Remarks on the Unimodular Fourier Multipliers on α-Modulation Spaces

on all Schwartz functions f ∈ S(R), where m is called the symbol or multiplier of T m . 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 X and Y be two function/distribution spaces with norms (or quasinorm) ‖ ⋅ ‖ X and ‖ ⋅ ‖ Y , respectively. A bounded functionm is called a Fourier multiplier from X to Y, if there exists a constant C > 0 such that


Introduction
Let F and F −1 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  ∈ S(R  ), 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  > 0 such that       ()     ≤           , for all  ∈ S(R  ).We use the above definition to avoid the situation where S(R  ) is not dense in  , , when  = ∞ or  = ∞.
In this paper, we will study the unimodular Fourier multipliers on the -modulation space  , , ( ∈ [0, 1]) (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 (, ) ∈ R × R  , the formal solution is given by  (, ) =   ()  0 +  ()  1 , where 1/2 (5) and the Klein-Gordon semigroups are defined by 2

Journal of Function Spaces
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  ∈ [0, 1].Modulation spaces are special -modulation spaces in the case  = 0, and the (inhomogeneous) Besov space   , can be regarded as the limit case of  , , as  → 1 (see [2]).So, for the sake of convenience, we can view the Besov spaces as special -modulation spaces and use  ,1 , to denote the inhomogeneous Besov space   , .It is known that  ||  is not bounded on any Lebesgue space   and Besov spaces   , , except for  = 2 or  = 1 and  = 1, (see [3,4]).However,  ||  is bounded on the modulation space   , =  ,0 , for all 1 ≤ , ≤ ∞,  ∈ R (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 0 ≤  ≤ 2,  ||  is bounded on   , for all 1 ≤ ,  ≤ ∞,  ∈ R. Furthermore, in the case  > 2, Miyachi et al. [6] showed that, for . The reader also can see [7][8][9][10][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 homoge-nous 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 () = (1+|| 2 ) 1/2 is not homogeneous.
We now present our main results.
Theorem 3. Let  > 0,  ∈ N,  ≥ [/2] + 1,  > 0,  ̸ = 1,  > 0, and holds for all  if and only if We list two examples to illustrate the assumptions in our theorems.First, the function satisfies the assumptions in Theorem 1 and Corollary 2 for  =  > 0, while () is not radial and not homogeneous.Another function is () = (||) with () = (1 +   )  (,  > 0).This function satisfies the assumptions in Theorem 3 for  = ,  ̸ = 1.One may also observe that if  = 1, there exists no  2 (R + ) 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  = 0.
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   1 ,  1 , 1 and   2 ,  2 , 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 Let S := S(R  ) be the Schwartz space and S  := S  (R  ) the space of all tempered distributions.We define the Fourier transform F and the inverse Fourier To describe the function spaces discussed in this note, we first give the partition of unity on frequency space for  ∈ [0, 1).We suppose  > 0 and  > 0 are two appropriate constants and choose a Schwartz function sequence Then {   ()} ∈Z  constitutes a smooth decomposition of unity.The frequency decomposition operators associated with above function sequence can be defined by for  ∈ Z  .Let 1 ≤ ,  ≤ ∞,  ∈ R, and  ∈ [0, 1); 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   , =  ,0 , and   () =  0  ().We introduce the dyadic decomposition of R  in order to define the Besov space.Let () be a smooth bump function supported in the ball { : || < 3/2} and be identically equal to 1 on the ball { : || ≤ 4/3}.We denote and a function sequence For all integers  ∈ N, we define the Littlewood-Paley operators Let 1 ≤ ,  ≤ ∞, and  ∈ R. For  ∈ S  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] one has We also recall some results obtained in [6,14], respectively. Lemma 5 (see Lemma 3.2 in [6]).Let  > 0 and let  be a Lemma 6 (see Theorem 1.1 in [14]).Let  > 0,  > 0, and Assume that  is a real-valued function of class where the constant  is independent of .

Sufficient Condition of the Boundedness of 𝑒 𝑖𝜇(𝐷)
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.Proof.We will state the proof for the cases  = 0, 1, 2; the other cases can be deduced by a similar argument and an easy induction.
We are in a position to give the proof of Theorem 1.
Proof of Theorem 1.In virtue of the above lemma, since Using Lemma 5, we know that Finally, we use Lemma 6 to complete the proof.
Finally, the conclusion is deduced by Theorem 1.

Sharpness of the Conditions for the Boundenness of 𝑒 𝑖𝜙(|𝜉|)
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.
Assume that  is a real-valued  [/2]+3 function which satisfies the assumptions of Theorem 3. Then one has for all  ∈ N.
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 1/|| (−1)/2 ≲ 1/|| (−1)/2 and (|| −/2 ) ≲ || −/2 .For the case || ≳ 2  , 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 || ≲ 2  , we define and notice that Then the inequality follows by an integration by parts.
Proof.For sufficiently large , we use estimate on some Δ  to estimate ◻   .Choose a  satisfying ⟨⟩ 1/(1−) ∼ 2  .An easy computation shows that Now, we give the asymptotic estimates of ‖ (||) ‖  , , and ‖‖  , , .These results can be verified by the same methods in [14].(67) Proof.In the case that 0 <  ≤ Λ  is trivial, we suppose that  ≥ Λ  in this proof.We only show the proof of (66).
Using Theorem Then the asymptotic formula follows.
For the third and fourth asymptotic formulas, let where   = { ∈ Z  : || ≤ , ⟨⟩ /(1−)  ∈ R  \ (0, )}.For the case  = 1, we have the following lemma; since its proof is similar to that of Lemma 10, we leave the detail to the reader.