- Open Access
-Boundedness of Weyl multipliers
© Huang and Wang; licensee Springer. 2014
- Received: 16 April 2014
- Accepted: 13 October 2014
- Published: 21 October 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.
- Hardy space
- twisted convolution
- Weyl multiplier
- square function
In order to define the space associated with ‘twisted convolution’, we first define the following version maximal operator in terms of twisted convolutions.
lies in , that is, , where denotes the Schwartz space. We set .
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.
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.
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.
(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.
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
, whenever .
This shows that and gives the proof of Lemma 2. □
Lemma 3 There exists such that for any -atom, we have , where .
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. □
It is easy to prove that the heat kernel has the following estimates (cf. ).
We have the following lemma, whose proof is standard (cf. ).
The operators and are isometries on .
- (ii)When , there exists a constant , such that
Now we can prove the following lemma.
- (1)if and only if its Lusin area integral . Moreover, we have
- (2)if and only if its Littlewood-Paley g-function . Moreover, we have
- (3)if and only if its -function , where . Moreover, we have
- (2)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)
- (3)By , we know when . In the following, we show there exists a constant such that for any atom of , we have
Therefore, when , we have . Then Lemma 7 is proved. □
In the following, we give the proof of Theorem 2.
This completes the proof of Theorem 2. □
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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