Multiplier conditions for Boundedness into Hardy spaces

TL;DR

This paper establishes explicit necessary and sufficient conditions for certain multilinear multiplier operators to be bounded into Hardy spaces, focusing on the vanishing of symbols and their derivatives on specific hyperplanes.

Contribution

It provides new explicit conditions for the boundedness of multilinear multipliers of Coifman-Meyer type into Hardy spaces, extending previous results.

Findings

Conditions involve vanishing of symbols and derivatives on hyperplanes

Applicable to linear and multilinear operators of Coifman-Meyer type

Results include intermediate types of operators

Abstract

In the present work, we find useful and explicit necessary and sufficient conditions for linear and multilinear multiplier operators of Coifman-Meyer type, finite sum of products of Calder\'on-Zygmund operators, and also of intermediate types to be bounded from a product of Lebesgue or Hardy spaces into a Hardy space. These conditions state that the symbols of the multipliers $\sigma(\xi_1,\dots , \xi_m)$ and their derivatives vanish on the hyperplane $\xi_1+\cdots+\xi_m=0$.

Equations

$$\Vert a{\Vert }_{L^{\infty }}\le 1,{\int }_{{\mathbb{R}}^n}{x}^{\alpha }a(x)dx=0$$

$$\Big{\|}\sum_{j=1}^{\infty}\lambda_{j}\chi_{Q_{j}}\Big{\|}_{L^{p}}<\infty.$$

$$T_{\sigma }(f_1,\dots ,f_m)(x)={\int }_{{\mathbb{R}}^{mn}}{e}^{2\pi ix\cdot ({\xi }_1+\cdots +{\xi }_m)}\sigma ({\xi }_1,\dots ,{\xi }_m)\widehat{f_1}({\xi }_1)\cdots \widehat{f_m}({\xi }_m)d{\xi }_1\cdots d{\xi }_m,$$

$${\Delta }_n=\{({\xi }_1,\dots ,{\xi }_m)\in {\mathbb{R}}^{mn}:{\xi }_1+\cdots +{\xi }_m=0\}.$$

$$\big{|}\partial^{\alpha_{1}}_{1}\cdots\partial^{\alpha_{m}}_{m}\sigma(\xi_{1},\dots,\xi_{m})\big{|}\leq C_{\alpha_{1},\dots,\alpha_{m}}(|\xi_{1}|+\cdots+|\xi_{m}|)^{-(|\alpha_{1}|+\cdots+|\alpha_{m}|)}$$

$$\frac{1}{p}=\frac{1}{p_1}+\cdots +\frac{1}{p_m}.$$

$${\int }_{{\mathbb{R}}^n}{x}^{\alpha }{T}_{\sigma }(a_1,\dots ,a_m)(x)dx=0$$

$${\mathcal{O}}_N({\mathbb{R}}^n)=\bigcap _{\beta \in {\mathbb{N}}_0^n,|\beta |\le N}\left\{f\in {\mathcal{C}}_{\mathrm{c}}^{\infty }({\mathbb{R}}^n):{\int }_{{\mathbb{R}}^n}{x}^{\beta }f(x)dx=0\right\}.$$

$$({\partial }_m^{\alpha }\sigma )({\xi }_1,\dots ,{\xi }_m)=0$$

$$\sigma ({\xi }_1,\dots ,{\xi }_m)=\sum _{j=1}^M{\sigma }_{j1}({\xi }_1)\cdots {\sigma }_{jm}({\xi }_m)$$

$$({\partial }_m^{\alpha }\sigma )({\xi }_1,\dots ,{\xi }_m)=0$$

$$\big{|}\partial_{1}^{\alpha_{1}}\cdots\partial_{m}^{\alpha_{m}}\sigma(\xi_{1},\ldots,\xi_{m})\big{|}\leq C_{\alpha_{1},\dots,\alpha_{m}}|\xi_{1}|^{-|\alpha_{1}|}\cdots|\xi_{m}|^{-|\alpha_{m}|}$$

$${\int }_{{\mathbb{R}}^n}{x}^{\alpha }{T}_{\sigma }(a_1,\dots ,a_m)(x)dx$$

$$|T_{\sigma }(a_1,\dots ,a_m)(x)|\lesssim b(x).$$

$$b(x)=\sum_{j=1}^{M}\prod_{k=1}^{m}\Big{[}|T_{\sigma_{jk}}(a_{k})(x)|\chi_{Q_{j}^{*}}(x)+\frac{|Q_{k}|^{1-\frac{1}{p_{k}}+\frac{N+1}{nm}}\chi_{(Q_{k}^{*})^{c}}(x)}{\big{(}|x-c_{k}|+\ell(Q_{k})\big{)}^{n+\frac{N+1}{m}}}\Big{]},$$

$$b(x)=|T_{\sigma}(a_{1},\ldots,a_{m})(x)|\chi_{\cup_{k=1}^{m}Q_{k}^{*}}(x)+\prod_{k=1}^{m}\frac{|Q_{k}|^{1-\frac{1}{p_{k}}+\frac{N+1}{nm}}\chi_{(Q_{k}^{*})^{c}}(x)}{\big{(}|x-c_{k}|+\ell(Q_{k})\big{)}^{n+\frac{N+1}{m}}}.$$

$${\Gamma }_{i,ϵ}({\mathbb{R}}^{mn})=\{({\xi }_1,\dots ,{\xi }_m)\in {\mathbb{R}}^{mn}:|{\xi }_i|\le ϵ\},{\Gamma }_{ϵ}({\mathbb{R}}^{mn})=\bigcup _{i=1}^m{\Gamma }_{i,ϵ}({\mathbb{R}}^{mn}).$$

$${\Gamma }_i({\mathbb{R}}^{mn})=\{({\xi }_1,\dots ,{\xi }_m)\in {\mathbb{R}}^{mn}:{\xi }_i=0\},\Gamma ({\mathbb{R}}^{mn})=\bigcup _{i=1}^m{\Gamma }_i({\mathbb{R}}^{mn}).$$

$$0<p_j<\infty ,1\le j\le m,0<p\le 1,\frac{1}{p}=\frac{1}{p_1}+\cdots +\frac{1}{p_m}.$$

$$({\partial }_m^{\alpha }\sigma )({\xi }_1,\dots ,{\xi }_m)=0,\forall ({\xi }_1,\dots ,{\xi }_m)\in {\Delta }_n\setminus \Gamma ({\mathbb{R}}^{mn}).$$

$${\int }_{{\mathbb{R}}^n}{x}^{\alpha }{T}_{\sigma }(a_1,\dots ,a_m)(x)dx=0$$

$${\int }_{{\mathbb{R}}^n}{x}^{\alpha }{T}_{\sigma }(a)(x)dx=0.$$

$$\left|{\int }_{{\mathbb{R}}^n}(-2\pi ix{)}^{\alpha }{T}_{\sigma }(a)(x)dx\right|=\left|{\partial }^{\alpha }\left[\widehat{T_{\sigma }(a)}\right](0)\right|=\underset{ϵ\to 0}{\lim }\left|{\int }_{{\mathbb{R}}^n}\sigma (\xi )\hat{a}(\xi ){\partial }^{\alpha }[{\phi }_{ϵ}](\xi )d\xi \right|$$

$$\hat{a}(\xi )=\sum _{\beta \le N}{C}_{\beta }{\partial }^{\beta }\hat{a}(0){\xi }^{\beta }+\mathrm{O}(|\xi {|}^{N+1})=\mathrm{O}(|\xi {|}^{N+1})$$

$$\left|{\int }_{{\mathbb{R}}^n}(-2\pi ix{)}^{\alpha }{T}_{\sigma }(a)(x)dx\right|$$

$$\le C_{\alpha }\underset{ϵ\to 0}{\lim }ϵ{\int }_{Q(0,ϵ)}\left|\sigma (\xi )[{\partial }^{\alpha }\phi {]}_{ϵ}(\xi )\right|d\xi$$

$$\le C_{\alpha }\underset{ϵ\to 0}{\lim }ϵ\Vert \sigma {\Vert }_{L^{\infty }}\Vert \partial \phi {\Vert }_{L^1}=0.$$

$${\partial }_2^{\alpha }\sigma ({\xi }_1,-{\xi }_1)=0.$$

$${\int }_{{\mathbb{R}}^n}{x}^{\alpha }{T}_{\sigma }(a_1,a_2)(x)dx=0,\forall |\alpha |\le N.$$

$$\underset{0<ϵ<1}{\sup }\underset{{\xi }_1\in {\mathbb{R}}^n\setminus B(0,2ϵ)}{\sup }\left|{\int }_{{\mathbb{R}}^n}g(\xi -{\xi }_1)\sigma ({\xi }_1,\xi -{\xi }_1){\partial }^{\alpha }[{\phi }_{ϵ}](\xi )d\xi \right|\le C_{\alpha },$$

$$\left|{\int }_{{\mathbb{R}}^n}g(\xi -{\xi }_1)\sigma ({\xi }_1,\xi -{\xi }_1){\partial }^{\alpha }[{\phi }_{ϵ}](\xi )d\xi \right|\le C_{\alpha },$$

Full text

The first author would like to thank the Simons Foundation. The fourth author is supported by Grant-in-Aid for Scientific Research (C), No. 16K05209, Japan Society for the Promotion of Science.

MSC 42B15, 42B30

1. Introduction

Hardy spaces are spaces of distributions on $\mathbb{R}^{n}$ whose smooth maximal functions lie in $L^{p}({\mathbb{R}}^{n})$, for $0<p<\infty$. These spaces coincide with $L^{p}({\mathbb{R}}^{n})$ if $1<p<\infty$. Let $0<p\leq 1$ and $N$ is a prescribed integer satisfying N\geq\big{\lfloor}n(\frac{1}{p}-1)\big{\rfloor}+1, where \big{\lfloor}s\big{\rfloor} denotes the largest integer less than or equal to $s$. An $L^{\infty}$ function $a$ is said to be $(p,\infty)$-atom, if $a$ is supported on some cube $Q$ and satisfies

[TABLE]

for all $\alpha\in{\mathbb{N}}_{0}^{n}$ such that $|\alpha|\leq N$, see [6], [19]. The space $H^{p}({\mathbb{R}}^{n})$ can be characterized as the set of all tempered distributions which can be expressed as a sum of the form $\sum_{j=1}^{\infty}\lambda_{j}a_{j}$, where $a_{j}$ are $(p,\infty)$-atoms and $(\lambda_{j})_{j=1}^{\infty}$ is a sequence of non-negative numbers such that

[TABLE]

In this note we study linear or multilinear multiplier operators that map products of Hardy spaces into other Hardy spaces. These operators have the form

[TABLE]

where $\sigma$ is a bounded function on ${\mathbb{R}}^{mn}$. Here $\widehat{f}(\xi)$ denotes the Fourier transform of a Schwartz function $f$ defined by $\int_{\mathbb{R}^{n}}f(x)e^{-2\pi ix\cdot\xi}dx$. We are interested in explicit conditions on the symbol $\sigma$ that characterize boundedness into a Hardy space. These conditions reflect the amount of cancellation the symbols contain. For instance, boundedness into $H^{1}({\mathbb{R}}^{n})$ for $m$-linear operators is characterized by the cancellation condition $\sigma(\xi_{1},\dots,\xi_{m})=0$ on the hyperplane $\Delta_{n}$, where $\Delta_{n}$ is given by

[TABLE]

For a multiindex $\alpha=(i_{1},\dots,i_{n})$ we set $\partial^{\alpha}_{k}=\partial^{i_{1}}_{\xi_{k1}}\cdots\partial^{i_{n}}_{\xi_{kn}}$, where $\xi_{k}=(\xi_{k1},\dots,\xi_{kn})\in\mathbb{R}^{n}$. A symbol $\sigma(\xi_{1},\dots,\xi_{m})$ on $\mathbb{R}^{mn}$ is called of Coifman-Meyer type if

[TABLE]

for sufficiently large $n$-tuples of nonnegative integers $\alpha_{j}$, henceforth called multiindices. Here $|\alpha|=i_{1}+\dots+i_{n}$ is the size of a multiindex $\alpha=(i_{1},\dots,i_{n})\in{\mathbb{N}}_{0}^{n}$. The associated operators $T_{\sigma}$ are called multilinear Calderón-Zygmund operators; these were initially introduced in [2] and were extensively studied in [14]. These operators map products $L^{p_{1}}({\mathbb{R}}^{n})\times\cdots\times L^{p_{m}}({\mathbb{R}}^{n})$ of Lebesgue spaces into another Lebesgue space $L^{p}({\mathbb{R}}^{n})$, where $1<p_{j}<\infty$, $j=1,2,\ldots,m$, and $0<p<\infty$ satisfy

[TABLE]

Boundedness into a Lebesgue space also holds if the initial spaces are Hardy spaces, as shown in [10]; the range $0<p_{i}<\infty$ is included in [10]. Additionally, it was shown by the authors [13] that $T_{\sigma}$ maps a product of Hardy spaces into another Hardy space if the action of $T_{\sigma}$ on atoms has vanishing moments, i.e.

[TABLE]

for all $(p_{j},\infty)$-atom $a_{j}$ and for all |\alpha|\leq\big{\lfloor}n(\frac{1}{p}-1)\big{\rfloor}. Remarkably, the cancellation condition (1.4) is only required to hold for all smooth functions with compact support $a_{j}\in\mathcal{O}_{N}(\mathbb{R}^{n})$, where

[TABLE]

We have the following theorem concerning operators associated with Coifman-Meyer symbols.

Theorem 1.1**.**

Let $\sigma$ be a bounded function on $\mathbb{R}^{mn}$ and \sigma\in\mathcal{C}^{\infty}\big{(}\mathbb{R}^{mn}\setminus\{(0,\dots,0)\}\big{)} that satisfies (1.2). Fix $0<p_{i}\leq\infty$, $0<p\leq 1$ that satisfy (1.3). Then the following two statements are equivalent:

1. (a)

$T_{\sigma}$* maps $H^{p_{1}}(\mathbb{R}^{n})\times\cdots\times H^{p_{m}}(\mathbb{R}^{n})$ to $H^{p}(\mathbb{R}^{n}).$* 2. (b)

For all multiindices $\alpha$ with |\alpha|\leq\big{\lfloor}n(\frac{1}{p}-1)\big{\rfloor} we have

[TABLE]

for all $(\xi_{1},\ldots,\xi_{m})\in\Delta_{n}\setminus\{(0,\dots,0)\}$.

We also consider symbols of the product form

[TABLE]

where the $\sigma_{jk}$’s are Fourier transforms of sufficiently smooth Calderón-Zygmund kernels on $\mathbb{R}^{n}$. For such symbols with $m=2$ it was shown in [4] (see also [11]) that the associated operators are bounded from a product of Hardy spaces into another Hardy space if and only if (1.4) holds. For symbols of the form (1.6) we prove the following analogous result:

Theorem 1.2**.**

Let $\sigma_{jk}$, $1\leq j\leq M,1\leq k\leq m$, be Fourier transforms of Calderón-Zygmund kernels on $\mathbb{R}^{n}$, and let $\sigma$ be a function on $\mathbb{R}^{mn}$ given by (1.6). Fix $0<p_{i}<\infty$, $0<p\leq 1$ that satisfy (1.3). Then the following two statements are equivalent:

1. (a)

$T_{\sigma}$* maps $H^{p_{1}}(\mathbb{R}^{n})\times\cdots\times H^{p_{m}}(\mathbb{R}^{n})$ to $H^{p}(\mathbb{R}^{n}).$* 2. (b)

For all multiindices $\alpha$ with |\alpha|\leq\big{\lfloor}n(\frac{1}{p}-1)\big{\rfloor} condition (1.5) holds, i.e.

[TABLE]

for all $(\xi_{1},\ldots,\xi_{m})\in(\mathbb{R}^{n}\setminus\{0\})^{m}\cap\Delta_{n}$.

Note that for symbols of both types (1.2) and (1.6) we always have

[TABLE]

for all $\alpha_{j}\in{\mathbb{N}}_{0}^{n}$ and all $\xi_{j}\in\mathbb{R}^{n}$, $j=1,\dots,m$, under the assumption that $|\alpha_{j}|>0$ if $\xi_{j}\neq 0$. It turns out that condition (1.7) suffices for the purposes of proving the equivalence between (a) and (b) in both Theorems 1.1 and 1.2, although it is not strong enough to imply boundedness on any product of Lebesgue spaces (see [9]).

Remark 1.3**.**

By symmetry, we note that in condition (1.5) the derivative $\partial_{m}^{\alpha}$ can be replaced by $\partial_{k}^{\alpha}$ for any $k\in\{1,\dots,m-1\}$ in Theorems 1.1 and 1.2.

Boundedness into $H^{p}({\mathbb{R}}^{n})$ for operators $T_{\sigma}$ is often expressed in terms of cancellation of the action of the operator on tuples of atoms. Let $x^{\alpha}=x_{1}^{i_{1}}\cdots x_{n}^{i_{n}}$ if $\alpha=(i_{1},\dots,i_{n})$. In order for the integral

[TABLE]

to be absolutely convergent, it is necessary for $T_{\sigma}(a_{1},\dots,a_{m})(x)$ to have decay, where $a_{j}$ are $(p_{j},\infty)$-atoms. Precisely, we assume that for any $m$-tuple of $(p_{j},\infty)$-atom $a_{j}$ there exists function $b\in L^{p_{j}}(\mathbb{R}^{n})$ which decays like $|x|^{-mn-N-1}$ as $|x|\to\infty$, such that for all $x\in\mathbb{R}^{n}$

[TABLE]

We note that condition (1.8) is valid for a large class of multilinear operators such as those in Theorems 1.1 and 1.2. Indeed, for operators with symbols of the form (1.6) we can take

[TABLE]

where $Q_{k}$ is a cube that contains the support of $a_{k}$, $\ell(Q_{k})$ denotes the length of $Q_{k}$.

Condition (1.8) is also valid for Coifman-Meyer multipliers (1.2). Indeed, we can choose

[TABLE]

See [13] for estimates (1.9) and (1.10).

To state the main equivalence result between cancellation of multipliers and cancellation of the action of an operator on $m$ tuples of atoms we introduce some notation. For $0<\epsilon<1$ and $1\leq i\leq m$, we denote

[TABLE]

We also define sets

[TABLE]

We will derive both Theorems 1.1 and  1.2 via the following general result.

Theorem 1.4**.**

Let $\sigma$ in L^{\infty}(\mathbb{R}^{mn})\cap\mathcal{C}^{\infty}\big{(}\mathbb{R}^{mn}\setminus\Gamma(\mathbb{R}^{mn})\big{)} satisfy (1.7). Assume that $T_{\sigma}$ satisfies (1.8) for all $a_{j}\in\mathcal{O}_{N}(\mathbb{R}^{n})$ and

[TABLE]

Then the following two statements are equivalent:

1. (a)

For all multiindices $\alpha$ with |\alpha|\leq\big{\lfloor}n(\frac{1}{p}-1)\big{\rfloor} condition (1.5) holds, i.e.

[TABLE] 2. (b)

For all $a_{i}\in\mathcal{O}_{N}(\mathbb{R}^{n})$, $1\leq i\leq m$, condition (1.4) holds, i.e.

[TABLE]

for all $\alpha$ with |\alpha|\leq\big{\lfloor}n(\frac{1}{p}-1)\big{\rfloor}.

Throughout this paper, we denote multiindices by letters $\alpha$, $\beta$, $\gamma$, etc and use the abbreviation $\alpha\leq\beta$ to denote that $\alpha_{j}\leq\beta_{j}$ for all $j$ if $\alpha=(\alpha_{1},\dots,\alpha_{n})$ and $\beta=(\beta_{1},\dots,\beta_{n})$. We also let $C$ denote a constant independent of crucial parameters whose value may vary on different occurrences.

2. The linear case

In the linear case, assumption (1.4) holds automatically via the following lemma:

Lemma 2.1**.**

For any $a\in\mathcal{O}_{N}(\mathbb{R}^{n})$ and $|\alpha|\leq N$, we have that

[TABLE]

Proof.

We write

[TABLE]

integrating by parts. Now, we notice that by the Taylor expansion and the vanishing moments of $a$,

[TABLE]

as $|\xi|\to 0$. Hence, we see that

[TABLE]

∎

As a result, the linear Fourier multipliers satisfying the suitable decay condition map product of Hardy spaces into Hardy spaces as is well known.

3. The bilinear case

For the sake of clarity of exposition, we first discuss the bilinear case of Theorem 1.4.

Theorem 3.1**.**

Let $\sigma\in L^{\infty}({\mathbb{R}}^{n}\times{\mathbb{R}}^{n})\cap\mathcal{C}^{\infty}({\mathbb{R}}^{n}\times{\mathbb{R}}^{n}\setminus\{(\xi,\eta):\,\,|\xi||\eta|=0\})$ satisfy $(\ref{decay})$ so that $T_{\sigma}$ satisfies $\eqref{EST-Tatoms}$. Then for a given $N\in{\mathbb{N}}_{0}$ the following conditions are equivalent:

1. (a)

For all $\alpha\in{\mathbb{N}}^{n}_{0}$ with $|\alpha|\leq N$ and $\xi_{1}\in{\mathbb{R}}^{n}\setminus\{0\}$, we have

[TABLE] 2. (b)

For any smooth functions $a_{1},a_{2}\in\mathcal{O}_{N}(\mathbb{R}^{n})$,

[TABLE]

To obtain Theorem 3.1 we need a couple of lemmas. Here and below by $B(x,r)$ the open ball centered at $x$ of radius $r>0$.

Lemma 3.2**.**

Assume that $\sigma$ is a bounded function on ${\mathbb{R}}^{n}\times{\mathbb{R}}^{n}$ and smooth away from the axes that satisfies (1.7). Fix $N\in\mathbb{N}_{0}$. Then for all $\alpha\in{\mathbb{N}}^{n}_{0}$ with $|\alpha|\leq N$ there is a constant $C_{\alpha}$ such that

[TABLE]

where $g$ is a smooth function with bounded derivatives $\partial^{\beta}g$ and $\partial^{\beta}g(0)=0$ for all $|\beta|\leq N$.

Proof.

Fix any $\epsilon<1$ and any $\xi_{1}\in{\mathbb{R}}^{n}\setminus B(0,2\epsilon)$. We will show that

[TABLE]

where $C_{\alpha}$ is independent of $\epsilon$ and $\xi_{1}$. Note that the function $\xi\mapsto\sigma(\xi_{1},\xi-\xi_{1})$ is smooth on the domain of integration $|\xi|<\epsilon$, since $\xi_{1}\notin B(0,2\epsilon)$ and thus $|\xi-\xi_{1}|\geq\epsilon$. With this in mind, involving the Taylor expansion of $g$, we notice that

[TABLE]

for any $\alpha\in{\mathbb{N}}^{n}_{0}$ with $|\alpha|\leq N$. Here we used assumption (1.7) and the fact that $\partial^{\beta}g$ are bounded and vanishing at [math] for all $|\beta|\leq N$. ∎

Lemma 3.3**.**

Given $a_{1},a_{2}\in\mathcal{O}_{N}(\mathbb{R}^{n})$ and $\sigma$ in $L^{\infty}({\mathbb{R}}^{n}\times{\mathbb{R}}^{n})\cap\mathcal{C}^{\infty}({\mathbb{R}}^{n}\times{\mathbb{R}}^{n}\setminus\{(\xi,\eta):\,\,|\xi||\eta|=0\})$ that satisfies $(\ref{decay})$, if $T_{\sigma}(a_{1},a_{2})$ has sufficient decay (1.8), then we have

[TABLE]

Proof.

First, we write

[TABLE]

using integration by parts. In view of the identity

[TABLE]

the expression on the right in (3) equals

[TABLE]

Now, we decompose (3.8) as $\lim_{\epsilon\to 0}({\rm I}_{\epsilon}+{\rm II}_{\epsilon})$, where

[TABLE]

For the first term, using the vanishing moment condition for $a_{1}$, we have that

[TABLE]

For the second term, inequality (3.3) gives us

[TABLE]

for any $\epsilon\in(0,1)$ and any $\xi_{1}\in{\mathbb{R}}^{n}\setminus B(0,2\epsilon)$ where the constant $C_{\alpha}$ is independent of $\epsilon$ and $\xi_{1}$. Recall $\partial_{2}$ the derivative with respect to the second variable of a function of two variables. Integrating by parts, we rewrite $\mathrm{II}_{\epsilon}$ as

[TABLE]

The Lebesgue dominated convergence theorem and the approximation to identity, combined with the fact that (3.9) holds and that $\widehat{a_{1}}\in L^{1}(\mathbb{R}^{n})$, yields

[TABLE]

This completes the proof of the lemma. ∎

Lemma 3.4**.**

There exists a function $\zeta\in\mathcal{C}^{\infty}_{0}(\mathbb{R}^{n})$ such that

[TABLE]

Proof.

The Fourier transform of the function \big{(}\frac{\cos|\xi|-1}{|\xi|}\big{)}^{n+1} on $\mathbb{R}^{n}$ is known to be compactly supported; see [1, Lemma 3.1] and bounded but may not be smooth. Let $\Phi$ be a smooth and compactly supported function with non-vanishing integral. Then \zeta=\Phi*\Big{(}\big{(}\frac{\cos|\xi|-1}{|\xi|}\big{)}^{n+1}\Big{)}^{\vee} lies in $\mathcal{C}^{\infty}_{0}(\mathbb{R}^{n})$ and satisfies $\widehat{\zeta}(\xi)\neq 0$ for all $0\neq\xi$ in a neighborhood of the origin, since $\widehat{\Phi}$ and $\cos|\xi|-1$ do not vanish near zero and $\cos|\xi|-1$ vanishes only at zero. It remains to dilate $\zeta$ to make it satisfy (3.10). ∎

Lemma 3.5**.**

Let $N\in{\mathbb{N}}$ be fixed and $F\in L^{\infty}({\mathbb{R}}^{n})$. Assume for all functions $G\in L^{\infty}_{\rm c}({\mathbb{R}}^{n})$ with $\widehat{G}\in L^{1}({\mathbb{R}}^{n})$ satisfying

[TABLE]

we have

[TABLE]

Then $F=0$ a.e..

Proof.

Denote

[TABLE]

First, we observe that if $G\in\Omega_{N}(\mathbb{R}^{n})$, then $G_{x_{0}}\in\Omega_{N}(\mathbb{R}^{n}),$ where $G_{x_{0}}=G(\cdot-x_{0})$ for given $x_{0}\in\mathbb{R}^{n}$. To check this observation for $G\in\Omega_{N}(\mathbb{R}^{n})$, we can easily see that $G_{x_{0}}$ is a bounded function with bounded support. Also $\widehat{G_{x_{0}}}(\xi)=e^{2\pi ix_{0}\cdot\xi}\widehat{G}(\xi)$; and hence $\widehat{G_{x_{0}}}\in L^{1}({\mathbb{R}}^{n}),$ since $\widehat{G}\in L^{1}({\mathbb{R}}^{n}).$ Next we want to show that

[TABLE]

In fact, we have

[TABLE]

Thus (3.11) is verified, and we are done with checking that $G_{x_{0}}\in\Omega_{N}(\mathbb{R}^{n})$.

As a consequence of the above observation, we claim that $\widehat{G}F=0$ a.e. and for all $G\in\Omega_{N}(\mathbb{R}^{n})$. Indeed, fix $G\in\Omega_{N}(\mathbb{R}^{n}).$ For each $x_{0}\in\mathbb{R}^{n},$ the above observation showed that $G_{x_{0}}=G(\cdot-x_{0})\in\Omega_{N}(\mathbb{R}^{n}).$ Therefore,

[TABLE]

i.e., \big{(}\widehat{G}F)^{\vee}(x_{0})=0 for each $x_{0}\in\mathbb{R}^{n},$ and for all $G\in\Omega_{N}(\mathbb{R}^{n})$. This completes our claim $\widehat{G}F=0$ a.e. and for all $G\in\Omega_{N}(\mathbb{R}^{n})$.

The rest of the proof is to verify that $F=0$ a.e. by showing $F=0$ a.e. on $B(0,1)$. By Lemma 3.4, we can find a function $\zeta\in C^{\infty}_{0}(\mathbb{R}^{n})$ such that $\widehat{\zeta}(0)=0$ and $\widehat{\zeta}(\xi)\neq 0$ for all $0<|\xi|<1.$ Define

[TABLE]

It is clear that $G\in C^{\infty}_{0}(\mathbb{R}^{n})$ and

[TABLE]

which satisfies condition $\partial^{\alpha}\widehat{G}(0)=0$ for all $|\alpha|\leq N.$ Thus $G\in\Omega_{N}(\mathbb{R}^{n})$. By our claim, we have $\widehat{G}F=0$ a.e. Noting that $\widehat{G}(\xi)\neq 0$ for $0<|\xi|<1,$ we deduce $F=0$ a.e. on $B(0,1)$. By a suitable dilation, we can show that $F=0$ a.e. on $\mathbb{R}^{n}$. ∎

Proof of Theorem 3.1.

We first assume (3.1), and then prove (3.2). This direction can be obtained easily by Lemma 3.3.

Next we consider the inverse implication, i.e., assume (3.2) and then prove (3.1). We first focus on the case of $\alpha=0$. By Lemma 3.3, condition (3.2) is equivalent to

[TABLE]

for all $H^{p_{1}}$-atoms $a_{1}$ and for all $H^{p_{2}}$-atoms $a_{2}$. Now Lemma 3.5 implies that

[TABLE]

Fix $\xi_{1}\in{\mathbb{R}}^{n},\xi_{1}\neq 0.$ Choose $a_{2}\in C^{\infty}_{0}(\mathbb{R}^{n})$, such that $\widehat{a_{2}}(-\xi_{1})>0$, and hence (3.12) deduces $\sigma(\xi_{1},-\xi_{1})=0$, which implies (3.1) for $\alpha=0$.

Next, we discuss the case of $|\alpha|\geq 1$ by induction on its order. Indeed, assume inductively that (3.1) holds for all $|\alpha|\leq k<N.$ We want to show that it also holds for $|\alpha|=k+1\leq N$. The inductive hypothesis together with Lemma 3.3 deduces

[TABLE]

Repeat the argument in the case $\alpha=0$, we obtain (3.1) for $|\alpha|=k+1.$ The proof of the theorem is now completed. ∎

4. The multilinear case

In this section we prove Theorem 1.4.

Lemma 4.1**.**

Let $N\in{\mathbb{N}}$. Let $\alpha$ be a multi-index with $|\alpha|\leq N$. Let $\sigma$ and $a_{i}$ be functions as stated in Theorem 1.4. Then we have

[TABLE]

Proof.

Recall the function $\varphi$ supported in the unit ball and $\widehat{\varphi}(0)=1.$ Fix $a_{j}\in\mathcal{O}(\mathbb{R}^{n})$, $1\leq j\leq m.$ Now we have

[TABLE]

Let

[TABLE]

and denote

[TABLE]

where $\Gamma_{i,\epsilon}(\mathbb{R}^{mn})$ is defined in (1.11). Also set $\Sigma^{1}_{\epsilon}=\mathbb{R}^{mn}\setminus\Sigma^{0}_{\epsilon}$, and hence $\mathbb{R}^{mn}=\Sigma^{0}_{\epsilon}\cup\Sigma^{1}_{\epsilon}$. The last integral in (4.2) can be decomposed into two parts: $\mathrm{I}_{\epsilon}+\mathrm{II}_{\epsilon},$ where

[TABLE]

and

[TABLE]

Next we will show that $\lim_{\epsilon\to 0}\mathrm{I}_{\epsilon}=0.$ Indeed, we can estimate

[TABLE]

Thus, it is enough to show that

[TABLE]

for all $1\leq i\leq m-1$, and

[TABLE]

Without loss of generality, we have only to prove (4.3) for $i=1.$ In this case, we have

[TABLE]

and hence

[TABLE]

which tends to [math] as $\epsilon$ approaches to [math].

Notice that $\varphi$ is supported in the unit ball, therefore $\varphi_{\epsilon}(\xi_{1}+\cdots+\xi_{m})$ survives only if $|\xi_{1}+\cdots+\xi_{m}|\leq\varepsilon$. Identity (4.4) can be proved similarly by making use of the fact that for all $(\xi_{1},\ldots,\xi_{m})\in\Delta^{m-1}_{\epsilon}$,

[TABLE]

and the vanishing moments of $a_{m}$.

Now we turn into $\mathrm{II}_{\epsilon}$ and rewrite it in the following form

[TABLE]

Fix $\xi_{1},\ldots,\xi_{m-1}$ so that $|\xi_{1}+\cdots+\xi_{m-1}|>2\epsilon$, and that $|\xi_{i}|>\epsilon$ for all $1\leq i\leq m-1.$ We easily see that the function $\xi\mapsto\sigma(\xi_{1},\ldots,\xi_{m-1},\xi-\xi_{1}-\cdots-\xi_{m-1})$ is smooth on $B(0,\epsilon)$. Integrating by parts, we have

[TABLE]

Thus

[TABLE]

An argument similar to Lemma 3.2 allows us to use Lebesgue dominated convergence theorem to pass the limit to inside the above integral, together with the use of the approximate identity, to obtain

[TABLE]

This identity completes the proof of the lemma. ∎

Proof of Theorem 1.4.

By Lemma 3.3, it is clear that if (1.5) is valid then (1.4) holds automatically. For the reverse direction, we use an analogous extension of Lemma 3.5 and repeat the proof of Theorem 3.1. ∎

5. Proof of Theorem 1.1

Let $N\in{\mathbb{N}}$ be fixed and let $\sigma$ be a bounded function in $\mathbb{R}^{n}$ that satisfies either (1.2) or (1.6), and let $T_{\sigma}$ be the multilinear multiplier operator associated to $\sigma$. As showed in [13], $T_{\sigma}$ is bounded from $H^{p_{1}}({\mathbb{R}}^{n})\times\cdots\times H^{p_{m}}({\mathbb{R}}^{n})$ to $H^{p}({\mathbb{R}}^{n})$, where $0<p\leq 1,0<p_{j}<\infty$ and $\frac{1}{p}=\frac{1}{p_{1}}+\cdots+\frac{1}{p_{m}}$, if (1.4) holds, i.e.,

[TABLE]

for all $a_{j}\in\mathcal{O}_{N}(\mathbb{R}^{n})$ and all 0<|\alpha|\leq\big{\lfloor}n(\frac{1}{p}-1)\big{\rfloor}. Therefore, the reverse direction from $(b)$ to $(a)$ of Theorem 1.1 follows from Theorem 1.4.

To obtain the other direction, since $T_{\sigma}$ satisfies (1.8), $|x|^{N}T_{\sigma}(a_{1},\ldots,a_{m})$ is an integrable function. Therefore if $T_{\sigma}(a_{1},\ldots,a_{m})\in H^{p}({\mathbb{R}}^{n})$, then (1.4) is valid. This is a consequence of a result in [19, p. 128, 5.4 (c)]. Similarly, we can prove Theorem 1.2 by repeating the above argument.

6. Remarks, Examples, and Applications

It is noteworthy to mention that our results are also valid for symbols of intermediate or mixed type, i.e., of the form

[TABLE]

where for each $\rho=1,\ldots,T$, $I^{\rho}_{1},\ldots,I^{\rho}_{G(\rho)}$ is a partition of $\{1,\ldots,m\}$ and each $T_{\sigma_{I_{g}^{\rho}}}$ is an $|I_{g}^{\rho}|$-linear Coifman-Meyer multiplier operator. We write $I^{\rho}_{1}+\cdots+I^{\rho}_{G(\rho)}=\{1,\ldots,m\}$ to denote such partitions. There is an analogous theorem for these general symbols.

Theorem 6.1**.**

Let $\sigma$ be as in (6.1). Fix $0<p_{i}<\infty$, $0<p\leq 1$ that satisfy (1.3). Then the following two statements are equivalent:

1. (a)

$T_{\sigma}$* maps $H^{p_{1}}(\mathbb{R}^{n})\times\cdots\times H^{p_{m}}(\mathbb{R}^{n})$ to $H^{p}(\mathbb{R}^{n}).$* 2. (b)

For all |\alpha|\leq\big{\lfloor}n(\frac{1}{p}-1)\big{\rfloor} condition (1.5) holds, i.e.

[TABLE]

for all $(\xi_{1},\ldots,\xi_{m})$ on the hyperplane $\Delta_{n}$ away from the points of singularity of $\sigma$.

For the sake of brevity we don’t include a proof of Theorem 6.1 in this note, but we point out that similar techniques can be used to obtain it.

Next, we provide examples of functions that satisfy conditions (3.1); some of these examples are inspired by those given in [7]: On $\mathbb{R}^{2}\times\mathbb{R}^{2}$ with coordinates $(\xi_{1},\eta_{2},\eta_{1},\eta_{2})$ consider the multipliers

[TABLE]

An alternative example is obtained by considering the multiplier

[TABLE]

It is easy to verify that for $(\xi_{1},\xi_{2})\neq(0,0)$ we have

[TABLE]

For higher order cancellation consider the examples

[TABLE]

and

[TABLE]

both of which satisfy:

[TABLE]

for $(k,l)\in\{(0,1),(1,0),(0,0)\}$. The symbols $\sigma_{1}$ and $\sigma_{3}$ are inspired by [7] and arise by expansions of the Hessian or by combinations of the Riesz transforms. Examples of $\sigma_{0}$ and $\sigma_{2}$ are of Coifman-Meyer type (case (i) in the introduction) while $\sigma_{1}$ and $\sigma_{3}$ are as in case (ii), i.e., sums of products of Calderón-Zygmund operators.

We generalize this example as follows:

[TABLE]

where each $n_{j}$ is positive integer. By the Leibniz rule we can check that

[TABLE]

as long as $k+l\leq N-1$.

Finally, we address the following question111posed by R. R. Coifman by personal communication and give a partial answer: Find a condition on a bilinear multiplier $B(f,g)$ such for any two sequences $f_{k}\to f$ weakly and $g_{k}\to g$ weakly, then $B(f_{k},g_{k})\to B(f,g)$ weakly. Suppose that $B$ is given in multiplier form by

[TABLE]

where $f,g$ are defined on $\mathbb{R}^{n}$ and $\sigma(\xi,\eta)$ is a Coifman-Meyer multiplier, i.e., it satisfies:

[TABLE]

for sufficiently large multiindices $\alpha$, $\beta$. We provide a condition on $\sigma$ so that the associated operator preserves weak convergence. Obviously the product $B(f,g)=fg$ does not preserve weak convergence because the symbol $\sigma(\xi_{1},\xi_{2})=1$ fails to satisfy condition $\rm(v)$ below.

Corollary 6.2**.**

Let $1<p<\infty$ and let $B$ be as above. Suppose that $f_{k}$, $g_{k}$, $f$, $g$, $k=1,2,\dots$ are functions on $\mathbb{R}^{n}$ that satisfy:

1. (i)

$\sup_{k}\|f_{k}\|_{L^{p}(\mathbb{R}^{n})}\leq C$. 2. (ii)

$\sup_{k}\|g_{k}\|_{L^{p^{\prime}}(\mathbb{R}^{n})}\leq C$. 3. (iii)

$f_{k}\to f$* weakly in $L^{p}({\mathbb{R}}^{n})$.* 4. (iv)

$g_{k}\to g$* weakly in $L^{p^{\prime}}({\mathbb{R}}^{n})$.* 5. (v)

$\sigma(\xi,-\xi)=0$* for all $\xi\neq 0$.* 6. (vi)

$B(f_{k},g_{k})$* converges a.e. to $B(f,g)$.*

Then $B(f_{k},g_{k})$ converges to $B(f,g)$ weakly in $H^{1}({\mathbb{R}}^{n})$ in the sense that

[TABLE]

for all functions $\varphi\in{\rm VMO}(\mathbb{R}^{n})$.

Proof.

The boundedness of $B$ from $L^{p}({\mathbb{R}}^{n})\times L^{p^{\prime}}({\mathbb{R}}^{n})$ to $H^{1}({\mathbb{R}}^{n})$ can proved by combining condition $\rm(v)$ with Theorem 3.1 ($N=1$) and the result in [13]; a version of this result was also proved by Dobyinski [5, Lemme 3.8]; see also [3]. It follows that

[TABLE]

Thus the sequence $B(f_{k},g_{k}),k=1,2,\ldots$ is uniformly bounded in $H^{1}({\mathbb{R}}^{n})$ and converges a.e. to $B(f,g)$. Then we obtain (6.2) as a consequence of the result in [16]. ∎