Hypersingular integral operators on modulation spaces for
© Cheng; licensee Springer 2012
Received: 12 February 2012
Accepted: 5 July 2012
Published: 23 July 2012
The purpose of this paper is to investigate the mapping properties of the hypersingular integral operators on general weighted modulation spaces for . Our results show that modulation spaces are good substitutions for Lebesgue spaces.
The modulation spaces , were first introduced by Feichtinger in  and . Their classical definition is based on the notion of short time Fourier transform (STFT). Recently, Kobayashi  extended the classical definition to general case for . For the more general definitions, involving different kinds of weight functions, both in the time and the frequency variables, we refer the readers to . During the last ten years, modulation spaces have not only become useful function spaces for time-frequency analysis, they have also been employed to study boundedness properties of pseudo-differential operators, Fourier multipliers, Fourier integral operators and well-posedness of solutions to PDE’s. For more details of the applications of these spaces, the readers can see [1–3, 9, 12, 16, 18–21] and references therein.
In this paper, we are mainly concerned about the boundedness of the hypersingular integral operators along curves on weighted modulation spaces for , and . From our results, we will see that modulation spaces are good substitutions for Lebesgue spaces.
where is a smooth, compactly supported, radial function which is equal to 1 in the unit ball.
It is well known that the -boundedness of the operator T defined by (1.2) is closely related to the values of the parameters α and β. In , Wainger proved that T is a bounded operator on if and only if . Interpolating this result with the result of weak type , Wainger obtained the -boundedness of the operator T. Inspired by Wainger, in , the authors considered the mapping properties of the operator T on weighted modulation for , and . In this paper, we extend such result to the case . Our result is as follows.
Theorem 1.1 Assume . Then the operator T defined by (1.2) is bounded on weighted modulation spaces for , and .
Recently, in , the authors have considered the mapping properties of the operator defined by (1.3) on weighted modulation spaces for , and . For the case , we prove the following result.
Theorem 1.2 Let or for and the operator be defined by (1.3). If , then the operator is bounded on for , and .
They proved that is bounded on if and only if . While , the operator is bounded on for . Inspired by Chen, Fan, Wang and Zhu, the authors of  investigated the boundedness of the operator defined by (1.4) on modulation spaces for , and . While for , we prove the following result.
Theorem 1.3 Suppose that are distinct positive numbers and the operator is defined as (1.4). If , then is bounded on for , and .
In what follows, we always denote C to be a positive constant that may be different at each place, but independent of the essential variables.
This paper is organized as follows. In Section 2, we give the definitions and basic properties of modulation spaces. Section 3 is devoted to the proofs of our main results.
2 Basic definition and important lemma
for every . For and , the weight function .
i.e., the Fourier transform applied to .
Recently, the above definition has been generated by Kobayashi in  to the case . In his definition, the function g is restricted to the space , which is defined by
We may choose a sufficiently small δ, such that the function space is not empty.
is finite, with obvious modifications for p or .
Different test functions define the same space and equivalent quasi-norms on .
- (2)Let and , then
If , then is dense in .
To prove our main results, we need the definition of the Wiener amalgam space .
The proof of Lemma 2.2 can be found in .
is bounded on weighted modulation spaces .
The proof of Lemma 2.3 can be found in , we omit here.
3 Proof of the main results
In this section, we will prove our main results. Firstly, we come to the proof of Theorem 1.1.
By Lemma 2.3, it suffices to prove . In accordance with Lemma 2.2, if we can prove for all , then we get the result. For , in , Wainger proved that if and only if . Thus when , we also have . Therefore, in the following, we only need to prove for all and .
Then for all and , we have . The proof of Theorem 1.1 is completed. □
Next, we give the proof of Theorem 1.2.
The following proof will be decomposed into two cases.
for and .
for all .
Hence for all . Using the same methods, we can also prove for all . Thus for all , we have . While for , in , Zielinski proved that if and only if . Therefore, for all and , we have , and then .
By Lemma 2.3, we complete the proof of Theorem 1.2. □
Finally, we turn our attention to the proof of Theorem 1.3.
So is the Fourier multiplier of the operator . Using the same methods as in , when , for all . By Lemma 2.2, we finish the proof of Theorem 1.3. □
The author would like to thank Professor Zhenqiu Zhang in Nankai University for his valuable comments and suggestions. This work is supported by the National Nature Science Foundation of China (No. 10971039) and NNSF (No. KJ2011A138, No. KJ2012A133) of Anhui Province in China.
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