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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 <p1. 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 ):suppφB(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 <p1, 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 [1], 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 <p1, so the H p also can be defined as {f S ( C n ): M f L p }, where 2 n 2 n + 1 <p1. The case of p=1 has been considered in [1].

Let 2 n 2 n + 1 <p1, 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 <p1, 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 [1], so we will consider the case 2 n 2 n + 1 <p<1 in this paper.

The boundedness of the Weyl multiplier has been considered by many authors (cf. [2] and [3]). 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 [4], 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 <p1.

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

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 [5]; 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 <p<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 [1], so we omit it.

Lemma 1 Let 2 n 2 n + 1 <p<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 <p<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 <p<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 Cσ ( 1 + σ ) 2 n + p p .

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

b ( 1 ) (z)= j = 1 η j ( 1 ) a j ( 1 ) (z),where  j = 1 | η j ( 1 ) | p 1 2 .

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 <C, where 2 n 2 n + 1 <p<1.

Proof Without loss of generality, we can assume that a(z) is an atom supported on B(0,r). Let φB, 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 dzC ( | 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. [[6], 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. [4])

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. [3]).

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 |zw|.

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. [3]).

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 <p<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 [7].

  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 [8]):

    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. [3])

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 [2], 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).

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Correspondence to Jizheng Huang.

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The authors declare that they have no competing interests.

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

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Keywords

  • Hardy space
  • twisted convolution
  • Weyl multiplier
  • square function