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Hypersingular integral operators on modulation spaces for
Journal of Inequalities and Applications volume 2012, Article number: 165 (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.
Define the convolution operator
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 .
The hypersingular integral operator along curves which is defined by
has been studied by many mathematicians (see [4–6, 8, 17, 22, 24]), where and . This operator is initially studied by Zielinski in his PhD thesis (see ). He showed that if , then is bounded on if and only if . This result was later improved by Chandrana in . He considered the general homogeneous curves or for and proved that if , then the operator is bounded on for
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
The following notations will be used throughout this paper. Let be the Schwartz space of all complex-valued rapidly decreasing infinitely differentiable functions on and be the topological dual of . For a function f in , its Fourier transform is defined by , and its inverse Fourier transform is . The translation and the modulation operators are defined by
for every . For and , the weight function .
Let g be a non-zero Schwartz function and and , the weighted modulation space is defined as the closure of the Schwartz class with respect to the norm
with obvious modifications for p or , where is the so-called short time Fourier transform (STFT), which is defined by
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
Definition 2.1 For , we define to be the spaces of all satisfying
We may choose a sufficiently small δ, such that the function space is not empty.
Definition 2.2 Given a , and , , we define the modulation space to be the space of all tempered distributions such that the quasi-norm
is finite, with obvious modifications for p or .
Lemma 2.1 Let and . Then
Different test functions define the same space and equivalent quasi-norms on .
Let and , then
If , then is dense in .
To prove our main results, we need the definition of the Wiener amalgam space .
Definition 2.3 Let and , we define to be the space of all tempered distributions such that
Lemma 2.2 Let and . Define the space by
The proof of Lemma 2.2 can be found in .
Lemma 2.3 Let , and . If , then the multiplier operator defined by
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.
Proof of Theorem 1.1 Our convolution operator T, which is defined by , may be realized on the transform side as a Fourier multiplier operator
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 .
Differentiating the function , and using the polar coordinate transformation, we have
Denote . Choose supported in such that for all r. We can rewrite as
Set , then
Let , then and for all . By Van der Corput lemma, we get
Then for all and , we have . The proof of Theorem 1.1 is completed. □
Next, we give the proof of Theorem 1.2.
Proof of Theorem 1.2 Using Fourier transformation, the operator can be written as
In the following, we only need to consider , the other half can be dealt with similarly. Take , then, by Lemma 2.2 and Lemma 2.3, it suffices to prove
for all . Firstly, we consider the case . Differentiate , and then
The following proof will be decomposed into two cases.
Case 1. . Then the first and second derivative of ϕ about s is
for and .
By Van der Corput lemma, we get and
for all .
Case 2. . In this case, the third derivative of ϕ satisfies
for all . Van der Corput lemma indicates that . Thus for , we have
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.
Proof of Theorem 1.3 By checking our proof in the following, we can assume and consider the operator
Using Fourier transformation, we get the Fourier multiplier of the operator is
Denote such that , then
Set , then
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. □
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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.
The author declares that they have no competing interests.
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Cheng, M. Hypersingular integral operators on modulation spaces for . J Inequal Appl 2012, 165 (2012). https://doi.org/10.1186/1029-242X-2012-165
- hypersingular integral operator
- modulation space
- Fourier multiplier