-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.
KeywordsHardy 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.
2 Atomic decomposition for
(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. □
3 The boundedness of the Weyl multiplier on
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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