. Observe that only one of $p_{j}^{\prime}s$ could be less than $2$ because $1/p=1/p_{1}+\dots+1/p_{n}<1$, and we will actually look at two cases $1<p_{1}<2\leq p_{2},\dots,p_{n}$ and $1<p_{2}<2\leq p_{1},p_{3},\dots,p_{n}$. Let $T_{m^{(1)}}^{*j}$ be the $j$th transpose of $T_{m^{(1)}}$, defined by the unique operator satisfying

[TABLE]

for $f_{1},\dots,f_{n},h\in S$. Then it is known in [36] that $T_{m^{(1)}}^{*j}=T_{(m^{(1)})^{*j}}$ where

[TABLE]

and then

[TABLE]

4.2.1. The case $1<p<p_{1}<2$

Let $2<p^{\prime},p_{1}^{\prime}<\infty$ be the conjugates of $p,p_{1}$, respectively. That is, $1/p+1/p^{\prime}=1/p_{1}+1/p_{1}^{\prime}=1$. Then $X^{p_{1}}=L^{p_{1}}$ and $1/p_{1}^{\prime}=1/p^{\prime}+1/p_{2}+\dots+1/p_{n}$. Therefore

[TABLE]

where the second inequality follows from Theorem 1.4 and the last one from (4.9).

4.2.2. The case $1<p<p_{2}<2$

Similarly, let $2<p^{\prime},p_{2}^{\prime}<\infty$ be the conjugates of $p,p_{2}$ and then

[TABLE]

5. Proof of Theorem 1.5

We use notations $\vec{\boldsymbol{f}}:=(f_{1},\dots,f_{n})$, $\vec{\boldsymbol{\xi}}:=(\xi_{1},\dots,\xi_{n})$, $\vec{\boldsymbol{l}}:=(l_{1},\dots,l_{n})$, $d\vec{\boldsymbol{\xi}}:=d\xi_{1}\cdots d\xi_{n}$, $d\vec{\boldsymbol{\eta}}:=d\eta_{1}\cdots d\eta_{n}$, ${\boldsymbol{\alpha}}:=({\alpha}_{1},\dots,{\alpha}_{n})$, ${\boldsymbol{|}}{\boldsymbol{\alpha}}{\boldsymbol{|}}:=|{\alpha}_{1}|+\dots+|{\alpha}_{n}|$, $\partial_{\vec{\boldsymbol{\xi}}}^{{\boldsymbol{\alpha}}}:=\partial_{\xi_{1}}^{\alpha_{1}}\cdots\partial_{\xi_{n}}^{\alpha_{n}}$, and $\partial_{\vec{\boldsymbol{\eta}}}^{{\boldsymbol{\alpha}}}:=\partial_{\eta_{1}}^{\alpha_{1}}\cdots\partial_{\eta_{n}}^{\alpha_{n}}$.

The proof is based on the decomposition technique by Bényi-Torres [2]. Throughout this section we regard $\phi_{0}=\Phi_{0}$, not the original meaning $\phi_{0}=\phi$, so that $\{\phi_{k}\}_{k\in\mathbb{N}_{0}}$ is inhomogeneous Littlewood-Paley partition of unity. We write $\mathcal{A}_{N}^{(m)}:=\|a\|_{{\boldsymbol{M}}_{n}\mathcal{S}_{1,1,N}^{m}}$ for simplicity.

5.1. Decomposition and reduction

By using Littlewood-Paley partition of unity, $a\in{\boldsymbol{M}}_{n}\mathcal{S}_{1,1}^{m}$ can be written as

[TABLE]

Then, due to the symmetry, it is enough to work only with $a^{(1)}$ and our actual goal is to show that if $s>\tau_{p,q}$ then

[TABLE]

for sufficiently large $N>0$ and ${1}/{p}={1}/{p_{1}}+\dots+{1}/{p_{n}}$.

Observe that

[TABLE]

Then each $a_{k}$ belongs to ${\boldsymbol{M}}_{n}\mathcal{S}_{1,1}^{m}$ and for $N\in\mathbb{N}_{0}$

[TABLE]

Let $\{\widetilde{\phi_{k}}\}_{k\in\mathbb{N}_{0}}$ be a collection of Schwartz functions so that $\widehat{\widetilde{\phi_{0}}}(\xi):=\widehat{\phi_{0}}(\xi)+\widehat{\phi_{1}}(\xi)(=\widehat{\Phi_{1}}(\xi))$ and $\widehat{\widetilde{\phi_{k}}}(\xi)=\widehat{\phi_{k}}(2\xi)+\widehat{\phi_{k}}(\xi)+\widehat{\phi_{k}}(2^{-1}\xi)$ for $k\geq 1$. By using Fourier series expansion and the fact that $\widehat{\widetilde{\phi_{k}}}=1$ on $Supp(\widehat{\phi_{k}})$ and $\widehat{{\Phi_{k+1}}}=1$ on $Supp(\widehat{\Phi_{k}})$, one can write

[TABLE]

where

[TABLE]

[TABLE]

It can be verified that for $l\in\mathbb{Z}^{d}$ and multi-index $\alpha$ one has

[TABLE]

[TABLE]

[TABLE]

We rewrite $a_{k}(x,\vec{\boldsymbol{\xi}})$ as

[TABLE]

where

[TABLE]

[TABLE]

Then

[TABLE]

and

[TABLE]

Therefore the proof of (5.1) can be deduced from the estimate that

[TABLE]

for sufficiently large $N>0$ and some $\mathcal{J}>d/\min{(1,p,q)}$.

5.2. Pointwise estimate of $T_{[A_{k,u}^{\vec{\boldsymbol{l}}}]}\vec{\boldsymbol{f}}$

Choose $N$ and $\sigma$ such that $N>s$, $N>d/\min{(1,p,q)}+d/\min{(p_{1},\dots,p_{n},q)}$, and $d/\min{(p_{1},\dots,p_{n},q)}<\sigma<N-d/\min{(1,p,q)}$. Let $\mathcal{J}:=N-\sigma(>d/\min{(1,p,q)})$. Then we will prove that

[TABLE]

We first see that

[TABLE]

Let $\phi_{0}^{*}:=\Phi_{2}$ and ${\phi_{k}^{*}}$ be Schwartz functions such that

[TABLE]

Setting

[TABLE]

one obtains that

[TABLE]

and a similar analysis reveals that for each $2\leq j\leq n$

[TABLE]

We now claim that

[TABLE]

By applying integration by parts and (5.2), one has

[TABLE]

where the second follows from the fact that the domain of the integral is actually \big{\{}\vec{\boldsymbol{\eta}}\in(\mathbb{R}^{d})^{n}:|\eta_{j}|\leq 2\text{ for }1\leq j\leq n\big{\}}. This yields that

[TABLE]

and for $u\geq 1$ one has

[TABLE]

by using the vanishing moment property of $\phi_{k+u}$. This proves (5.8).

Finally, (5.5), (5.2), (5.7), and (5.8) establish (5.4).

5.3. Proof of (5.3)

We observe that

[TABLE]

and this yields, with the support condition of $\widehat{\phi_{h}}$, that for $h\in\mathbb{N}_{0}$

[TABLE]

By assuming $A_{k,u}^{\vec{\boldsymbol{l}}}=0$ for $k<0$ and applying a change of variables, the last expression is

[TABLE]

That is, for $h\in\mathbb{N}_{0}$

[TABLE]

Moreover, a proper use of Calderón’s reproducing formula proves that

[TABLE]

[TABLE]

[TABLE]

5.3.1. The case $0<p<\infty$ or $p=q=\infty$

From (5.10) one has

[TABLE]

It follows from (5.9) that the Fourier transform of $T_{[A_{v+h-u-3-\lfloor\log_{2}{n}\rfloor,u}^{\vec{\boldsymbol{l}}}]}\vec{\boldsymbol{f}}$ is supported on \big{\{}|\xi|\leq 2^{v+h}\big{\}}. We choose $t>0$ such that $s>t-d/2>\tau_{p,q}$ and apply Lemma 2.7 (1) to obtain

[TABLE]

where we applied a change of variables and (5.4) in the last two inequalities. Since $s-t+d/2>0$ and $N-s>0$, the left hand side of (5.3) is majored by a constant multiple of

[TABLE]

Moreover, using (2.4),

[TABLE]

Now let $S_{Q}:=S_{Q}^{\gamma,q}(\{\mathfrak{M}_{\sigma,2^{k}}(f_{1})_{k}\}_{k\in\mathbb{N}_{0}})$ and apply (3.4) and (2.2) for $0<r<\min{(1,p)}$ to show that the last expression is

[TABLE]

where Hölder’s inequality, Corollary 3.3, Corollary 3.4 (with $\mu=0$), and (5.11)-(5.13) are applied.

Combining all together the proof of (5.3) ends for $0<p<\infty$ or $p=q=\infty$.

5.3.2. The case $p=\infty$ and $0<q<\infty$

Suppose $\sigma>d/q$. First of all, by using (5.3) for the case $p=q=\infty$ and the embedding $F_{\infty}^{s+m,q}\hookrightarrow F_{\infty}^{s+m,\infty}$ one has

[TABLE]

Now we fix a dyadic cube $P\in\mathcal{D}$ with $l(P)<1$. Then it follows from (5.10) that

[TABLE]

We choose $t>0$ such that $s>t-d/2>\tau_{q}$ and apply Lemma 2.7 (2) with $\mu=1$. Then

[TABLE]

We deal with only the case $v-u-3-\lfloor\log_{2}{n}\rfloor\leq-1$ since the other case follows in a similar and simpler way. The supremum in the last expression is less than a constant times the sum of

[TABLE]

[TABLE]

We see that

[TABLE]

and by using (5.4), (5.11), (5.12), and the embedding $F_{\infty}^{s+m,q}\hookrightarrow F_{\infty}^{s+m,\infty}$,

[TABLE]

This proves that the term corresponding to (5.16) in (5.3.2) is dominated by a constant times

[TABLE]

because

[TABLE]

Similarly, (5.4) yields that for $N>s$

[TABLE]

where Lemma 2.6 (2) and (5.13) are applied in the last inequality. This implies that the term corresponding to (5.15) in (5.3.2) is also bounded by a constant times

[TABLE]

This completes the proof of (5.3) for $p=\infty$ and $0<q<\infty$.

Appendix A The proof of (1.5)

Suppose $0<t<1$ and $\sigma>d/t>d$. For $Q\in\mathcal{D}$ let $S_{Q}$ be the subset of $Q$ for $BMO$ norm equivalence of $f$ in Corollary 1.2. For $k\in\mathbb{Z}$ let $\widetilde{\phi_{k}}:=\phi_{k-1}+\phi_{k}+\phi_{k+1}$ so that $\widehat{\widetilde{\phi_{k}}}(\xi)\widehat{\phi_{k}}(\xi)=\widehat{\phi_{k}}(\xi)$. Then by applying (2.4), (3.4), Lemma 2.5 (1), Corollary 1.2, and Lemma 3.1 (1), one obtains

[TABLE]

where we used the fact that $\|g\|_{\dot{F}_{1}^{0,2}}\approx\|g\|_{H^{1}}$.

Appendix B Proof of Lemma 2.2, 2.3, and 2.4

B.1. Proof of Lemma 2.2

Since the case $t=s$ is trivial, we only consider the case $t<s$. Let $\Psi_{0}\in S$ satisfy

[TABLE]

Then we note that $f=\Psi_{k}\ast f$.

First, assume $s=\infty$ and $0<t<\infty$. If $1<t<\infty$, then it follows from Hölder’s inequality that

[TABLE]

If $0<t\leq 1$ then we apply Nikolskii’s inequality to obtain

[TABLE]

and thus

[TABLE]

This proves

[TABLE]

Now assume $0<t<s<\infty$. Then one has

[TABLE]

by applying (B.1).

B.2. Proof of Lemma 2.3

We only care about the case $0<t<s<\infty$ as the other cases can be done similarly. By applying Minkowski’s inequality with $s/t>1$, one has

[TABLE]

and a standard computation (see [16, Appendix B]) yields that

[TABLE]

Therefore,

[TABLE]

B.3. Proof of Lemma 2.4

Suppose $0<\epsilon<\sigma-d/t$. Let

[TABLE]

Then one has

[TABLE]

which concludes the proof since $\sigma-d/t-\epsilon>0$.