Remarks on the Unimodular Fourier Multipliers on α-Modulation Spaces

Abstract

We study the boundedness properties of the Fourier multiplier operator 𝑒^{𝑖𝜇(𝐷)} on 𝛼-modulation spaces 𝑀^𝑠,𝛼_𝑝,𝑞(0≤𝛼<1) and Besov spaces 𝐵^𝑠_𝑝,𝑞(𝑀^𝑠,1_𝑝,𝑞). We improve the conditions for the boundedness of Fourier multipliers with compact supports and for the boundedness of 𝑒^{𝑖𝜇(𝐷)} on 𝑀^𝑠,𝛼_𝑝,𝑞. If 𝜇 is a radial function 𝜙(|𝜉|) and 𝜙 satisfies some size condition, we obtain the sufficient and necessary conditions for the boundedness of 𝑒^{𝑖𝜙(|𝐷|)} between 𝑀^𝑠1_𝑝^,𝛼_1,𝑞1 and 𝑀^𝑠2_𝑝^,𝛼_2,𝑞2.

1. Introduction

Let ℱ and ℱ⁻¹ denote the Fourier transform and the inverse Fourier transform, respectively. For a bounded function 𝑚, the Fourier multiplier operator associated with 𝑚 is defined by 𝑇_𝑚𝑓(𝑥)=𝑚(𝐷)𝑓=ℱ⁻¹𝑚ℱ𝑓 (1) on all Schwartz functions 𝑓∈𝒮(ℝ^𝑛), 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 function 𝑚 is called a Fourier multiplier from 𝑋 to 𝑌, if there exists a constant 𝐶>0 such that 𝑇_𝑚𝑓_𝑌≤𝐶𝑓_𝑋, (2) for all 𝑓∈𝒮(ℝ^𝑛). We use the above definition to avoid the situation where 𝒮(ℝ^𝑛) is not dense in 𝑀^𝑠,𝛼_𝑝,𝑞 when 𝑝=∞ or 𝑞=∞.

In this paper, we will study the unimodular Fourier multipliers on the 𝛼-modulation space 𝑀^𝑠,𝛼_𝑝,𝑞(𝛼∈[0,1]) (see Section 2 for the definition of 𝑀^𝑠,𝛼_𝑝,𝑞). Particularly, we will focus on the unimodular Fourier multipliers with symbol 𝑒^{𝑖𝜇(𝜉)} for real-valued functions 𝜇. These multipliers arise when one solves the Cauchy problem for some dispersive equations. For example, for the Cauchy problem of (linear) Klein-Gordon equations 𝑢_𝑡𝑡+()𝑢()𝐼−Δ𝑢=0,0,𝑥=𝑢₀,𝑢_𝑡(0,𝑥)=𝑢₁, (3)(𝑡,𝑥)∈ℝ×ℝ^𝑛, the formal solution is given by 𝑢(𝑡,𝑥)=𝐾^′(𝑡)𝑢₀+𝐾(𝑡)𝑢₁, (4) where 𝐾(𝑡)=𝐺(𝑡)−𝐺(−𝑡)2𝑖1+|𝐷|^21/2 (5) and the Klein-Gordon semigroups are defined by 𝐺(𝑡)=𝑒^{𝑖𝑡(1+|𝐷|2)1/2}. (6)

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 𝐵^𝑠_𝑝,𝑞 is also a popular working frame in harmonic analysis and partial differential equations. In 1992, Gröbner introduced the 𝛼-modulation space 𝑀^𝑠,𝛼_𝑝,𝑞 [2], that is an intermediate space between these two types of spaces with respect to the parameters 𝛼∈[0,1]. Modulation spaces are special 𝛼-modulation spaces in the case 𝛼=0, and the (inhomogeneous) Besov space 𝐵^𝑠_𝑝,𝑞 can be regarded as the limit case of 𝑀^𝑠,𝛼_𝑝,𝑞 as 𝛼→1 (see [2]). So, for the sake of convenience, we can view the Besov spaces as special 𝛼-modulation spaces and use 𝑀^𝑠,1_𝑝,𝑞 to denote the inhomogeneous Besov space 𝐵^𝑠_𝑝,𝑞.

It is known that 𝑒^{𝑖|𝐷|𝛽} is not bounded on any Lebesgue space 𝐿^𝑝 and Besov spaces 𝐵^𝑠_𝑝,𝑞, except for 𝑝=2 or 𝛽=1 and 𝑛=1, (see [3, 4]). However, 𝑒^{𝑖|𝐷|𝛽} is bounded on the modulation space 𝑀^𝑠_𝑝,𝑞=𝑀^𝑠,0_𝑝,𝑞 for all 1≤𝑝,𝑞≤∞, 𝑠∈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≤𝛽≤2, 𝑒^{𝑖|𝐷|𝛽} is bounded on 𝑀^𝑠_𝑝,𝑞 for all 1≤𝑝,𝑞≤∞, 𝑠∈ℝ. Furthermore, in the case 𝛽>2, Miyachi et al. [6] showed that, for 1≤𝑝,𝑞≤∞ and 𝑠₁,𝑠₂∈ℝ, 𝑒^{𝑖|𝐷|𝛽} is bounded from 𝑀^𝑠1_𝑝,𝑞 to 𝑀^𝑠2_𝑝,𝑞 if and only if 𝑠₁−𝑠₂≥(𝛽−2)𝑛|1/𝑝−1/2|. The reader also can see [7–11] for more results in this topic.

Since the 𝑎-modulation space 𝑀^𝑠,𝛼_𝑝,𝑞 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 𝑒^{𝑖𝜇(𝐷)} on function spaces 𝑀^𝑠,𝛼_𝑝,𝑞 and establish a sufficient and necessary boundedness theorem by assuming that 𝜇 is a homogenous function. Thus, it will be interesting to study 𝑒^{𝑖𝜇(𝐷)} when 𝜇 is not a homogenous function. This motivates us to seek some sharp condition to ensure the boundedness on 𝑀^𝑠,𝛼_𝑝,𝑞 for the unimodular multiplier 𝑒^{𝑖𝜇(𝐷)} when 𝜇 is not homogenous. In this note, we will focus on the case that 𝜇 is a radial but not homogeneous function. We remark that, for a radial function 𝜇, the operator 𝑒^{𝑖𝜇(𝐷)} not only is a generation for the Schrödinger semigroup 𝑒^{𝑖|𝐷|𝛽}, but also works for the Klein-Gordon semigroup with symbol 𝑒^𝑖𝜇, where 𝜇(𝜉)=(1+|𝜉|²)^1/2 is not homogeneous.

We now present our main results.

Theorem 1. Let 𝛿>0, 𝐿∈ℕ, 𝐿≥[𝑛/2]+1, 𝛽>0, and 𝒮_𝑝=𝒮_𝑝𝛽=1𝑝−12max𝛽−2𝑛+2𝛼𝑛,0. (7) Assume that 𝜇 is a real-valued function of class 𝐶^𝑁(𝑁≥𝐿,[𝑛/2]+3) on ℝ^𝑛⧵{0} which satisfies 𝜕^𝛾𝜇(𝜉)≤𝐶_𝛾𝜉^{𝛿−|𝛾|}𝜉𝛾𝜕,0<≤1,=𝐿,_𝛾𝜇(𝜉)≤𝐶_𝛾𝜉^{𝛽−|𝛾|},𝜉𝛾≤𝑛>1,2≤2+3. (8) Suppose also that 1≤𝑝,𝑞≤∞, 𝑠_𝑖∈ℝ, 𝛼∈[0,1], for 𝑖=1,2, and satisfies 𝑠₁−𝑠₂≥|𝒮_𝑝|. Then one has 𝑒^{𝑖𝜇(𝐷)}𝑓_{𝑀𝑠2,𝛼𝑝,𝑞}≤𝐶𝑓_𝑀^𝑠₁^,𝛼_𝑝,𝑞, (9) where the constant 𝐶 is independent of 𝑓.

Corollary 2. Let 𝛿>0, 𝐿∈ℕ, 𝐿≥[𝑛/2]+1, 𝜃>0, 𝜆>0, and 𝒮_𝑝𝛽=1𝑝−12max𝛽−2𝑛+2𝛼𝑛,0. (10) Assume that 𝜙 is a real-valued 𝐶^𝑁(ℝ⧵{0})(𝑁≥𝐿,[𝑛/2]+3) function satisfying 𝜕^𝑘𝜙(𝑟)≤𝐶_𝑘|𝑟|^𝛿−𝑘|𝑟|𝜕,0<≤1,𝑘=𝐿,_𝑘𝜙(𝑟)≤𝐶_𝑘|𝑟|^{𝜃−|𝑘|},|𝑟||𝑘|≤𝑛>1,1≤2+3. (11) Let ℎ∶ℝ^𝑛⧵{0}→ℝ⧵{0} be a smooth positive homogeneous function with degree 𝜆. Suppose 1≤𝑝,𝑞≤∞, 𝑠_𝑖∈ℝ, 𝛼∈[0,1] for 𝑖=1,2, and satisfies 𝑠₁−𝑠₂≥|𝒮_𝑝(𝜆𝜃)|. Then one has 𝑒^{𝑖𝜙(ℎ(𝐷))}𝑓_{𝑀𝑠2,𝛼𝑝,𝑞}≤𝐶𝑓_𝑀^𝑠₁^,𝛼_𝑝,𝑞, (12) where the constant 𝐶 is independent of 𝑓.

Theorem 3. Let 𝛿>0, 𝐿∈ℕ, 𝐿≥[𝑛/2]+1, 𝛽>0, 𝛽≠1, 𝜆>0, and 𝒮_𝑝=𝒮_𝑝𝛽=1𝑝−12max𝛽−2𝑛+2𝛼𝑛,0. (13) Assume that 𝜙 is a real-valued 𝐶^𝑁(ℝ⁺)(𝑁≥𝐿,[𝑛/2]+3) function which satisfies 𝜕^𝑘𝜙(𝑟)≤𝐶_𝑘|𝑟|^𝛿−𝑘,0<|𝑟|≤1,𝑘=𝐿,𝜕^𝑘𝜙(𝑟)≤𝐶_𝑘|𝑟|^{𝛽−|𝑘|},|𝑟||𝑘|≤𝑛>1,1≤2+3,𝜕^𝑘𝜙(𝑟)∼|𝑟|^𝛽−𝑘,|𝑟|>1,𝑘=1,2. (14)(15)(16) Let 1≤𝑝_𝑖,𝑞_𝑖≤∞, 𝑠_𝑖∈ℝ, 𝛼∈[0,1] for 𝑖=1,2. Then 𝑒^{𝑖𝜙(|𝐷|)}𝑓_{𝑀𝑠2𝑝22,𝛼,𝑞}≲𝑓_𝑀^𝑠_1𝑝11^,𝛼_,𝑞 (17) holds for all 𝑓 if and only if 1𝑝₂≤1𝑝₁,𝑠₂−𝑛𝛼𝑝₂+max𝒮_𝑝2,𝒮_𝑝′1=𝑠₁−𝑛𝛼𝑝₁,1𝑞₂≤1𝑞₁ (18) or 1𝑝₂≤1𝑝₁,𝑠₂−𝑛𝛼𝑝₂+max𝒮_𝑝2,𝒮_𝑝′1<𝑠₁−𝑛𝛼𝑝₁,𝑠₂−𝑛𝛼𝑝₂+max𝒮_𝑝2,𝒮_𝑝′1+𝑛(1−𝛼)𝑞₂<𝑠₁−𝑛𝛼𝑝₁+𝑛(1−𝛼)𝑞₁. (19)

We list two examples to illustrate the assumptions in our theorems. First, the function 𝜇(𝜉)=(1+∑^𝑛_𝑗=1𝜆_𝑗|𝜉|^𝜆_𝑗)^𝜃(𝜆_𝑗≥0,𝜆>0,𝜃>0) satisfies the assumptions in Theorem 1 and Corollary 2 for 𝛽=𝜆𝜃>0, while 𝜇(𝜉) is not radial and not homogeneous. Another function is 𝜇(𝜉)=𝜇(|𝜉|) with 𝜇(𝑟)=(1+𝑟^𝜆)^𝜃(𝜆,𝜃>0). This function satisfies the assumptions in Theorem 3 for 𝛽=𝜆𝜃, 𝜆𝜃≠1. One may also observe that if 𝛽=1, there exists no 𝐶²(ℝ⁺) function 𝜙(𝑟), 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 𝛼=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 𝑒^{𝑖𝜇(𝐷)}, in which we do not need to assume lower order derivatives of 𝜇(𝜉) near 0. This new results will be used to achieve a more general result for the boundedness of 𝑒^{𝑖𝜇(𝐷)} on spaces 𝑀^𝑠,𝛼_𝑝,𝑞. In Section 4, by assuming radial condition on 𝜇(𝜉), we deduce a dual estimate of 𝑒^{𝑖𝜙(|𝐷|)}, and then we use the method in [14] to give a sharp result for the boundedness of 𝑒^{𝑖𝜙(|𝐷|)} between 𝑀^𝑠1_𝑝^,𝛼_1,𝑞1 and 𝑀^𝑠2_𝑝^,𝛼_2,𝑞2.

2. Preliminaries

We start this section by recalling some notations. Let 𝐶 be a positive constant that may depend on the indices 𝑛,𝑝_𝑖,𝑞_𝑖,𝑠_𝑖,𝛼,𝛽. The notation 𝑋≲𝑌 denotes the statement that 𝑋≤𝐶𝑌, the notation 𝑋∼𝑌 means the statement 𝑋≲𝑌≲𝑋, and the notation 𝑋⋍𝑌 denotes the statement 𝑋=𝐶𝑌. For a multi-index 𝑘=(𝑘₁,𝑘₂,…,𝑘_𝑛)∈ℤ^𝑛, we denote |𝑘|_∞∶=sup_{𝑖=1,2,…,𝑛}|𝑘_𝑖| and ⟨𝑘⟩∶=(1+|𝑘|²)^1/2.

Let 𝒮∶=𝒮(ℝ^𝑛) be the Schwartz space and 𝒮^′∶=𝒮^′(ℝ^𝑛) the space of all tempered distributions. We define the Fourier transform ℱ𝑓 and the inverse Fourier ℱ⁻¹𝑓 of 𝑓∈𝒮(ℝ^𝑛) by ℱ𝑓(𝜉)=𝑓(𝜉)=_ℝ𝑛𝑓(𝑥)𝑒^{−2𝜋𝑖𝑥⋅𝜉}ℱ𝑑𝑥,₋₁𝑓(𝑥)=𝑓(−𝑥)=_ℝ𝑛𝑓(𝜉)𝑒^{2𝜋𝑖𝑥⋅𝜉}𝑑𝜉. (20)

To describe the function spaces discussed in this note, we first give the partition of unity on frequency space for 𝛼∈[0,1). We suppose 𝑐>0 and 𝐶>0 are two appropriate constants and choose a Schwartz function sequence {𝜂^𝛼_𝑘}_𝑘∈ℤ^𝑛 satisfying 𝜂^𝛼_𝑘(𝜉)≥0,if𝜉−𝑘⟨𝑘⟩^{𝛼/(1−𝛼)}<𝑐⟨𝑘⟩^{𝛼/(1−𝛼)};supp𝜂^𝛼_𝑘⊂𝜉∶𝜉−𝑘⟨𝑘⟩^{𝛼/(1−𝛼)}<𝐶⟨𝑘⟩^{𝛼/(1−𝛼)};_{𝑘∈ℤ𝑛}𝜂^𝛼_𝑘(𝜉)≡1,∀𝜉∈ℝ^𝑛;𝜕^𝛾𝜂^𝛼_𝑘(𝜉)≤𝐶_|𝛼|⟨𝑘⟩^{−𝛼|𝛾|/(1−𝛼)},∀𝜉∈ℝ^𝑛,𝛾∈ℤ⁺∪{0}^𝑛. (21) Then {𝜂^𝛼_𝑘(𝜉)}_𝑘∈ℤ^𝑛 constitutes a smooth decomposition of unity. The frequency decomposition operators associated with above function sequence can be defined by □^𝛼_𝑘∶=ℱ⁻¹𝜂^𝛼_𝑘ℱ (22) for 𝑘∈ℤ^𝑛. Let 1≤𝑝,𝑞≤∞, 𝑠∈ℝ, and 𝛼∈[0,1); the 𝛼-modulation space associated with above decomposition is defined by 𝑀^𝑠,𝛼_𝑝,𝑞ℝ^𝑛=𝑓∈𝒮^′ℝ^𝑛∶𝑓_𝑀^𝑠,𝛼_{𝑝,𝑞(ℝ𝑛)}=_{𝑘∈ℤ𝑛}⟨𝑘⟩^{𝑠𝑞/(1−𝛼)}□^𝛼_𝑘𝑓^𝑞_𝑝^1/𝑞<∞ (23) with the usual modifications when 𝑞=∞. For the sake of simplicity, in this note, we always denote 𝑀^𝑠_𝑝,𝑞=𝑀^𝑠,0_𝑝,𝑞 and 𝜂_𝑘(𝜉)=𝜂⁰_𝑘(𝜉).

We introduce the dyadic decomposition of ℝ^𝑛 in order to define the Besov space. Let 𝜑(𝜉) be a smooth bump function supported in the ball {𝜉∶|𝜉|<3/2} and be identically equal to 1 on the ball {𝜉∶|𝜉|≤4/3}. We denote 𝜓(𝜉)=𝜑(𝜉)−𝜑(2𝜉), (24) and a function sequence 𝜓_𝑗(𝜉)=𝜓2^−𝑗𝜉𝜓,𝑗∈ℕ₀(𝜉)=1−_𝑗∈ℕ𝜓_𝑗(𝜉)=𝜑(𝜉). (25) For all integers 𝑗∈ℕ, we define the Littlewood-Paley operators Δ_𝑗𝑓=𝜓_𝑗(𝜉)𝑓(𝜉),Δ₀𝑓=𝜑(𝜉)𝑓(𝜉). (26) Let 1≤𝑝,𝑞≤∞, and 𝑠∈ℝ. For 𝑓∈𝒮^′ we set the the (inhomogeneous) Besov space space norm by 𝑓_{𝐵𝑠𝑝,𝑞}=^∞_𝑗=02^𝑗𝑠𝑞Δ_𝑗𝑓^𝑞_𝐿𝑝^1/𝑞. (27) The (inhomogeneous) Besov space is the space of all tempered distributions 𝑓 for which the quantity ‖𝑓‖_{𝐵𝑠𝑝,𝑞} is finite. We recall the following embedding results.

Lemma 4 (embedding [12, 13]). Let 1≤𝑝_𝑖,𝑞_𝑖≤∞, 𝑠_𝑖∈ℝ, for 𝑖=1,2, 𝛼∈[0,1]. If 1𝑝₂≤1𝑝₁,𝑠₂−𝑛𝛼𝑝₂=𝑠₁−𝑛𝛼𝑝₁,1𝑞₂≤1𝑞₁ (28) or 1𝑝₂≤1𝑝₁,𝑠₂−𝑛𝛼𝑝₂<𝑠₁−𝑛𝛼𝑝₁,𝑠₂−𝑛𝛼𝑝₂+𝑛(1−𝛼)𝑞₂<𝑠₁−𝑛𝛼𝑝₁+𝑛(1−𝛼)𝑞₁, (29) one has 𝑀^𝑠1_𝑝^,𝛼_1,𝑞1⊆𝑀^𝑠2_𝑝^,𝛼_2,𝑞2. (30)

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

Lemma 5 (see Lemma 3.2 in [6]). Let 𝜖>0 and let 𝜇 be a 𝐶^[𝑛/2]+1 function on ℝ^𝑛⧵{0} satisfying 𝜕^𝛾𝜇(𝜉)≤𝐶_𝛾𝜉^𝜖−𝛾 (31) for |𝛾|≤[𝑛/2]+1. Then ℱ⁻¹[𝜂𝑒^{𝑖𝜇(𝜉)}]∈𝐿¹(ℝ^𝑛) for each 𝜂∈𝐶^∞_𝑐(ℝ^𝑛).

Lemma 6 (see Theorem  1.1 in [14]). Let 𝛽>0, 𝜖>0, and 𝒮_𝑝=𝒮_𝑝𝛽=1𝑝−12max𝛽−2𝑛+2𝛼𝑛,0. (32) Assume that 𝜇 is a real-valued function of class 𝐶^[𝑛/2]+3 on ℝ^𝑛⧵{0} which satisfies 𝜕^𝛾𝜇(𝜉)≤𝐶_𝛾𝜉^{𝜖−|𝛾|}𝜉𝛾≤𝑛,0<≤1,2𝜕+1,_𝛾𝜇(𝜉)≤𝐶_𝛾𝜉^{𝛽−|𝛾|},𝜉𝛾≤𝑛>1,2≤2+3. (33) Suppose also that 1≤𝑝,𝑞≤∞, 𝑠_𝑖∈ℝ, 𝛼∈[0,1], for 𝑖=1,2, and they satisfy 𝑠₁−𝑠₂≥|𝒮_𝑝|. Then we have 𝑒^{𝑖𝜇(𝐷)}𝑓_{𝑀𝑠2,𝛼𝑝,𝑞}≤𝐶𝑓_𝑀^𝑠₁^,𝛼_𝑝,𝑞, (34) where the constant 𝐶 is independent of 𝑓.

3. 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 𝑒^{𝑖𝜇(𝐷)} on 𝛼-modulation spaces.

Lemma 7 (derivative lemma). Let 𝐿∈ℤ, 𝐿≥0, 𝛿>0. Suppose that 𝑚∈𝐶^𝐿(ℝ^𝑛⧵{0}) satisfying 𝜕^𝛾𝑚(𝜉)≤𝐶_𝛾𝜉^{𝛿−|𝛾|},0<𝜉≤1,𝛾=𝐿. (35) Then the limit 𝑎=lim_𝜉→0𝑚(𝜉)=𝑚(0) (36) exists, and for any 𝜖∈(0,1)∩(0,𝛿], we have 𝜕^𝛾(𝑚−𝑎)(𝜉)≤𝐶_𝛾𝜉^{𝜖−|𝛾|},0<𝜉≤1,𝛾≤𝐿. (37)

Proof. We will state the proof for the cases 𝐿=0,1,2; the other cases can be deduced by a similar argument and an easy induction.
For 𝐿=0, one can observe directly that 𝑎=0=lim_𝜉→0𝑚(𝜉).
For 𝐿=1, fix 𝜖∈(0,1)∩(0,𝛿). For any 𝜉₁,𝜉₂∈𝐵(0,1), |𝜉₁|≤|𝜉₂|, we can find a simple piecewise smooth curve Γ∶[0,𝑙]⟶ℝ^𝑛, (38) which is jointed by two curves Γ₁and Γ₂ with length 𝑙=|Γ|, where Γ₁ is the straight line connecting points 𝜉₁ and (|𝜉₂|/|𝜉₁|)𝜉₁ and Γ₂ is shortest curve on the great circle connecting points (|𝜉₂|/|𝜉₁|)⋅𝜉₁ and 𝜉₂, such that Γ(0)=𝜉₁,Γ𝜉₂−𝜉₁=𝜉₂𝜉₁𝜉₁,Γ(𝑙)=𝜉₂,Γ^′(𝑡)=1 (39) for all 𝑡∈[0,𝑙]. We have 𝑚𝜉₂−𝑚𝜉₁=|𝑚(Γ(𝑙(Γ(0≤𝑚(Γ(𝑙Γ𝜉))−𝑚))|))−𝑚₂−𝜉₁+𝑚Γ𝜉₂−𝜉₁(Γ(0,𝑚Γ𝜉−𝑚))₂−𝜉₁(Γ(0=−𝑚))_{|𝜉2|−|𝜉1|0}(∇𝑚)(Γ(𝑡))⋅Γ^′(𝑡)≲𝑑𝑡_{|𝜉2|−|𝜉1|0}Γ(𝑡≲|(∇𝑚)())|𝑑𝑡_{|𝜉2|−|𝜉1|0}𝜉₁+𝑡^𝜖−1=𝜉𝑑𝑡_2𝜖−𝜉₁^𝜖𝜖,𝑚(Γ(𝑙))−𝑚Γ𝜉₂−𝜉₁=^𝑙_{𝜉2−𝜉1}(∇𝑚)(Γ(𝑡))⋅Γ^′(𝑡)≲𝑑𝑡_{𝑙𝜉2−𝜉1}Γ(𝑡≲|(∇𝑚)())|𝑑𝑡_{𝑙𝜉2−𝜉1}𝜉₂^𝜖−1𝑑𝑡≲𝜉₂^𝜖. (40) Hence, 𝑚𝜉₂−𝑚𝜉₁≲𝜉₂^𝜖−𝜉₁^𝜖𝜖+𝜉₂^𝜖⟶0 (41) as 𝜉₁,𝜉₂→0. So the limit 𝑎=lim_𝜉→0𝑚(𝜉)=𝑚(0) (42) exists and 𝑚(𝜉)−𝑎=lim_𝜂→0𝑚(𝜉)𝜂≤𝜉−𝑚^𝜖𝜖+𝜉^𝜖≲𝜉^𝜖. (43) For 𝐿=2, fix 𝜖∈(0,1)∩(0,𝛿). For any 𝜉∈𝐵(0,1), let Γ be the straight line connecting 𝜉 and 𝜉/|𝜉|, such that Γ(0)𝜉=𝜉=𝜉,Γ1−𝜉,Γ^′(𝑡)=1. (44) For any 𝑗=1,2,…,𝑛, we have 𝜕_𝑗𝑚(𝜉)−𝜕_𝑗𝑚𝜉𝜉=𝜕_𝑗𝑚Γ1−𝜉−𝜕_𝑗𝑚(Γ(0=))_01−|𝜉|∇𝜕_𝑗𝑚(Γ(𝑡))⋅Γ^′(𝑡)≲𝑑𝑡_01−|𝜉|∇𝜕_𝑗𝑚(Γ(𝑡))𝑑𝑡≲₀^1−|𝜉|𝜉+𝑡^𝜖−2=𝜉𝑑𝑡1−^𝜖−1.𝜖−1 (45) It follows that 𝜕_𝑗𝑚(𝜉)≤𝜕_𝑗𝑚(𝜉)−𝜕_𝑗𝑚𝜉𝜉+𝜕_𝑗𝑚𝜉𝜉≲1−𝜉^𝜖−1𝜉𝜖−1+1≲^𝜖−1. (46) Now we finish the proof by repeating the 𝐿=1 case.

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

Proof of Theorem 1. In virtue of the above lemma, since 𝜕^𝛾𝜇(𝜉)≤𝐶_𝛾𝜉^{𝛿−|𝛾|},0<𝜉≤1,𝛾=𝐿, (47) and 𝐿≥[𝑛/2]+1, for fixed 𝜖∈(0,1)∩(0,𝛿], we have 𝜕^𝛾𝜇−𝜇(0)(𝜉)≤𝐶_𝛾𝜉^{𝜖−|𝛾|},0<𝜉≤1 (48) for any |𝛾|≤[𝑛/2]+1.
Using Lemma 5, we know that ℱ⁻¹𝜂𝑒^{𝑖𝜇(𝜉)}=ℱ⁻¹𝜂𝑒^{𝑖(𝜇−𝜇(0))(𝜉)}∈𝐿¹. (49)
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 𝜕^𝑘𝜙−𝜙(0)(𝑟)≤𝐶_𝑘|𝑟|^𝜖−𝑘,0<|𝑟|≤1 (50) for fixed 𝜖∈(0,1)∩(0,𝛿] and any 𝑘≤[𝑛/2]+1. The assumption of ℎ(𝜉) implies that ℎ(𝜉)∼𝜉^𝜆. (51) It follows then 𝜕^𝛾𝜙∘ℎ(𝜉)≲𝜉^{𝜆𝜖−|𝛾|},0<𝜉≤1, (52) for any |𝛾|=[𝑛/2]+1.
On the other hand, one can deduce that 𝜕^𝛾𝜙∘ℎ(𝜉)≲𝜉^{𝜆𝜃−|𝛾|},𝜉≥1, (53) for 2≤|𝛾|≤[𝑛/2]+3.
Finally, the conclusion is deduced by Theorem 1.

4. 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 𝑒^{𝑖𝜙(|𝜉|)} under some size condition on 𝜙. By combining the dual estimate with Theorem 1, we get the simultaneous asymptotic estimates of ‖𝑓‖_𝑀^𝑠,𝛼_𝑝,𝑞 and ‖𝑒^{𝑖𝜇(𝐷)}‖_𝑀^𝑠,𝛼_𝑝,𝑞. 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 𝐵^𝑠_𝑝,𝑞). Suppose 𝛽>0, 𝛽≠1. Assume that 𝜙 is a real-valued 𝐶^[𝑛/2]+3 function which satisfies the assumptions of Theorem 3. Then one has ℱ⁻¹𝜓_𝑗(𝜉)𝑒^{𝑖𝜙(|𝜉|)}_𝐿∞≤𝐶2^{𝑗(𝛽−2)(−𝑛/2)} (54) for all 𝑗∈ℕ.

Proof. Using the change of variables, we have ℱ⁻¹𝜓_𝑗(𝜉)𝑒^{𝑖𝜙(|𝜉|)}_𝐿∞=2^𝑗𝑛ℱ⁻¹𝜓(𝜉)𝑒^{𝑖𝜙(2𝑗|𝜉|)}_𝐿∞. (55) Use the polar coordinates, ℱ⁻¹𝜓(𝜉)𝑒^{𝑖𝜙(2𝑗|𝜉|)}𝑥)=_ℝ𝑛(𝜓(𝜉)𝑒^{𝑖𝜙(2𝑗|𝜉|)}𝑒^{2𝜋𝑖𝑥⋅𝜉}=𝑑𝜉_∞0𝜓(𝑟)𝑒^{𝑖𝜙(2𝑗𝑟)}𝑟^𝑛−1_𝑆𝑛−1𝑒^{2𝜋𝑖𝑟𝑥⋅𝜉′}𝑑𝜎𝜉^′𝑑𝑟=^∞₀𝜓(𝑟)𝑒^{𝑖𝜙(2𝑗𝑟)}𝑟^𝑛−1𝑑𝜎(−𝑟𝑥)𝑑𝑟. (56) Recall that the Fourier transform of the area measure satisfies ()𝑒𝑑𝜎−𝑟𝑥=𝐶^{2𝜋𝑖𝑟|𝑥|}|𝑟𝑥|^(𝑛−1)/2𝑒+𝐶^{−2𝜋𝑖𝑟|𝑥|}|𝑟𝑥|^(𝑛−1)/2+𝑂|𝑟𝑥|^−𝑛/2. (57) The support of 𝜓 yields 1/|𝑟𝑥|^(𝑛−1)/2≲1/|𝑥|^(𝑛−1)/2 and 𝑂(|𝑟𝑥|^−𝑛/2)≲|𝑥|^−𝑛/2.
For the case |𝑥|≳2^𝑗𝛽, we only need to show that ^∞₀𝜓(𝑟)𝑒^{𝑖[𝜙(2𝑗𝑟)±2𝜋𝑖𝑟|𝑥|]}𝑟^𝑛−1𝑑𝑟≲2^{−𝑗𝛽(𝑛/2)}, (58) which is a direct conclusion by the fact that 𝑑²𝜙2^𝑗𝑟±2𝜋𝑖𝑟|𝑥|𝑑𝑟²≳2^𝑗𝛽 (59) and the Van der Corput lemma.
For the case that |𝑥|≲2^𝑗𝛽, we define ∑𝐿∶=^𝑛𝑗=1𝜉_𝑗𝜕_𝑗𝑖2^𝑗𝜉𝜙^′2^𝑗𝜉+∑^𝑛_𝑗=12𝜋𝑥_𝑗𝜉_𝑗 (60) and notice that 𝐿𝑒^{𝑖[𝜙(2𝑗𝑟)±2𝜋𝑖𝑟|𝑥|]}=𝑒^{𝑖[𝜙(2𝑗𝑟)±2𝜋𝑖𝑟|𝑥|]}. (61) Then the inequality _ℝ𝑛𝜓(𝜉)𝑒^{𝑖𝜙(2𝑗|𝜉|)}𝑒^{2𝜋𝑖𝑥⋅𝜉}𝑑𝜉≲2^{𝑗𝛽−𝑛/2} (62) follows by an integration by parts.
Hence, for all 𝑗∈ℕ, 𝐹⁻¹𝜓_𝑗(𝜉)𝑒^{𝑖𝜙(|𝜉|)}_𝐿∞≲2^{𝑗(𝐵−2)(−𝑛/2)}. (63)

Lemma 9 (dual estimate for 𝑀^𝑠,𝛼_𝑝,𝑞, 𝛼∈[0,1)). Suppose 𝛽>0, 𝛽≠1. Assume that 𝜙 is a real-valued 𝐶^[𝑛/2]+3 function satisfying the assumptions of Theorem 3. Then there exists a sufficiently large constant 𝑅 such that ℱ⁻¹𝜂^𝛼_𝑘(𝜉)𝑒^{𝑖𝜙(|𝜉|)}_𝐿∞≤𝐶⟨𝑘⟩^{((𝛽−2)/(1−𝛼))(−𝑛/2)}, (64) for all 𝑘∈ℤ^𝑛 with ⟨𝑘⟩^{𝛼/(1−𝛼)}𝑘∈ℝ^𝑛⧵𝐵(0,𝑅), where the constant 𝐶 is independent of 𝑘.

Proof. For sufficiently large 𝑘, we use estimate on some Δ_𝑗 to estimate □^𝛼_𝑘. Choose a 𝑗 satisfying ⟨𝑘⟩^1/(1−𝛼)∼2^𝑗. An easy computation shows that ℱ⁻¹𝜂^𝛼_𝑘(𝜉)𝑒^{𝑖𝜙(|𝜉|)}_𝐿∞≲ℱ⁻¹𝜓_𝑗(𝜉)𝜂^𝛼_𝑘(𝜉)𝑒^{𝑖𝜙(|𝜉|)}_𝐿∞≲ℱ⁻¹𝜓_𝑗(𝜉)𝑒^{𝑖𝜙(|𝜉|)}_𝐿∞≲2^{𝑗(𝛽−2)(−𝑛/2)}∼⟨𝑘⟩^{((𝛽−2)/(1−𝛼))(−𝑛/2)}. (65)

Now, we give the asymptotic estimates of ‖𝑒^{𝑖𝜙(|𝐷|)}𝑓‖_𝑀^𝑠,𝛼_𝑝,𝑞 and ‖𝑓‖_𝑀^𝑠,𝛼_𝑝,𝑞. These results can be verified by the same methods in [14].

Lemma 10 (asymptotic estimates of ‖𝑒^{𝑖𝜙(|𝐷|)}𝑓‖_𝑀^𝑠,𝛼_𝑝,𝑞(𝛼∈[0,1))). Suppose 𝛽>0, 𝛽≠1. Assume that 𝜙 is a real-valued function which satisfies the assumptions of Theorem 3. we can find function sequences {𝑓_𝜆}, {ℎ_𝑘}, and {𝑔_𝑁} such that 𝑒^{𝑖𝜙(|𝐷|)}𝑓_{𝜆𝑀𝑠,𝛼𝑝,𝑞}∼𝜆^{𝑛(1−(1/𝑝))}𝑒𝑎𝑠𝜆⟶0,_{𝑖𝜙(|𝐷|)}ℎ_{𝑘𝑀𝑠,𝛼𝑝,𝑞}∼⟨𝑘⟩^{(𝑠+𝒮𝑝+𝛼𝑛(1−1/𝑝))/(1−𝛼)}|𝑘|𝑒𝑎𝑠⟶∞,_{𝑖𝜙(|𝐷|)}𝑔_{𝑁𝑀𝑠,𝛼𝑝,𝑞}∼𝑁^{((𝑠+𝒮𝑝+𝛼𝑛(1−1/𝑝))/(1−𝛼))+(𝑛/𝑞)}𝑎𝑠𝑁⟶∞,𝑠−𝑛𝛼𝑝+𝒮_𝑝+𝑛(1−𝛼)𝑞𝑒>−𝑛𝛼,_{𝑖𝜙(|𝐷|)}𝑔_{𝑁𝑀𝑠,𝛼𝑝,𝑞}∼(ln𝑁)^1/𝑞𝑎𝑠𝑁⟶∞,𝑠−𝑛𝛼𝑝+𝒮_𝑝+𝑛(1−𝛼)𝑞=−𝑛𝛼. (66) Moreover, we can easily verify that 𝑓_{𝜆𝑀𝑠,𝛼𝑝,𝑞}∼𝜆^{𝑛(1−(1/𝑝))}ℎ𝑎𝑠𝜆⟶0,_{𝑘𝑀𝑠,𝛼𝑝,𝑞}∼⟨𝑘⟩^{(𝑠+𝛼𝑛(1−1/𝑝))/(1−𝛼)}|𝑘|𝑔𝑎𝑠⟶∞,_{𝑁𝑀𝑠,𝛼𝑝,𝑞}∼𝑁^{((𝑠+𝛼𝑛(1−1/𝑝)/(1−𝛼))+(𝑛/𝑞)}𝑎𝑠𝑁⟶∞,𝑠−𝑛𝛼𝑝+𝑛(1−𝛼)𝑞𝑔>−𝑛𝛼,_{𝑁𝑀𝑠,𝛼𝑝,𝑞}∼(ln𝑁)^1/𝑞𝑎𝑠𝑁⟶∞,𝑠−𝑛𝛼𝑝+𝑛(1−𝛼)𝑞=−𝑛𝛼. (67)

Proof. In the case that 0<𝛽≤Λ_𝛼 is trivial, we suppose that 𝛽≥Λ_𝛼 in this proof. We only show the proof of (66).
Denote by ℎ(𝜉) a 𝐶^∞_𝑐(ℝ^𝑛) function with sufficiently small support near zero. Let 𝑓_𝜆=ℎ𝜉𝜆,𝜆≥1. (68) Using Theorem 1, we deduce that 𝑓_𝜆𝑀𝑝^{|𝑠−|𝒮}_{𝑝,𝑞,𝛼}≲𝑒^{𝑖𝜙(|𝐷|)}𝑓_{𝜆𝑀𝑠,𝛼𝑝,𝑞}≲𝑓_𝜆𝑀𝑝^|𝑠+|𝒮_{𝑝,𝑞,𝛼}. (69) Notice that 𝑓_𝜆𝑀𝑝^{|𝑠−|𝒮}_{𝑝,𝑞,𝛼}=𝑓_𝜆𝑀𝑝^|𝑠+|𝒮_{𝑝,𝑞,𝛼}=□^𝛼₀𝑓_𝜆𝐿𝑝=𝜆^𝑛‖ℎ(𝜆𝑥)‖_𝐿^𝑝=𝜆^{𝑛(1−(1/𝑝))}‖ℎ‖_𝐿^𝑝; (70) so we have 𝑒^{𝑖𝜙(|𝐷|)}𝑓_{𝜆𝑀𝑠,𝛼𝑝,𝑞}∼𝜆^{𝑛(1−(1/𝑝))}as𝜆⟶0. (71) For the second equation, we denote ℎ_𝑘(𝜉)=ℎ𝜉−⟨𝑘⟩^{𝛼/(1−𝛼)}𝑘⟨𝑘⟩^{𝛼/(1−𝛼)} (72) and let 𝑓_𝑘(𝜉)=ℎ_𝑘(𝜉), where 𝑘∈ℤ^𝑛 with ⟨𝑘⟩^{𝛼/(1−𝛼)}𝑘∈Γ⧵𝐵(0,𝑅).
If 1≤𝑝≤2, we use Theorem 1 and Lemma 9 to deduce that 𝑓_{𝑘𝑀𝑝′}^{𝑠+(1/𝑝−1/2)(𝛽−2)𝑛,𝛼}_,𝑞≲𝑒^{𝑖𝜙(|𝐷|)}𝑓_{𝑘𝑀𝑠,𝛼𝑝,𝑞}≲𝑓_𝑘𝑀𝑝^{𝑠+𝒮,𝛼}_𝑝,𝑞. (73) If 2≤𝑝≤∞, we deduce that 𝑓_𝑘𝑀𝑝^{𝑠+𝒮,𝛼}_𝑝,𝑞≲𝑒^{𝑖𝜙(|𝐷|)}𝑓_{𝑘𝑀𝑠,𝛼𝑝,𝑞}≲𝑓_{𝑘𝑀𝑝′}^{𝑠+(1/𝑝−1/2)(𝛽−2)𝑛,𝛼}_,𝑞. (74) By a direct calculation, we obtain 𝑓_𝑘𝑀𝑝^{𝑠+𝒮,𝛼}_𝑝,𝑞=⟨𝑘⟩^{(𝑠+𝒮𝑝)/(1−𝛼)}ℎ_𝑘𝐿𝑝=⟨𝑘⟩^{(𝑠+𝒮𝑝+𝛼𝑛(1−1/𝑝))/(1−𝛼)}‖ℎ‖_𝐿^𝑝,𝑓_{𝑘𝑀𝑝′}^{𝑠+(1/𝑝−1/2)(𝛽−2)𝑛,𝛼}_,𝑞=⟨𝑘⟩^{(𝑠+(1/𝑝−1/2)(𝛽−2)𝑛)/(1−𝛼)}ℎ_{𝑘𝐿𝑝′}=⟨𝑘⟩^{(𝑠+𝒮𝑝+𝛼𝑛(1−1/𝑝))/(1−𝛼)}‖ℎ‖_𝐿𝑝′. (75) Then the asymptotic formula follows.
For the third and fourth asymptotic formulas, let 𝑓_𝑁(𝜉)=_{𝑘∈𝐴𝑁}ℎ_𝑘(𝜉), (76) where 𝐴_𝑁={𝑘∈ℤ^𝑛∶|𝑘|≤𝑁,⟨𝑘⟩^{𝛼/(1−𝛼)}𝑘∈ℝ^𝑛⧵𝐵(0,𝑅)}. By a direct calculation, we obtain 𝑓_𝑁𝑀𝑝^{𝑠+𝒮,𝛼}_𝑝,𝑞≃_{𝑘∈𝐴𝑁}⟨𝑘⟩^{((𝑠+𝒮𝑝+𝛼𝑛(1−1/𝑝))/(1−𝛼))𝑞1/𝑞}≃𝑓_{𝑁𝑀𝑝′}^{𝑠+(1/𝑝−1/2)(𝛽−2)𝑛,𝛼}_,𝑞. (77) As above, Theorem 1 and Lemma 9 yield 𝑒^{𝑖𝜙(|𝐷|)}𝑓_{𝑁𝑀𝑠,𝛼𝑝,𝑞}∼_{𝑘∈𝐴𝑁}⟨𝑘⟩^{((𝑠+𝒮𝑝+𝛼𝑛(1−1/𝑝))/(1−𝛼))𝑞1/𝑞}. (78) Finally, the claim follows by _{𝑘∈𝐴𝑁}⟨𝑘⟩^{((𝑠+𝒮𝑝+𝛼𝑛(1−1/𝑝))/(1−𝛼))𝑞1/𝑞}∼𝑁^{((𝑠+𝒮𝑝+𝛼𝑛(1−1/𝑝))/(1−𝛼))+(𝑛/𝑞)}, (79) for 𝑠−(𝑛𝛼/𝑝)+𝒮_𝑝+(𝑛(1−𝛼)/𝑞)>−𝑛𝛼, and _{𝑘∈𝐴𝑁}⟨𝑘⟩^{((𝑠+𝒮𝑝+𝛼𝑛(1−1/𝑝))/(1−𝛼))𝑞1/𝑞}∼(ln𝑁)^1/𝑞 (80) for 𝑠−(𝑛𝛼/𝑝)+𝒮_𝑝+(𝑛(1−𝛼)/𝑞)=−𝑛𝛼.

For the case 𝛼=1, 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 ‖𝑒^{𝑖𝜙(|𝐷|)}𝑓‖_{𝐵𝑠𝑝,𝑞}). Suppose 𝛽>0, 𝛽≠1. Assume that 𝜙 is a real-valued function which satisfies the assumptions of Theorem 3. We can find function sequences {𝑓_𝜆}, {ℎ_𝑘}, and {𝑔_𝑁} such that 𝑒^{𝑖𝜙(|𝐷|)}𝑓_{𝜆𝐵𝑠𝑝,𝑞}∼𝜆^{𝑛(1−(1/𝑝))}𝑒𝑎𝑠𝜆⟶0,_{𝑖𝜙(|𝐷|)}ℎ_{𝑗𝐵𝑠𝑝,𝑞}∼2^{𝑗[𝑠+𝒮𝑝+𝑛(1−1/𝑝)]}𝑗𝑒𝑎𝑠⟶∞,_{𝑖𝜙(|𝐷|)}𝑔_{𝑁𝐵𝑠𝑝,𝑞}∼2^{𝑁[𝑠+𝒮𝑝+𝑛(1−1/𝑝)]}𝑛𝑎𝑠𝑁⟶∞,𝑠−𝑝+𝒮_𝑝𝑒>−𝑛,_{𝑖𝜙(|𝐷|)}𝑔_{𝑁𝐵𝑠𝑝,𝑞}∼𝑁^1/𝑞𝑛𝑎𝑠𝑁⟶∞,𝑠−𝑝+𝒮_𝑝=−𝑛. (81) Moreover, we can easily verify that 𝑓_{𝜆𝐵𝑠𝑝,𝑞}∼𝜆^{𝑛(1−(1/𝑝))}ℎ𝑎𝑠𝜆⟶0,_{𝑗𝐵𝑠𝑝,𝑞}∼2^{𝑗[𝑠+𝑛(1−1/𝑝)]}𝑗𝑔𝑎𝑠⟶∞,_{𝑁𝐵𝑠𝑝,𝑞}∼2^{𝑁[𝑠+𝑛(1−1/𝑝)]}𝑛𝑎𝑠𝑁⟶∞,𝑠−𝑝𝑔>−𝑛,_{𝑁𝐵𝑠𝑝,𝑞}∼𝑁^1/𝑞𝑛𝑎𝑠𝑁⟶∞,𝑠−𝑝=−𝑛. (82)

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

Proof of Theorem 3. As in [14], if 𝑒^{𝑖𝜙(|𝐷|)}𝑓_{𝑀𝑠2𝑝22,𝛼,𝑞}≲𝑓_𝑀^𝑠_1𝑝11^,𝛼_,𝑞 (83) holds for all 𝑓, 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 𝑒^{𝑖𝜙(|𝐷|)} from 𝑀^𝑠1,𝛼1_𝑝1,𝑞1 to 𝑀^𝑠2,𝛼2_𝑝2,𝑞2.

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).