#### Abstract

We study the boundedness properties of the Fourier multiplier operator on -modulation spaces and Besov spaces . We improve the conditions for the boundedness of Fourier multipliers with compact supports and for the boundedness of on . If is a radial function and satisfies some size condition, we obtain the sufficient and necessary conditions for the boundedness of between and .

#### 1. Introduction

Let and denote the Fourier transform and the inverse Fourier transform, respectively. For a bounded function , the Fourier multiplier operator associated with is defined by on all Schwartz functions , where is called the symbol or multiplier of . Fourier multipliers arise naturally from the formal solution of linear partial differential equations and from the summabilities of Fourier series. The boundedness properties of a Fourier multiplier in various function or distribution spaces contribute an important research topic in harmonic analysis, as well as many significant applications in partial differential equations.

Let and be two function/distribution spaces with norms (or quasinorm) and , respectively. A bounded function is called a Fourier multiplier from to , if there exists a constant such that for all . We use the above definition to avoid the situation where is not dense in when or .

In this paper, we will study the unimodular Fourier multipliers on the -modulation space (see Section 2 for the definition of ). Particularly, we will focus on the unimodular Fourier multipliers with symbol for real-valued functions . These multipliers arise when one solves the Cauchy problem for some dispersive equations. For example, for the Cauchy problem of (linear) Klein-Gordon equations , the formal solution is given by where and the Klein-Gordon semigroups are defined by

The modulation spaces were introduced by Feichtinger [1] in 1983 by the short-time Fourier transform. Now, people have recognized that the modulation spaces are very important function spaces, since they play more and more significant roles not only in harmonic analysis, but also in the study of partial differential equations. On the other hand, Besov space is also a popular working frame in harmonic analysis and partial differential equations. In 1992, Gröbner introduced the -modulation space [2], that is an intermediate space between these two types of spaces with respect to the parameters . Modulation spaces are special -modulation spaces in the case , and the (inhomogeneous) Besov space can be regarded as the limit case of as (see [2]). So, for the sake of convenience, we can view the Besov spaces as special -modulation spaces and use to denote the inhomogeneous Besov space .

It is known that is not bounded on any Lebesgue space and Besov spaces , except for or and , (see [3, 4]). However, is bounded on the modulation space for all , (see Bényi et al. [5]). Hence, the modulation spaces play an alternative role in the study of unimodular Fourier multipliers. In [5], the authors proved that if , is bounded on for all , . Furthermore, in the case , Miyachi et al. [6] showed that, for and , is bounded from to if and only if . The reader also can see [7–11] for more results in this topic.

Since the -modulation space is an extension of the classical modulation space and it is a natural bridge connecting the modulation spaces and the Besov spaces (see [12, 13]), in a recent paper [14], we study the boundedness of on function spaces and establish a sufficient and necessary boundedness theorem by assuming that is a homogenous function. Thus, it will be interesting to study when is not a homogenous function. This motivates us to seek some sharp condition to ensure the boundedness on for the unimodular multiplier when is not homogenous. In this note, we will focus on the case that is a radial but not homogeneous function. We remark that, for a radial function , the operator not only is a generation for the Schrödinger semigroup , but also works for the Klein-Gordon semigroup with symbol , where is not homogeneous.

We now present our main results.

Theorem 1. *Let , , , , and
**
Assume that is a real-valued function of class on which satisfies
**
Suppose also that , , , for , and satisfies . Then one has
**
where the constant is independent of .*

Corollary 2. *Let , , , , , and
**
Assume that is a real-valued function satisfying
**
Let be a smooth positive homogeneous function with degree . Suppose , , for , and satisfies . Then one has
**
where the constant is independent of .*

Theorem 3. *Let , , , , , , and
**
Assume that is a real-valued function which satisfies
**
Let , , for . Then
**
holds for all if and only if
**
or
*

We list two examples to illustrate the assumptions in our theorems. First, the function satisfies the assumptions in Theorem 1 and Corollary 2 for , while is not radial and not homogeneous. Another function is with . This function satisfies the assumptions in Theorem 3 for , . One may also observe that if , there exists no function , which satisfies the size condition (16). If the reader checks the main theorems in [5, 6], it is not difficult to see that our theorems are a substantial improvement and extension of the known results, even in the case .

The paper is organized as follows. In Section 2, we recall some definitions and basic properties. In Section 3, we obtain an improvement of results in [5, 6] by studying more general Fourier multipliers , in which we do not need to assume lower order derivatives of near 0. This new results will be used to achieve a more general result for the boundedness of on spaces . In Section 4, by assuming radial condition on , we deduce a dual estimate of , and then we use the method in [14] to give a sharp result for the boundedness of between and .

#### 2. Preliminaries

We start this section by recalling some notations. Let be a positive constant that may depend on the indices . The notation denotes the statement that , the notation means the statement , and the notation denotes the statement . For a multi-index , we denote and .

Let be the Schwartz space and the space of all tempered distributions. We define the Fourier transform and the inverse Fourier of by

To describe the function spaces discussed in this note, we first give the partition of unity on frequency space for . We suppose and are two appropriate constants and choose a Schwartz function sequence satisfying Then constitutes a smooth decomposition of unity. The frequency decomposition operators associated with above function sequence can be defined by for . Let , , and ; the -modulation space associated with above decomposition is defined by with the usual modifications when . For the sake of simplicity, in this note, we always denote and .

We introduce the dyadic decomposition of in order to define the Besov space. Let be a smooth bump function supported in the ball and be identically equal to 1 on the ball . We denote and a function sequence For all integers , we define the Littlewood-Paley operators Let , and . For we set the the (inhomogeneous) Besov space space norm by The (inhomogeneous) Besov space is the space of all tempered distributions for which the quantity is finite. We recall the following embedding results.

Lemma 4 (embedding [12, 13]). *Let , , for , . If
**
or
**
one has
*

We also recall some results obtained in [6, 14], respectively.

Lemma 5 (see Lemma 3.2 in [6]). *Let and let be a function on satisfying
**
for . Then for each .*

Lemma 6 (see Theorem 1.1 in [14]). *Let , , and
**
Assume that is a real-valued function of class on which satisfies
**
Suppose also that , , , for , and they satisfy . Then we have
**
where the constant is independent of .*

#### 3. Sufficient Condition of the Boundedness of

The goal of this section is to prove Theorem 1 and Corollary 2. We will start with the following derivative lemma for showing that the lower order derivative near 0 does not interrupt the boundedness of on -modulation spaces.

Lemma 7 (derivative lemma). *Let , , . Suppose that satisfying
**
Then the limit
**
exists, and for any , we have
*

*Proof. *We will state the proof for the cases ; the other cases can be deduced by a similar argument and an easy induction.

For , one can observe directly that .

For , fix . For any , , we can find a simple piecewise smooth curve
which is jointed by two curves and with length , where is the straight line connecting points and and is shortest curve on the great circle connecting points and , such that
for all . We have
Hence,
as . So the limit
exists and
For , fix . For any , let be the straight line connecting and , such that
For any , we have
It follows that
Now we finish the proof by repeating the case.

We are in a position to give the proof of Theorem 1.

*Proof of Theorem 1. *In virtue of the above lemma, since
and , for fixed , we have
for any .

Using Lemma 5, we know that

Finally, we use Lemma 6 to complete the proof.

*Proof of Corollary 2. *By the assumptions of Corollary 2, we can use Lemma 7 to deduce that
for fixed and any . The assumption of implies that
It follows then
for any .

On the other hand, one can deduce that
for .

Finally, the conclusion is deduced by Theorem 1.

#### 4. Sharpness of the Conditions for the Boundenness of

In this section, we give the proof of Theorem 3. The key point is that we can obtain a dual estimate for under some size condition on . By combining the dual estimate with Theorem 1, we get the simultaneous asymptotic estimates of and . Then the proof can be finished by the method in [14]. We first start with the dual estimate on Besov spaces.

Lemma 8 (dual estimate for ). *Suppose , . Assume that is a real-valued function which satisfies the assumptions of Theorem 3. Then one has
**
for all .*

*Proof. *Using the change of variables, we have
Use the polar coordinates,
Recall that the Fourier transform of the area measure satisfies
The support of yields and .

For the case , we only need to show that
which is a direct conclusion by the fact that
and the Van der Corput lemma.

For the case that , we define
and notice that
Then the inequality
follows by an integration by parts.

Hence, for all ,

Lemma 9 (dual estimate for , ). *Suppose , . Assume that is a real-valued function satisfying the assumptions of Theorem 3. Then there exists a sufficiently large constant such that
**
for all with , where the constant is independent of .*

*Proof. *For sufficiently large , we use estimate on some to estimate . Choose a satisfying . An easy computation shows that

Now, we give the asymptotic estimates of and . These results can be verified by the same methods in [14].

Lemma 10 (asymptotic estimates of ). *Suppose , . Assume that is a real-valued function which satisfies the assumptions of Theorem 3. we can find function sequences , , and such that
**
Moreover, we can easily verify that
*

*Proof. *In the case that is trivial, we suppose that in this proof. We only show the proof of (66).

Denote by a function with sufficiently small support near zero. Let
Using Theorem 1, we deduce that
Notice that
so we have
For the second equation, we denote
and let , where with .

If , we use Theorem 1 and Lemma 9 to deduce that
If , we deduce that
By a direct calculation, we obtain
Then the asymptotic formula follows.

For the third and fourth asymptotic formulas, let
where . By a direct calculation, we obtain
As above, Theorem 1 and Lemma 9 yield
Finally, the claim follows by
for , and
for .

For the case , we have the following lemma; since its proof is similar to that of Lemma 10, we leave the detail to the reader.

Lemma 11 (asymptotic estimates of ). *Suppose , . Assume that is a real-valued function which satisfies the assumptions of Theorem 3. We can find function sequences , , and such that
**
Moreover, we can easily verify that
*

Now, we give the sketch of proof for Theorem 3.

*Proof of Theorem 3. *As in [14], if
holds for all , we can use Lemmas 10 and 11 to obtain (18) or (19).

On the other hand, if (18) or (19) holds, we can use the embedding lemma (Lemma 4), the dual estimates (Lemma 8 and Lemma 9), and Theorem 1 to obtain the boundedness of from to .

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The authors would like to thank the anonymous referee for helpful comments. This work is partially supported by the NSF of China (Grant no. 11271330).