Open Access

H p -Boundedness of Weyl multipliers

Journal of Inequalities and Applications20142014:422

https://doi.org/10.1186/1029-242X-2014-422

Received: 16 April 2014

Accepted: 13 October 2014

Published: 21 October 2014

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 < p 1 . Then we consider the boundedness of the Weyl multiplier on H p ( C n ) .

MSC:42B30, 42B25, 42B35.

Keywords

Hardy space twisted convolution Weyl multiplier square function

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

Let 2 n 2 n + 1 < p 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 < p 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 [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 2 n + 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 ) d z .
(4)
Let H = Δ + | x | 2 be the Hermite operator, then we have τ ( L f ) = 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 < p 1 .

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 ) d z = 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 | 2 σ and a j ( z ) ω ¯ ( ξ j , z ) d z = 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 ) d z = 0 for some ξ, | ξ z 0 | 2 σ . 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 ) d z = 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 C σ .
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
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 φ B , t > 0 , then
φ t × a ( z ) = [ φ t ( z w ) φ t ( z ) ] a ( w ) ω ¯ ( z , w ) d w + φ t ( z ) a ˆ ( i z ) .
If | z | > 2 r 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 d z | z | > 2 r | I 1 + I 2 | p d z 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 d z C 1 | z | > 2 r r 2 n ( p 1 ) + p | z | ( 2 n + 1 ) p d z C 1 .
By Hardy’s inequality (cf. [[6], Theorem 7.22, p.341]), we get
| z | > 2 r | I 2 | p d z C 2 | z | > 2 r | a ˆ ( i z ) | p | z | 2 n p d z 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 | 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. [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 > 4 n ,
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. □

Declarations

Acknowledgements

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

Authors’ Affiliations

(1)
College of Sciences, North China University of Technology

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© Huang and Wang; licensee Springer. 2014

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