- Open Access
-Boundedness of Weyl multipliers
Journal of Inequalities and Applicationsvolume 2014, Article number: 422 (2014)
In this paper, we will define Hardy spaces associated with twisted convolution operators and give the atomic decomposition of , where . Then we consider the boundedness of the Weyl multiplier on .
MSC:42B30, 42B25, 42B35.
The ‘twisted translation’ on is defined on measurable functions by
and the ‘twisted convolution’ of two functions f and g on can be defined as
In order to define the space associated with ‘twisted convolution’, we first define the following version maximal operator in terms of twisted convolutions.
Let , and for , . Given and a tempered distribution f, define the grand maximal function
Definition 1 Let f be a tempered distribution on and , we say that f belongs to the Hardy space if and only if the grand maximal function
lies in , that is, , where denotes the Schwartz space. We set .
Let , an atom for centered at is a function which satisfies:
The atomic norm of f can be defined as
where the infimum is taken over all decompositions of in the sense of and are atoms.
In this paper, we will first give the atomic decomposition of as follows.
Theorem 1 Let and , then if and only if , where are atoms. Moreover, .
Remark 2 The case has been proved in , so we will consider the case 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 . We first give some notations for Weyl multipliers. On consider the 2n linear differential operators
Together with the identity they generate a Lie algebra which is isomorphic to the dimensional Heisenberg algebra. The only nontrivial commutation relations are
The operator L defined by
is nonnegative, self-adjoint, and elliptic. Therefore it generates a diffusion semigroup . There exists an irreducible projective representation W of into a separable Hilbert space such that
Given a function f in its Weyl transform is a bounded operator on defined by
Let be the Hermite operator, then we have or more generally
We say that a bounded operator M on is a Weyl multiplier on if the operator initially defined on by
extends to a bounded operator on . In , the author considered multipliers of the form and proved the -boundedness of .
In this paper, we will prove the following.
Theorem 2 Let
Suppose that the function ϕ satisfies
with k, m positive integers such that , where when n is odd and when n is even. Then is a Weyl multiplier on , where .
Remark 3 The case 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 , we mean that there exists a constant such that .
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
The local Hardy space has been defined in ; let and write
Proposition 1 When , the following conditions are equivalent:
(I3) , where , , , and , whenever .
Consider a partition of into a mesh of balls , , and construct a partition of unity such that . The proof of the following lemma is quite similar to Theorem 2.2 in , so we omit it.
Lemma 1 Let , assume , then , , moreover, there exists such that
By Proposition 1 and Lemma 1, we know that every element in can be written as , where
is supported in , and ;
whenever , there exists such that and .
This is not yet the atomic decomposition for . In order to obtain it we must first replace condition (III) with a centered cancellation property.
Lemma 2 Let and be a function supported on , such that
(I2) for some ξ, . If σ is sufficiently small, can be decomposed as , where
, whenever .
Proof Write , where
Then the function satisfies (b) and (c); on the other hand
Let , since , we have , hence
Since and , we have
Let and , then
Then we have . Let σ be small enough such that . By Proposition 1, we have
The functions are as in (b) and (c). We can now decompose the function whose support is contained in a ball , with , as we did for , thus
where , satisfy (b) and (c), and . Moreover,
where the are as in (b) and (c) and
So we can construct sequences and such that
, where the satisfy (b) and (c), and also
, where the satisfy (b) and (c), and also
This shows that and gives the proof of Lemma 2. □
Lemma 3 There exists such that for any -atom, we have , where .
Proof Without loss of generality, we can assume that is an atom supported on . Let , , then
If and , we have , so that
Let , and , then . Therefore
First we have
By Hardy’s inequality (cf. [, Theorem 7.22, p.341]), we get
We also have
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
In order to prove Theorem 2, we need to give some characterizations for . Let be the heat kernel of , then we can get (cf. )
It is easy to prove that the heat kernel has the following estimates (cf. ).
Lemma 4 There exists a positive constant such that
Let be the twisted convolution kernel of , then
Lemma 5 There exist constants such that
In the following, we define the Lusin area integral operator by
and the Littlewood-Paley g-function
We also consider the -function associated with L defined by
We have the following lemma, whose proof is standard (cf. ).
The operators and are isometries on .
When , there exists a constant , such that
Now we can prove the following lemma.
Lemma 7 Let and , then we have:
if and only if its Lusin area integral . Moreover, we have
if and only if its Littlewood-Paley g-function . Moreover, we have
if and only if its -function , where . Moreover, we have
Proof (1) By Lemma 6, we know there exists a constant such that, for any atom of , we have
For the reverse, by Theorem 1, we can prove similarly to Proposition 4.1 in .
Firstly, we can prove is uniformly bounded on atoms of . For the reverse, we can prove the following inequality (cf. Theorem 7. 28 in ):(8)
Then (2) follows from part (1) and (8).
By , we know when . In the following, we show there exists a constant such that for any atom of , we have
We assume is supported in , then
By part (1), we have . We can prove (cf. )
Therefore, when , we have . 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
where , then, by Lemma 7, when ,
This completes the proof of Theorem 2. □
Mauceri G, Picardello M, Ricci F: A Hardy space associated with twisted convolution. Adv. Math. 1981, 39: 270–288. 10.1016/0001-8708(81)90004-9
Thangavelu S:Littlewood-Paley-Stein theory on and Weyl multipliers. Rev. Mat. Iberoam. 1990, 6: 75–90.
Jizheng H: The boundedness of Weyl multiplier on Hardy spaces associated with twisted convolution. Bull. Sci. Math. 2009, 133: 588–596. 10.1016/j.bulsci.2009.04.003
Thangavelu S Math. Notes 42. In Lectures on Hermite and Laguerre Expansions. Princeton University Press, Princeton; 1993.
Goldberg D: A local version of real Hardy spaces. Duke Math. J. 1979, 46: 27–42. 10.1215/S0012-7094-79-04603-9
Garcia-Cuerva J, Rubio de Francia JL North-Holland Mathematics Studies 116. In Weighted Norm Inequalities and Related Topics. North-Holland, Amsterdam; 1985.
Duong XT, Ouhabaz EM, Yan LX: Endpoint estimates for Riesz transforms of magnetic Schrödinger operators. Ark. Mat. 2006, 44: 261–275. 10.1007/s11512-006-0021-x
Folland GB, Stein EM: Hardy Spaces on Homogeneous Groups. Princeton University Press, Princeton; 1982.
This work was supported by the National Natural Science Foundation of China (Grant No. 11471018) and Beijing Natural Science Foundation (Grant No. 1142005).
The authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.