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Unimodular Fourier multipliers with a time parameter on modulation spaces
Journal of Inequalities and Applications volume 2014, Article number: 43 (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.
Denote the frequency-uniform decomposition operators
where constitute a so-called unity partition with good properties, and ψ is a bump-like function. is a subspace of the distribution space with the norm
where . Strictly speaking, only when , (1) becomes a norm. If , we simply write instead of .
Definition 1.1 Given as in the modulation spaces, a function Banach space X and a sequence Banach space Y, the space consists of all distributions ( precisely) for which
Another class of spaces mentioned in the paper is
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.
where 2π is unessential and the Fourier transform is normalised to be
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.
Let us recall the main results relating with our study. Reference  (Theorem 1 as its main result) proved that , . Moreover, we have the following estimate of under the norm (see Corollary 5 in  and also [24, 27]):
The case is well discussed in  where has been replaced with a real-valued homogeneous function of degree α.
Furthermore,  (Theorem 1 and Theorem 2) gave the following estimate:
However, when , the multiplier loses a regularity , precisely
The purpose of the paper is to study the unimodular multipliers with a time parameter and to estimate
where is a real-valued function satisfying the Mihilin type condition
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.
Theorem 1.1 If the real-valued function mentioned above satisfies , then
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 .
Simply stated, we will do three things mainly in the paper:
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 )
Let be an integer, then
By checking the proof Proposition 1.11 in , for any we have
where , is an integer. From the Sobolev embedding theorem,
where (or ), .
By (2), it follows a generalised version of Bernstein’s theorem that will be used in Remark 3.2.
Lemma 2.3 Let be an integer, then
where , .
Lemma 2.4 , for all . More precisely
If , whose Fourier transform is supported in a compact subset Ω of , then uniformly for .
Now we assume that the multiplier m is bounded from to , , namely
If , where Ω is a compact subset of then
has finitely many elements and
and from Nikol’skij-Triebel’s inequality,
Then we have
Replacing f with , we have
which implies a representation of the multipliers on the modulation spaces,
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.
Remark 2.1 When , one only has
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 .
It is easy to see that
which is a polynomial about t multiplied by with functional coefficients; then
where . Equation (8) yields
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.
Lemma 3.1 If , , for and , then
where is selected corresponding to α.
L is assumed to be large, but we will see that L can be chosen as small as .
Proof As in , from (9) we get
Consider the following two cases.
Case 1: or when n is odd. With Lemma 2.2 and (10), it follows that
where . Take . whenever or if n is odd, , so must be square integrable.
Case 2: . Decompose (8) as a polynomial into
We have , and near 0
Unlike , we go back to the second step of (2). For any we have
Take . For the lower terms, we have
To make it finite, we have to find and an integer satisfying
For the leading term, we have
As above, we have to find and so that
where , are both regarded as two variables. The mission is to find a solution to (15) such that , are small enough, and then
Obviously, if n is odd and , then is a solution, while if n is even and , then is a solution. Now we try to find a solution on . Since , it is only need to consider (13) that is equivalent to . It is easy to see that there exists satisfying the inequality, since runs through at least whenever n is odd or even. This implies the following result:
As a result, , can be restrained on and
As is illustrated by Figure 1, in domain I, (15) is reduced to (13), while in domain II, it is reduced to
which implies .
Noticing that , we will only focus on , i.e. . Figure 1 shows that the point is the optimal one. Now (16) can be modified as
which is determined by the coordinates of point A. □
Remark 3.1 If we take where , , then can be smaller. Thus one can see
Remark 3.2 We introduce another approach which is easier but less accurate. Applying Lemma 2.3, we have
where , , . To ensure , we choose satisfying
Obviously there exists such r and a real number . Then
We still have when as (16).
Remark 3.3 The inequality (9) is the key to the proof. Actually, given any multiplier , satisfying
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
Thus we can state the classical result here that
This technique makes a linear alteration of the phase without affecting the norm. More generally,
where has the same support as ψ. Therefore, we have the following results.
Theorem 4.1 If with the assumption of Lemma 3.1, then
Proof It is true when obviously (consider (11) at infinity), while with (22) it holds in the case . □
Now it is sufficient to consider the case , for Theorem 4.1. Denote
where and the implicative constant is unessential. Then .
Lemma 4.1 Let be supported in Ω. If and its Hessian matrix Hp is nonsingular in Ω, then there exists a large a such that
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.
does not have any critical point out of the domain where defined as (20). With the classical argument of oscillatory integrals (see ) and replacing p with in Lemma 4.1, we have for
where p satisfies, additionally to the assumptions of Lemma 3.1, , and the Hessian matrix Hp is nonsingular on . The last assumption yields
We have the following.
Theorem 4.2 If from Lemma 3.1 satisfies and the Hessian matrix Hp is nonsingular on , then
where and η is the same as in Lemma 3.1.
Proof The discussion above shows that for large k
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 .
Proof Notice that , , . With the interpolation theorems, we have
where , and does not depend on k or t. □
On the other hand,
for Lemma 4.1. We obtain
where is a function as in Theorem 4.2. Equation (26) implies the sharpness of s in the case . Replacing with in Lemma 4.2, we get the lower bound
When , we have . However, (27) will not be applied to the Cauchy problem.
Remark 4.2 Reference  provided an alternative method for the sharpness argument. It can be summarised in the following steps:
establish the identity or ,
calculate with the identity for ,
estimate with oscillatory integrals,
where . Combining the results listed above, we conclude . Notice that the dilation properties of modulation spaces are employed in the step (ii). Generally, it is advocated to find a one-parameter group acting on multipliers so that
Remark 4.3 (25) is also the key to get the Strichartz estimates since .
5 Application to Cauchy problem
Theorem 4.2 provides the time-space estimate  for the linear dispersive equations, namely
where and is estimated by Theorem 4.2. When , forms a semigroup named the dispersive semigroup.
Now we consider the Cauchy problem for the nonlinear dispersive equation,
where the complex variable function . If the initial condition is , we have where v is the solution to (28). The Cauchy problem can be written in an equivalent integral form,
In this section, we shall apply the estimate of the multipliers into the local well-posedness of the equation on the modulation spaces. Define two operators on :
Assuming , we immediately have
where . Restrained to the ball ,
With (29), we obtain a quantitative form of the local well-posedness for (28) (see ).
Corollary 5.1 Assume p is the same as in Theorem 4.2 and . If
then (28) has a unique solution where the constant C depends on k and n.
Proof For (29), ℑ is a contraction mapping on , if R satisfies
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.
Remark 5.3 Our method can also be applied in the estimate of the multipliers on Hardy type modulation spaces where we shall use , rather than . First of all, one should return to Remark 2.1, then use Nikol’skij-Triebel’s inequality (see Lemma 14 in ), say . Then one employs the following Bernstein type theorem: if integer , then
where , . In the particular case when ,
Once again, it leads to the partial derivatives of multipliers.
Cordero E, Nicola F: Sharpness of some properties of Wiener amalgam and modulation spaces. Bull. Aust. Math. Soc. 2009, 80: 105-116.
Feichtinger, HG: Modulation spaces on locally compact Abelian group. Technical Report. University of Vienna, Vienna (1983). Also in Wavelets and Their Applications, pp. 99-140. Allied Publishers (2003)
Feichtinger HG: Modulation spaces: looking back and ahead. Sampl. Theory Signal Image Process., Int. J. 2006, 5: 109-140.
Kobayashi M:Modulation spaces for . J. Funct. Spaces Appl. 2006,4(3):329-341.
Kobayashi M: Dual of modulation spaces. J. Funct. Spaces Appl. 2007,5(1):1-8.
Kobayashi M, Sugimoto M: The inclusion relation between Sobolev and modulation spaces. J. Funct. Anal. 2011, 260: 3189-3208.
Okoudjou KA: Embeddings of some classical Banach spaces into modulation spaces. Proc. Am. Math. Soc. 2004,132(6):1639-1647.
Sugimoto M, Tomita N: The dilation property of modulation spaces and their inclusion relation with Besov spaces. J. Funct. Anal. 2007, 248: 79-106.
Tomita N: On the Hörmander multiplier theorem and modulation spaces. Appl. Comput. Harmon. Anal. 2009, 26: 408-415.
Triebel H: Modulation spaces on the Euclidean n -space. Z. Anal. Anwend. 1983,2(5):443-457.
Wang BX, Huo ZH, Hao CC, Guo ZH: Harmonic Analysis Method for Nonlinear Evolution Equations. World Scientific, Singapore; 2011.
Gröchenig K: Foundations of Time-Frequency Analysis. Birkhäuser, Boston; 2001.
Chen JC, Zhong Y: Modulation space estimates for the fractional integral operators. Sci. China Math. 2011,54(7):1478-1489.
Bényi A, Grochenig K, Okoudjou KA, Rogers LG: Unimodular Fourier multipliers for modulation spaces. J. Funct. Anal. 2007, 246: 366-384.
Feichtinger HG: Generalized amalgams, with applications to Fourier transform. Can. J. Math. 1990, 42: 395-409.
Chen, JC, Fan, DS, Sun, LJ, Zhang, CJ: Estimates for unimodular multipliers on modulation Hardy spaces. J. Funct. Spaces Appl. (2013, in press)
Chen JC, Fan DS, Sun LJ: Asymptotic estimates for unimodular Fourier multipliers on modulation spaces. Discrete Contin. Dyn. Syst. 2012,32(2):467-485.
Mizuhara T: On Fourier multipliers of homogeneous Besov spaces. Math. Nachr. 1994, 133: 155-161.
Hirschman II: On multiplier transformations. Duke Math. J. 1959, 26: 211-242.
Maz’ya V, Shaposhnikova TO: Theory of Sobolev Multipliers: With Applications to Differential and Integral Operators. Springer, Berlin; 2011.
Grafakos L: Classic and Modern Fourier Analysis. Prentice Hall, New York; 2003.
Triebel H: Theory of Function Spaces. Birkhäuser, Basel; 1983.
Hörmander L:Estimates for translation invariant operators in -spaces. Ann. Math. 1960, 104: 93-140.
Bényi A, Okoudjou KA: Local well-posedness of nonlinear dispersive equations on modulation spaces. Bull. Lond. Math. Soc. 2009, 41: 549-558.
Miyachi A, Nicola F, Rivetti S, Tabacco A, Tomita N: Estimates for unimodular Fourier multipliers on modulation space. Proc. Am. Math. Soc. 2009, 137: 3869-3883.
Feichtinger HG, Narimani G: Fourier multipliers of classical modulation spaces. Appl. Comput. Harmon. Anal. 2006,21(3):349-359.
Wang B, Zhao L, Boling G:Isomeric decomposition operators, function spaces and applications to nonlinear evolution equations. J. Funct. Anal. 2006,233(1):1-39.
Hörmander L 3. In The Analysis of Linear Partial Differential Operators. Springer, Berlin; 1985.
Domar Y:On the spectral synthesis problem for -dimensional of , . Ann. Math. 1971, 9: 23-37.
Littman W: Fourier transforms of surface-carried measures and differentiability of surface averages. Bull. Am. Math. Soc. 1963, 69: 766-770.
Stein EM: Harnonic Analysis. Princeton University Press, Princeton; 1993.
Cordero E, Nicola F: Some new Strichartz estimates for the Schrödinger equation. J. Differ. Equ. 2008, 245: 1945-1974.
Borup L, Nielsen M: Boundedness for pseudodifferential operators on multivariate α -modulation space. Ark. Mat. 2006, 44: 241-259.
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.
The author declares that they have no competing interests.
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Song, C. Unimodular Fourier multipliers with a time parameter on modulation spaces. J Inequal Appl 2014, 43 (2014). https://doi.org/10.1186/1029-242X-2014-43
- modulation spaces
- Fourier multipliers
- nonlinear dispersive equations
- oscillatory integrals