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# $H p$-Boundedness of Weyl multipliers

## Abstract

In this paper, we will define Hardy spaces $H p ( C n )$ associated with twisted convolution operators and give the atomic decomposition of $H p ( C n )$, where $2 n 2 n + 1 . Then we consider the boundedness of the Weyl multiplier on $H p ( C n )$.

MSC:42B30, 42B25, 42B35.

## 1 Introduction

The ‘twisted translation’ $τ w$ on $C n$ is defined on measurable functions by

$( τ w f ) ( z ) = exp ( 1 2 ∑ j = 1 n ( w j z j + w ¯ j z ¯ j ) ) f ( z ) = f ( z + w ) exp ( i 2 Im ( z ⋅ w ¯ ) )$

and the ‘twisted convolution’ of two functions f and g on $C n$ can be defined as

$( f × g ) ( z ) = ∫ C n f ( w ) τ − w g ( z ) d w = ∫ C n f ( z − w ) g ( w ) ω ¯ ( z , w ) d w ,$

where $ω(z,w)=exp( i 2 Im(z⋅ w ¯ ))$.

In order to define the space $H p ( C n )$ associated with ‘twisted convolution’, we first define the following version maximal operator in terms of twisted convolutions.

Let $B={φ∈ C ∞ ( C n ):suppφ⊂B(0,1), ∥ φ ∥ ∞ ≤1, ∥ ∇ φ ∥ ∞ ≤2}$, and for $t>0$, $φ t (z)= t − 2 n φ( z t )$. Given $σ∈(0,+∞]$ and a tempered distribution f, define the grand maximal function

$M σ f(z)= sup φ ∈ B , 0 < t < σ | φ t ×f(z)|,$
(1)

where $B(0,1)={z∈ C n :|z|<1}$.

Definition 1 Let f be a tempered distribution on $C n$ and $2 n 2 n + 1 , we say that f belongs to the Hardy space $H p ( C n )$ if and only if the grand maximal function

$M σ f(z)= sup φ ∈ B , 0 < t < σ | φ t ×f(z)|$

lies in $L p ( C n )$, that is, $H p ( C n )={f∈ S ′ ( C n ): M σ f∈ L p }$, where $S( C n )$ denotes the Schwartz space. We set $∥ f ∥ H p = ∥ M σ f ∥ L p$.

Remark 1 From Theorem A in , we know that for some σ, $0<σ<∞$, $M σ f∈ L p$ if and only if $M ∞ f∈ L p$, when $2 n 2 n + 1 , so the $H p$ also can be defined as ${f∈ S ′ ( C n ): M ∞ f∈ L p }$, where $2 n 2 n + 1 . The case of $p=1$ has been considered in .

Let $2 n 2 n + 1 , an atom for $H p ( C n )$ centered at $z∈ C n$ is a function $a(z)$ which satisfies:

• $( 1 ) supp a ? B ( z , r ) , ( 2 ) ? a ? 8 = ( 2 r ) - 2 n p , ( 3 ) ? a ( w ) ? ¯ ( z , w ) d w = 0 .$

The atomic norm of f can be defined as

$∥ f ∥ atom =inf { ( ∑ j = 1 ∞ | λ j | p ) 1 p : f = ∑ λ j a j } ,$

where the infimum is taken over all decompositions of $f=∑ λ j a j$ in the sense of $S ′ ( C n )$ and $a j$ are atoms.

In this paper, we will first give the atomic decomposition of $H p ( C n )$ as follows.

Theorem 1 Let $f∈ S ′ ( C n )$ and $2 n 2 n + 1 , then $f∈ H p ( C n )$ if and only if $f=∑ λ j a j$, where $a j$ are atoms. Moreover, $∥ f ∥ H p ∼ ∥ f ∥ atom$.

Remark 2 The case $p=1$ has been proved in , so we will consider the case $2 n 2 n + 1 in this paper.

The boundedness of the Weyl multiplier has been considered by many authors (cf.  and ). In this paper, we will consider the boundedness of the Weyl multiplier on $H p ( C n )$. We first give some notations for Weyl multipliers. On $C n$ consider the 2n linear differential operators

$Z j = ∂ ∂ z j + 1 4 z ¯ j , Z ¯ j = ∂ ∂ z ¯ j − 1 4 z j ,j=1,2,…,n.$
(2)

Together with the identity they generate a Lie algebra $h n$ which is isomorphic to the $2n+1$ dimensional Heisenberg algebra. The only nontrivial commutation relations are

$[ Z j , Z ¯ j ]=− 1 2 I,j=1,2,…,n.$
(3)

The operator L defined by

$L=− 1 2 ∑ j = 1 n ( Z j Z ¯ j + Z ¯ j Z j )$

is nonnegative, self-adjoint, and elliptic. Therefore it generates a diffusion semigroup ${ T t L } t > 0 = { e − t L } t > 0$. There exists an irreducible projective representation W of $C n$ into a separable Hilbert space $H W$ such that

$W(z+v)=ω(z,v)W(z)W(v).$

Given a function f in $L 1 ( C n )$ its Weyl transform $τ(f)$ is a bounded operator on $H W$ defined by

$τ(f)= ∫ C n f(z)W(z)dz.$
(4)

Let $H=−Δ+ | x | 2$ be the Hermite operator, then we have $τ(Lf)=τ(f)H$ or more generally

$τ ( ϕ ( L ) f ) =τ(f)ϕ(H).$
(5)

We say that a bounded operator M on $L 2 ( R n )$ is a Weyl multiplier on $L p ( R n )$ if the operator $T M$ initially defined on $L 1 ∪ L p$ by

$τ( T M f)=τ(f)M$

extends to a bounded operator on $L p ( C n )$. In , the author considered multipliers of the form $ϕ(H)$ and proved the $L p$-boundedness of $ϕ(H)$.

In this paper, we will prove the following.

Theorem 2 Let

$Δ + ϕ(N)=ϕ(N+1)−ϕ(N)and Δ − ϕ(N)=ϕ(N)−ϕ(N+1).$

Suppose that the function ϕ satisfies

$| Δ − k Δ + m ϕ ( N ) | ≤C N − ( k + m )$
(6)

with k, m positive integers such that $k+m=0,1,…,ν$, where $ν=n+1$ when n is odd and $ν=n+2$ when n is even. Then $ϕ(H)$ is a Weyl multiplier on $H p ( C n )$, where $2 n 2 n + 1 .

Remark 3 The case $p=1$ has been proved in .

Throughout the article, we will use A and C to denote the positive constants, which are independent of main parameters and may be different at each occurrence. By $B 1 ∼ B 2$, we mean that there exists a constant $C>1$ such that $1 C ≤ B 1 B 2 ≤C$.

The paper is organized as follows. In Section 2, we will give the proof of Theorem 1. Theorem 2 will be proved in Section 3.

## 2 Atomic decomposition for $H p ( C n )$

The local Hardy space $h p ( C n )$ has been defined in ; let $f∈ S ′ ( C n )$ and write

$f σ ∗ (z)= sup φ ∈ B , 0 < t < σ | φ t ∗f(z)|.$

Proposition 1 When $2 n 2 n + 1 , the following conditions are equivalent:

• (I1) $f∈ h p ( C n )$;

• (I2) $f σ ∗ (z)∈ L p ( C n )$;

• (I3) $f=∑ λ j a j$, where $∑ j = 1 n | λ j | p <∞$, $supp a j ⊂B( z j , r j )$, $∥ a j ∥ ∞ ≤ ( 2 r j ) − 2 n p$, and $∫ a j (z)dz=0$, whenever $r j <σ$.

Consider a partition of $C n$ into a mesh of balls $B j =B( z j , σ 2 )$, $j=1,2,…$ , and construct a $C ∞$ partition of unity $φ j$ such that $supp φ j ⊂B( z j ,σ)$. The proof of the following lemma is quite similar to Theorem 2.2 in , so we omit it.

Lemma 1 Let $2 n 2 n + 1 , assume $M σ f(z)∈ L p$, then $g j (z)=f(z) φ j (z) ω ¯ ( z j ,z)∈ h p$, $j=1,2,…$ , moreover, there exists $C>0$ such that

$∑ j = 1 ∞ ∥ g j ∥ h p ≤C ∥ M σ f ( z ) ∥ p .$

By Proposition 1 and Lemma 1, we know that every element in $H p ( C n )$ can be written as $f=∑ λ j a j$, where

1. (I)

$( ∑ j = 1 ∞ | λ j | p ) 1 p ≤C ∥ f ∥ H p$;

2. (II)

$a j$ is supported in $B( z j , r j )$, and $∥ a j ∥ ∞ ≤ ( 2 r j ) − 2 n p$;

3. (III)

whenever $r j <σ$, there exists $ξ j$ such that $| ξ j − z j |≤2σ$ and $∫ a j (z) ω ¯ ( ξ j ,z)dz=0$.

This is not yet the atomic decomposition for $H p$. In order to obtain it we must first replace condition (III) with a centered cancellation property.

Lemma 2 Let $2 n 2 n + 1 and $a(z)$ be a function supported on $B=B( z 0 ,r)$, $r<σ$ such that

(I1) $∥ a ∥ ∞ ≤ ( 2 r ) − 2 n p$;

(I2) $∫ a j (z) ω ¯ (ξ,z)dz=0$ for some ξ, $|ξ− z 0 |≤2σ$. If σ is sufficiently small, $a(z)$ can be decomposed as $a(z)=∑ λ j α j (z)$, where

1. (a)

$∑ j = 1 ∞ | λ j | p ≤C$;

2. (b)

$supp α j ⊂B( z j , r j )$, $∥ α j ∥ ∞ ≤ ( 2 r j ) − 2 n p$;

3. (c)

$∫ α j (z) ω ¯ ( z j ,z)dz=0$, whenever $r j <σ$.

Proof Write $a(z)= g ( 1 ) (z)+ b ( 1 ) (z)$, where

$b ( 1 ) (z)= ( 1 | B | ∫ B a ( w ) ω ¯ ( z 0 , w ) d w ) χ B (z)ω( z 0 ,z).$

Then the function $1 2 g ( 1 )$ satisfies (b) and (c); on the other hand

$| b ( 1 ) ( z ) | = 1 | B | | ∫ B a ( w ) ( ω ¯ ( z 0 , w − z 0 ) − ω ¯ ( ξ , w − z 0 ) ) d w | ≤ 1 | B | ∫ B | a ( w ) | | ( ω ¯ ( z 0 , w − z 0 ) − ω ¯ ( ξ , w − z 0 ) ) | d w ≤ ∥ a ∥ ∞ | B | ∫ B | ( ω ¯ ( z 0 , w − z 0 ) − ω ¯ ( ξ , w − z 0 ) ) | d w ≤ C ⋅ r 1 − 2 n p ⋅ σ .$

Let $q= 2 n p 2 n − p$, since $2 n 2 n + 1 , we have $q>1$, hence

$∥ b ( 1 ) ( z ) ∥ q ≤C ( ∫ B | r 1 − 2 n p ⋅ σ | q d z ) 1 q ≤Cσ.$

Since $supp b ( 1 ) (z)⊂B( z 0 ,σ)$ and $ω ¯ ( z 0 ,z) b ( 1 ) (z)∈ h p$, we have

$∥ ω ¯ ( z 0 , z ) b ( 1 ) ( z ) ∥ h p ≤ C ( σ ) ∥ ( ω ¯ ( z 0 , z ) b ( 1 ) ( z ) ) 1 ∗ ∥ p = C ( σ ) ( ∫ B | ( ω ¯ ( z 0 , z ) b ( 1 ) ( z ) ) 1 ∗ | p d z ) 1 p .$

Let $l= q p$ and $I= { ∫ B | ( ω ¯ ( z 0 , z ) b ( 1 ) ( z ) ) 1 ∗ | p d z } 1 p$, then

$I p ≤ { ∫ B | ( ω ¯ ( z 0 , z ) b ( 1 ) ( z ) ) 1 ∗ | q d z } 1 l ⋅ ( 2 r ) 2 n l ′ = ∥ ( ω ¯ ( z 0 , z ) b ( 1 ) ( z ) ) 1 ∗ ∥ q q l ⋅ ( 2 r ) 2 n l ′ ,$

thus

$I≤ ∥ ( ω ¯ ( z 0 , z ) b ( 1 ) ( z ) ) 1 ∗ ∥ q ⋅ ( 2 r ) 2 n ( q − p ) p q ≤C⋅σ⋅ ( 1 + σ ) 2 n + p p .$

Then we have $∥ ω ¯ ( z 0 , z ) b ( 1 ) ( z ) ∥ h p ≤C⋅σ⋅ ( 1 + σ ) 2 n + p p$. Let σ be small enough such that $C⋅σ⋅ ( 1 + σ ) 2 n + p p < 1 2$. By Proposition 1, we have

The functions $a j ( 1 ) (z)$ are as in (b) and (c). We can now decompose the function $a j ( 1 ) (z)$ whose support is contained in a ball $B( z j , r j )$, with $r j <σ$, as we did for $a(z)$, thus

$b ( 1 ) (z)= g ( 2 ) (z)+ b ( 2 ) (z),$

where $g ( 2 ) (z)=∑ λ j ( 2 ) α j ( 2 ) (z)$, $α j ( 2 ) (z)$ satisfy (b) and (c), and $∑ j = 1 ∞ | λ j ( 2 ) | p ≤ 1 2$. Moreover,

$b ( 2 ) (z)=∑ η j ( 2 ) a j ( 2 ) (z),$

where the $a j ( 2 ) (z)$ are as in (b) and (c) and

$∑ j = 1 ∞ | η j ( 2 ) | p ≤ 1 4 .$

So we can construct sequences $b ( k )$ and $g ( k )$ such that

$b ( k ) = g ( k + 1 ) + b ( k + 1 ) ,$

$g ( k + 1 ) =∑ λ j ( k + 1 ) α j ( k + 1 ) (z)$, where the $α j ( k + 1 ) (z)$ satisfy (b) and (c), and also

$∑ j = 1 ∞ | λ j ( k + 1 ) | p ≤ 1 2 k − 1 ,$

$b ( k + 1 ) =∑ η j ( k + 1 ) a j ( k + 1 ) (z)$, where the $a j ( k + 1 ) (z)$ satisfy (b) and (c), and also

$∑ j = 1 ∞ | η j ( k + 1 ) | p ≤ 1 2 k + 1 .$

This shows that $a(z)= ∑ k g ( k ) (z)$ and gives the proof of Lemma 2. □

Lemma 3 There exists $C>0$ such that for any $H p$-atom, we have $∥ M ∞ a ( z ) ∥ p , where $2 n 2 n + 1 .

Proof Without loss of generality, we can assume that $a(z)$ is an atom supported on $B(0,r)$. Let $φ∈B$, $t>0$, then

$φ t ×a(z)=∫ [ φ t ( z − w ) − φ t ( z ) ] a(w) ω ¯ (z,w)dw+ φ t (z) a ˆ (−iz).$

If $|z|>2r$ and $φ t ×a(z)≠0$, we have $t>|z|−r> 1 2 z$, so that

$| φ t ×a(z)|≤ C 1 r r 2 n ( p − 1 ) p | z | 2 n + 1 + C 2 | a ˆ ( − i z ) | | z | 2 n .$

Let $I 1 =r r 2 n ( p − 1 ) p | z | 2 n + 1$, and $I 2 = | a ˆ ( − i z ) | | z | 2 n$, then $| φ t ×a(z)|≤C( I 1 + I 2 )$. Therefore

$∫ | z | > 2 r | M ∞ a(z) | p dz≤ ∫ | z | > 2 r | I 1 + I 2 | p dz≤C ( ∫ | z | > 2 r | I 1 | p d z + ∫ | z | > 2 r | I 2 | p d z ) .$

First we have

$∫ | z | > 2 r | I 1 | p dz≤ C 1 ∫ | z | > 2 r r 2 n ( p − 1 ) + p | z | ( 2 n + 1 ) p dz≤ C 1 ′ .$

By Hardy’s inequality (cf. [, Theorem 7.22, p.341]), we get

$∫ | z | > 2 r | I 2 | p dz≤ C 2 ∫ | z | > 2 r | a ˆ ( − i z ) | p | z | 2 n p dz≤ C 2 ′ .$

We also have

$∫ | z | ≤ 2 r | M ∞ a ( z ) | p d z ≤ ( ∫ | z | ≤ 2 r | M ∞ a ( z ) | d z ) 1 p ⋅ ( 2 r ) 2 n ( 1 − p ) ≤ C ∥ a ∥ ∞ p ( 4 r ) 2 n p ( 4 r ) 2 n ( 1 − p ) ≤ C .$

This completes the proof of Lemma 3. □

Proof of Theorem 1 By Lemma 2 and Lemma 3, we can obtain the proof of Theorem 1. □

## 3 The boundedness of the Weyl multiplier on $H p ( C n )$

In order to prove Theorem 2, we need to give some characterizations for $H p ( C n )$. Let $K t L (z)$ be the heat kernel of ${ T t L } t > 0$, then we can get (cf. )

$K t (z)= ( 4 π ) − n ( sinh t ) − n e − 1 4 | z | 2 ( coth t ) .$
(7)

It is easy to prove that the heat kernel $K t (z)$ has the following estimates (cf. ).

Lemma 4 There exists a positive constant $C>0$ such that

1. (i)

$| K t (z)|≤C t − n e − C | z | 2 t$;

2. (ii)

$|∇ K t (z)|≤C t − n − 1 2 e − C | z | 2 t$.

Let $Q t k (z)$ be the twisted convolution kernel of $Q t k = t 2 k ∂ s k T s L | s = t 2$, then

$Q t k (z)= t 2 k ∂ s k K s (z) | s = t 2 .$

Lemma 5 There exist constants $C, C k >0$ such that

1. (i)

$| Q t k (z)|≤ C k t − 2 n e − C t − 2 | z | 2$;

2. (ii)

$|∇ Q t k (z)− Q t k (w)|≤ C k t − 2 n − 1 e − C t − 2 | z | 2 |z−w|$.

In the following, we define the Lusin area integral operator by

$( S L k f ) (z)= ( ∫ 0 + ∞ ∫ | z − w | < t | Q t k f ( w ) | 2 d w d t t 2 n + 1 ) 1 / 2$

and the Littlewood-Paley g-function

$G L k (f)(z)= ( ∫ 0 ∞ | Q t k f ( z ) | 2 d t t ) 1 / 2 .$

We also consider the $g λ ∗$-function associated with L defined by

$g λ , k ∗ f(x)= ( ∫ 0 ∞ ∫ C n ( t t + | z − w | ) 2 λ n | Q t k f ( w ) | 2 d w d t t 2 n + 1 ) 1 / 2 .$

We have the following lemma, whose proof is standard (cf. ).

Lemma 6

1. (i)

The operators $S L k$ and $G L k$ are isometries on $L 2 ( C n )$.

2. (ii)

When $λ>1$, there exists a constant $C>0$, such that

$C − 1 ∥ f ∥ L 2 ≤ ∥ g λ , k ∗ f ∥ L 2 ≤C ∥ f ∥ L 2 .$

Now we can prove the following lemma.

Lemma 7 Let $2 n 2 n + 1 and $f∈ S ′ ( C n )$, then we have:

1. (1)

$f∈ H p ( C n )$ if and only if its Lusin area integral $S L k f∈ L p ( C n )$. Moreover, we have

$∥ f ∥ H p ∼ ∥ S L k f ∥ L p .$
2. (2)

$f∈ H p ( C n )$ if and only if its Littlewood-Paley g-function $G L k f∈ L p ( C n )$. Moreover, we have

$∥ f ∥ H p ∼ ∥ G L k f ∥ L p .$
3. (3)

$f∈ H p ( C n )$ if and only if its $G λ ∗$-function $G λ , k ∗ f∈ L p ( C n )$, where $λ>4$. Moreover, we have

$∥ f ∥ H p ∼ ∥ G λ , k ∗ f ∥ L p .$

Proof (1) By Lemma 6, we know there exists a constant $C>0$ such that, for any atom $a(x)$ of $H p ( C n )$, we have

$∥ S L k a ∥ L p ≤C.$

For the reverse, by Theorem 1, we can prove similarly to Proposition 4.1 in .

1. (2)

Firstly, we can prove $G L k$ is uniformly bounded on atoms of $H p ( C n )$. For the reverse, we can prove the following inequality (cf. Theorem 7. 28 in ):

$∥ S L k + 1 f ∥ L p ≤C ∥ G L k f ∥ L p .$
(8)

Then (2) follows from part (1) and (8).

1. (3)

By $S L k f(z)≤ ( 1 2 ) 2 λ n g λ , k ∗ f(z)$, we know $f∈ H p ( C n )$ when $g λ , k ∗ f∈ L p ( C n )$. In the following, we show there exists a constant $C>0$ such that for any atom $a(z)$ of $H p ( C n )$, we have

$∥ g λ , k ∗ a ∥ L p ≤C.$

We assume $a(z)$ is supported in $B( z 0 ,r)$, then

$g λ , k ∗ a ( z ) 2 ≤C S L k a ( z ) 2 + ∑ k = 1 ∞ 2 − 2 k λ n S L 2 k a ( z ) 2 .$

Then

$∥ g λ , k ∗ a ∥ L p ≤ C 1 ∥ S L k a ∥ L p + C 2 ∑ k = 1 ∞ 2 − k λ n ∥ S L 2 k a ∥ L p .$

By part (1), we have $∥ S L k a ∥ L p ≤C$. We can prove (cf. )

$∥ S L 2 k a ∥ L p ≤C 2 4 k n .$
(9)

Therefore, when $λ>4$, we have $∥ g λ , k ∗ a ∥ L p ≤C$. Then Lemma 7 is proved. □

In the following, we give the proof of Theorem 2.

Proof of Theorem 2 Firstly, by Lemma 4.1 in , we get

$G L k + 1 (F)(z)≤C G k n , 1 ∗ (f)(z),$

where $F(z)= T ϕ f(z)$, then, by Lemma 7, when $k>4n$,

$∥ ϕ ( L ) f ∥ H p ≤C ∥ G L k + 1 ( F ) ( z ) ∥ L p ≤C ∥ G k n , 1 ∗ ( f ) ( z ) ∥ L p ≤C ∥ f ∥ H p .$

This completes the proof of Theorem 2. □

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## Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 11471018) and Beijing Natural Science Foundation (Grant No. 1142005).

## Author information

Correspondence to Jizheng Huang.

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The authors declare that they have no competing interests.

### Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

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