Unimodular Fourier multipliers with a time parameter on modulation spaces
© Song; licensee Springer. 2014
Received: 29 October 2013
Accepted: 6 January 2014
Published: 27 January 2014
In this paper, we mainly study the boundedness of unimodular Fourier multipliers with a time parameter on the modulation spaces where is a differentiable real-valued function, namely we estimate under the multiplier norm, denoted by . The sharpness of s and the regularity lost are also discussed when the multiplier acts on functions in modulation spaces. Meanwhile the lower bound of the multiplier is shown. Finally, we present a discussion of the relationship between the main result and well-posedness results for nonlinear PDEs already existing in the literature.
MSC:42B15, 42B35, 42C15.
The modulation spaces have been well known as the ‘right’ spaces in time-frequency analysis. Refer to [1–12]. Recently Hardy type modulation spaces have been proposed in . In this paper, we discuss these spaces in a brief manner, and then study the Fourier multipliers on them.
1.1 Modulation spaces
In this subsection we will introduce modulation spaces and Wiener spaces briefly. We adopt the definitions and notation in  mainly.
where . Strictly speaking, only when , (1) becomes a norm. If , we simply write instead of .
where X is or any other known space for which , , makes sense. Actually, , where is the q-summable sequence with the weight . Moreover, Hardy type modulation spaces , are introduced in  where is the Hardy space.
1.2 Fourier multipliers on modulation spaces
In this section, we review the previous work about Fourier multipliers on modulation spaces. Then we introduce the main contribution of the present paper.
The set of the Fourier multipliers from the function space X to Y is denoted with as a subspace of the bounded linear operator space and is the embedding mapping. Fourier multipliers and the corresponding operators are seldom distinct. Notice and denote , [20–22].
The Fourier multipliers discussed in this paper generally do not preserve most of the Lebesgue spaces or even the Besov spaces (see [18, 23]). This is the motivation to study the boundedness properties on other function spaces. The Fourier multipliers for the modulation spaces have been developed in many papers [14, 17, 24–26] where the so-called unimodular Fourier multipliers were studied and applied into PDEs. One of the most famous examples is as occurs in the Schrödinger equation.
The case is well discussed in  where has been replaced with a real-valued homogeneous function of degree α.
where α is extended to and for large N.
Denote where s represents the regularity that the multiplier gains (or loses when ). The main result of this paper can be stated in the following form. It indeed contains most results of the previous work.
where and will be determined in Section 3 when α is small. (It was shown that when .)
It will be restated in Theorem 4.2 in more precise form. In Theorem 3.1 we estimate the lower bound that the η can reach. Furthermore, we will discuss the sharpness of s in Section 4.2, namely the maximum value for s is such that is controlled by .
refine η so that when α is small,
discuss the sharpness of s with a lemma,
show the lower bound of .
Feature (i) is the main contribution of the paper. We obtain the smallest value of η among the related research. As far as this author knows, (iii) is seldom discussed in the previous works. Meanwhile, we propose a generic process that is different from previous papers, and which allows us to handle the estimate of the unimodular multipliers (to be compared with [16, 17]).
1.3 Notations and organisation
Throughout this paper, means that there exists a positive constant C such that for all x in an abstract space where A, B are two non-negative functions in the space, while is used to denote . means that is less than a small number. will be simply written as , if there is no ambiguity. is shortened as , if n is arbitrary and kept uniform. Notice that throughout this paper. Finally, we define . In fact, is regarded as a vector. denotes the homogeneous Sobolev space, and .
This paper is organised as follows. In Section 2, we mainly discuss the representation of the Fourier multipliers on the modulation spaces. In Section 3, we refine the bound of the part near 0 of the multipliers as the main contribution of the paper. In Section 4, we study the boundedness of the unimodular Fourier multipliers by oscillatory integral theory (Lemma 4.1). We also give the sharpness argument as regards the regularity lost by the multipliers and prove the lower bound. The result in the previous section is applied to the local well-posedness of the dispersive equations in Section 5.
2 Representation of Fourier multipliers
In this section, we will discuss the representation of Fourier multipliers that is the fundamental step for the estimate and shown in (4) and (5). Several well-known lemmata are listed as follows which will be used in the sequel.
Lemma 2.1 (Sobolev embedding )
, , .
Lemma 2.2 (Bernstein theorem )
where (or ), .
By (2), it follows a generalised version of Bernstein’s theorem that will be used in Remark 3.2.
where , .
If , whose Fourier transform is supported in a compact subset Ω of , then uniformly for .
By this fact, we only need to study , but in this paper we are only interested in the following class of multipliers.
Lemma 2.6 (also see )
Denote for short. When , it is written as . The representation of multipliers on modulation spaces leads us to Hormander’s multipliers.
where . One has to apply (6) in extending the results to Hardy type spaces.
It remains to calculate for the estimate of , while the Bernstein theorem suggests calculating . Thus the primary task is to calculate .
This implies the basic thought of the paper (see Lemma 3.1 and Remark 3.2).
3 Near the origin
We shall show the estimates of by distinguishing between the cases when k near the origin and k near infinity. In this section, we focus on the former.
The following lemma generalises Lemma 9 in  as the main contribution of the paper. It shows a more precise upper bound.
where is selected corresponding to α.
L is assumed to be large, but we will see that L can be chosen as small as .
Consider the following two cases.
where . Take . whenever or if n is odd, , so must be square integrable.
which is determined by the coordinates of point A. □
We still have when as (16).
which yields , and we have the same result.
Remark 3.4 A simple example is (or ). It occurs in the wave equation () or the Schrödinger equation ().
4 Near infinity
In this section we come to the infinite case with oscillatory integrals. The sharpness and the lower bound will be discussed in the second subsection.
4.1 Oscillatory integrals
where has the same support as ψ. Therefore, we have the following results.
Proof It is true when obviously (consider (11) at infinity), while with (22) it holds in the case . □
where and the implicative constant is unessential. Then .
One also can get it with oscillatory integrals  considering the two cases and separatively. Strictly speaking, it should be ensured that for the inequality holds, and it indeed does in this context.
We have the following.
where and η is the same as in Lemma 3.1.
Then we combine this estimate with Lemma 3.1 to complete the proof. □
Remark 4.1 An example for such is the homogeneous function of degree α.
4.2 Sharpness and lower bound
The multiplier will lose a regularity when in Theorem 4.2. To show an easy proof of the sharpness, we employ a lemma implicit in  as the key of the argument. In particular, we only need to estimate the upper bound of to get the lower bound of .
where m is any Fourier multiplier whose inverse is .
where , and does not depend on k or t. □
When , we have . However, (27) will not be applied to the Cauchy problem.
establish the identity or ,
calculate with the identity for ,
estimate with oscillatory integrals,
Remark 4.3 (25) is also the key to get the Strichartz estimates since .
5 Application to Cauchy problem
where and is estimated by Theorem 4.2. When , forms a semigroup named the dispersive semigroup.
With (29), we obtain a quantitative form of the local well-posedness for (28) (see ).
then (28) has a unique solution where the constant C depends on k and n.
Now (30) suffices for the previous two inequalities to hold. □
Remark 5.1 The assumption is indispensable for Theorem 4.2 (see ). can be chosen as . Then will be replaced with in (29). The key point is that F has Lipschitz type property. Note that when as is shown in the literature.
Remark 5.2 The results can be generalised to α-modulation spaces analogically where one should consider the so-called α-covering rather than the uniform decomposition. See  for the basic concepts.
Once again, it leads to the partial derivatives of multipliers.
This work is partially supported by NSF of China (Grant No. 11271330, 10931001) and NSFZJ of China (Grant No. Y604563). The author wishes to thank Professor Fan who checked the whole article.
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