Estimates for Unimodular Multipliers on Modulation Hardy Spaces

Abstract

It is known that the unimodular Fourier multipliers 𝑒^{𝑖𝑡|Δ|𝛼/2},𝛼>0, are bounded on all modulation spaces 𝑀^𝑠_𝑝,𝑞 for 1≤𝑝,𝑞≤∞. We extend such boundedness to the case of all 0<𝑝,𝑞≤∞ and obtain its asymptotic estimate as t goes to infinity. As applications, we give the grow-up rate of the solution for the Cauchy problems for the free Schrödinger equation with the initial data in a modulation space, as well as some mixed norm estimates. We also study the 𝑀^𝑠_𝑝1,𝑞→𝑀^𝑠_𝑝2,𝑞 boundedness for the operator 𝑒^{𝑖𝑡|Δ|𝛼/2}, for the case 0<𝛼≤2 and 𝛼≠1. Finally, we investigate the boundedness of the operator 𝑒^{𝑖𝑡|Δ|𝛼/2} for 𝛼>0 and obtain the local well-posedness for the Cauchy problem of some nonlinear partial differential equations with fundamental semigroup 𝑒^{𝑖𝑡|Δ|𝛼/2}.

1. Introduction

A Fourier multiplier is a linear operator 𝐻_𝜇 whose action on a test function 𝑓 on ℝ^𝑛 is formally defined by 𝐻_𝜇𝑓(𝑥)=1(2𝜋)^𝑛_ℝ^𝑛𝜇(𝜉)𝑓(𝜉)𝑒^{𝑖2𝜋𝜉⋅𝑥}𝑑𝜉. (1) The function 𝜇 is called the symbol or multiplier of 𝐻_𝜇.

In this paper, we will study the unimodular Fourier multipliers with symbol 𝑒^{𝑖𝑡|𝜉|𝛼} for 𝑡∈ℝ₊. They arise when one solves the Cauchy problem for dispersive equations. For example, for the solution 𝑢(𝑡,𝑥) of the Cauchy problem 𝑖𝜕_𝑡𝑢+|Δ|^𝛼/2𝑢=0,(𝑡,𝑥)∈ℝ₊×ℝ^𝑛,𝑢(0,𝑥)=𝑢₀(𝑥), (2) we have the formula 𝑢(𝑡,𝑥)=(𝑒^{𝑖𝑡|Δ|𝛼/2}𝑢₀)(𝑥). Here Δ=Δ_𝑥 is the Laplacian and 𝑒^{𝑖𝑡|Δ|𝛼/2} is the multiplier operator with symbol 𝑒^{𝑖𝑡|𝜉|𝛼} (see [1] for its definition). The cases 𝛼=1,2,3 are of particular interest because they correspond to the (half-) wave equation, the Schrödinger equation, and (essentially) the Airy equation, respectively.

Unimodular Fourier multipliers generally do not preserve any Lebesgue space 𝐿^𝑝, except for 𝑝=2. The 𝐿^𝑝-spaces are not the appropriate function spaces for the study of these operators and the so-called modulation spaces are good alternative classes for the study of unimodular Fourier multipliers. The modulation spaces 𝑀^𝑠_𝑝,𝑞(ℝ^𝑛) were first introduced by Feichtinger [2–4] to measure smoothness of a function or distribution in a way different from 𝐿^𝑝 spaces, and they are now recognized as a useful tool for studying pseudodifferential operators [5–7]. We will recall the precise definition of modulation spaces in Section 2 below.

Recently, the boundedness of unimodular Fourier multipliers 𝑒^{𝑖𝑡|Δ|𝛼/2} on the modulation spaces has been investigated in [1, 8–15]. Particularly, one has the following results.

Theorem A (see [11]). Let 𝑠∈ℝ,1≤𝑝, 𝑞≤∞, 𝛼>1/2and 𝛼≠1. One has, for 𝑡≥1, 𝑒^{𝑖𝑡|Δ|𝛼/2}𝑓_{𝑀𝑠𝑝,𝑞}⪯𝑡^{𝑛|1/2−1/𝑝|}𝑓_𝑀^𝑠+𝛾_𝑝,𝑞+𝑓_{𝑀𝑠𝑝,𝑞}, (3) where (1𝛾=𝑛𝛼−2)2−1𝑝. (4)

Here (and throughout this paper), we use the notation 𝐴⪯𝐵 to mean that there is a positive constant 𝐶 independent of all essential variables such that 𝐴≤𝐶𝐵.

Theorem B (see [15]). Let 0<𝑝<1, 0<𝑞≤∞, 𝛼>𝑛(1/𝑝−1) and 𝑠∈ℝ. Then 𝑒^{𝑖𝑡|Δ|𝛼/2} is bounded from 𝑀^𝑠_𝑝,𝑞 to 𝑀_𝑝,𝑞 if and only if {}𝑛1𝑠≥max0,𝛼−2𝑝−12. (5)

In this paper, we use a different method from [15] to prove the following theorem, which, in particular, uses the modulation Hardy spaces 𝜇^𝑠_𝑝,𝑞 that will be later defined in Section 2.

Theorem 1. Let 𝑠∈ℝ, 0<𝑞≤∞, 𝑡≥1. For a positive 𝛼≠1, denote 𝛾=𝑛(𝛼−2)|1/𝑝−1/2|. Let 𝛼₀=2/(2+𝑛) if n is even and 𝛼₀=3/(3+𝑛) if 𝑛 is odd.(i)Assume 𝛼>𝛼₀. If 𝑝>𝑛/(𝑛+𝛼) and 𝑛≥2, one has 𝑒^{𝑖𝑡|Δ|𝛼/2}𝑓_{𝑀𝑠𝑝,𝑞(ℝ𝑛)}⪯𝑡^{𝑛|1/𝑝−1/2|}𝑓_𝑀^𝑠+𝛾_{𝑝,𝑞(ℝ𝑛)}+𝑓_{𝑀𝑠𝑝,𝑞(ℝ𝑛)}. (6)Particularly, the above inequality holds for all 𝑝>0 if 𝛼is a positive even number. (ii) For any 0<𝑝≤∞, one has 𝑒^{𝑖𝑡|Δ|𝛼/2}𝑓_{𝑀𝑠𝑝,𝑞(ℝ𝑛)}⪯𝑡^{𝑛|1/𝑝−1/2|}𝑓_𝑀^𝑠+𝛾_{𝑝,𝑞(ℝ𝑛)}+𝑓_{𝜇𝑠−𝛿𝑝,𝑞(ℝ𝑛)}, (7)for any 𝛿≥0.
Here (1𝛾=𝑛𝛼−2)𝑝−12. (8)(iii) Assume 𝑛=1. If 𝛼>1/2, then 𝑒^{𝑖𝑡|Δ|𝛼/2}𝑓_{𝑀𝑠𝑝,𝑞(ℝ)}⪯𝑡^{|1/𝑝−1/2|}𝑓_𝑀^𝑠+𝛾_{𝑝,𝑞(ℝ)}+𝑓_{𝑀𝑠𝑝,𝑞(ℝ)} (9)for all 𝑝>2/3.

We want to make a few remarks on Theorem 1. First, (iii) in Theorem 1 says that when 𝑛=1, compared to the case 𝑛≥2 in (i), one obtains a larger range of 𝛼 and a smaller range of 𝑝. We do not know if there is a unified formula regarding 𝛼 and 𝑝 for all dimension 𝑛≥1. Second, in the proof we will see that, in the low frequency parts of the definition of 𝑀^𝑠_𝑝,𝑞, the fractional Schrödinger semigroup has a growth 𝑡^{𝑛|1/𝑝−1/2|} when 𝑡 is growing, but it gains an arbitrary regularity. In the high frequency part, the semigroup can be controlled by max1,𝑡|𝑘|^{𝛼−2𝑛|1/𝑝−1/2|}, (10) at each piece of its decomposition with frequency 𝑘. This phenomenon was also more precisely observed in [1, 15] (see also [11]). Thirdly, the case 𝛼=1 was studied in [8, 16].

Since the 𝐿^𝑝 norm is dominated by the 𝐻^𝑝 norm and the Riesz transforms are bounded on 𝐻^𝑝, by the Riesz transform characterization of the 𝐻^𝑝 (see Section 2), we easily obtain the following corollary.

Corollary 2. Let 𝑠∈ℝ, 0<𝑝,𝑞≤∞and 𝛼>0,𝛼≠1. One has for 𝑡≥1𝑒^{𝑖𝑡|Δ|𝛼/2}𝑓_{𝜇𝑠𝑝,𝑞}⪯𝑡^{𝑛|1/𝑝−1/2|}𝑓_𝜇^𝑠+𝛾_𝑝,𝑞, (11) where (1𝛾=𝑛𝛼−2)𝑝−12. (12)

Our next result shows that the asymptotic factor𝑡^{𝑛|1/𝑝−1/2|} in Theorem 1 is the best for all 𝑝>0, at least for 𝛼=2.

Theorem 3. Let 𝛼=2. The asymptotic factor 𝑡^{𝑛|1/𝑝−1/2|} in Theorem 1 is the best. Precisely, for 𝑡>1, if 𝑒^𝑖𝑡|Δ|𝑓_{𝑀𝑠𝑝,𝑞}⪯𝑡^𝛿𝑓_{𝑀𝑠𝑝,𝑞}, (13) then 1𝛿≥𝑛𝑝−12. (14)

In the next theorem, we state some mixed norm estimates.

Theorem 4. Let 0<𝛼≤2and 𝛼≠1. For 0<𝑝₁≤1≤𝑝₂≤∞, suppose 𝑛𝑟(1/2−1/𝑝₂)>1.(i)If 𝑟≥𝑞, then 𝑒^{𝑖𝑡|Δ|𝛼/2}𝑓_{𝑀𝑠𝑝2,𝑞𝐿𝑟(0,∞)}⪯𝑓_𝑀𝑝1^{𝑠+(2−𝛼)/𝑟}_,𝑞. (15)(ii)If 𝑟≤𝑞, then 𝑒^{𝑖𝑡|Δ|𝛼/2}𝑓_{𝑀𝑠𝑝2,𝑞𝐿𝑟(0,∞)}⪯𝑓_𝑀𝑝1^{𝑠+(2−𝛼)/𝑟}_,𝑟. (16)

We consider the following linear Cauchy problem with negative power: 𝑖𝜕_𝑡𝑢+Δ_𝑥^−𝛼/2𝑢=0,(𝑡,𝑥)∈ℝ₊×ℝ^𝑛𝑢(),𝛼>0,0,𝑥=𝑢₀(𝑥). (17)

We give the grow-up rate of the solution to the above Cauchy problem in the modulation spaces.

Theorem 5. Assume 𝛼>0 and 𝛾=𝑛(𝛼+2)|1/2−1/𝑝|.(i)Let 1≤𝑝<2. One has that for any 0<𝑟<𝑛𝑝/(𝑛+𝛼([𝑛/2]+1)(2−𝑝))‖𝑢(𝑡)‖_𝑀^𝑠_{𝑝,𝑞(ℝ}^𝑛₎⪯(1+𝑡)^{([𝑛/2]+1)|2/𝑝−1|}𝑢_0𝜇^𝑠−𝛾_{𝑟,𝑞(ℝ𝑛)}+𝑢_{0𝑀𝑠𝑝,𝑞(ℝ𝑛)}. (18)(ii)For any 𝑟≤2≤𝑝, one has ‖𝑢(𝑡)‖_𝑀^𝑠_{𝑝,𝑞(ℝ}^𝑛₎⪯𝑢_{0𝜇𝑠𝑟,𝑞(ℝ𝑛)}+(1+𝑡)^{𝑛|1/𝑝−1/2|}𝑢_0𝑀^𝑠−𝛾_{𝑝,𝑞(ℝ𝑛)}. (19)

Now, we study the following Cauchy problem of the nonlinear dispersive equations (NDE): 𝑖𝜕𝑢−|Δ|𝜕𝑡^−𝛼/2(𝑢)𝑢()𝑢+𝐹=0,0,𝑥=𝑢₀(𝑥), (20) where 𝐹(𝑢)=|𝑢|^2𝑘𝑢 for some positive integer 𝑘. For 𝑇>0, the space 𝑌=𝐶([0,𝑇],𝑀^𝑠_𝑝,1) is defined by 𝐶[0,𝑇],𝑀^𝑠_𝑝,1=𝑢(𝑡,𝑥)∶‖𝑢‖_{𝐶([0,𝑇],𝑀}^𝑠_𝑝,1)=sup_{0≤𝑡≤𝑇}‖𝑢(𝑡,⋅)‖_𝑀^𝑠_𝑝,1<∞. (21)

We obtain the quantitative forms about the solution to the above Cauchy problem of the nonlinear dispersive equations.

Theorem 6. Let 2>𝑝>1, 𝛾=𝑛(𝛼+2)|1/2−1/𝑝|, 𝛼>0 and assume 𝑛𝑝𝑛+𝛼([𝑛/2]+1)2−𝑝>1. (22)
Assume 𝑢₀∈𝑀^𝑠−𝛾_𝑟,1∩𝑀^𝑠_𝑝,1 for any 1<𝑟<𝑛𝑝]).𝑛+𝛼([𝑛/2+12−𝑝 (23)
There exists 𝑇>0 such that the above Cauchy system (NDE) has a unique solution 𝑢∈𝐶([0,𝑇],𝑀^𝑠_𝑝,1), where 𝑇 depends on the norm ‖𝑢₀‖_𝑀^𝑠−𝛾_𝑟,1 and ‖𝑢₀‖_{𝑀𝑠𝑝,1}.

According to the inclusions of modulation space (see Proposition 2.5 in [13]), we know the space of initial data 𝑀^𝑠−𝛾_𝑟,1∩𝑀^𝑠_𝑝,1=𝑀^𝑠−𝛾_𝑟,1 if 𝑟≤𝑝.

Theorem 7. Let 𝑝≥2,𝛾=𝑛(𝛼+2)|1/2−1/𝑝|. Assume 𝑝2𝑘+1≥1 (24) and 𝑢₀∈𝑀^𝑠_𝑟,1 for any 𝑝2𝑘+1≤𝑟<2. (25)
There exists 𝑇>0 such that the above Cauchy system (NDE) has a unique solution 𝑢∈𝐶([0,𝑇],𝑀^𝑠_𝑝,1), where 𝑇 depends on the norm ‖𝑢₀‖_{𝑀𝑠𝑟,1}.

The rest of the paper is organized as follows. In Section 2, we recall or establish some necessary lemmas and known results. Sections 3 and 4 are devoted to the proofs of Theorems 1 and 3, respectively. Finally, in Section 5, we give some applications including the 𝑀^𝑠_𝑝1,𝑞→𝑀^𝑠_𝑝2,𝑞 boundedness for the operator 𝑒^{𝑖𝑡|Δ|𝛼/2} in the case 𝛼≤2 and 𝛼≠1, including negative 𝛼.

2. Preliminaries

2.1. The Definitions

The modulation space is originally defined by Feichtinper in 1983 on the locally compact Ablian groups 𝐺. When 𝐺=ℝ^𝑛, the modulation space can be equivalently defined by using the unit-cube decomposition to the frequency space (see Appendix A in [13], also [14, 17]). The following definition is based on the unit-cube decomposition introduced in [13].

Let 𝜎 be a fixed nonnegative-valued function in 𝒮(ℝ^𝑛) with support in the cube [−4/5,4/5]^𝑛 and satisfy 𝜎(𝜉)=1 for any 𝜉 in the cube [−2/5,2/5]^𝑛. By a standard constructive method, we may assume that for all 𝜉∈ℝ^𝑛, _{𝑘∈ℤ𝑛}𝜎_𝑘(𝜉)=1, (26) where 𝜎_𝑘 is the 𝑘-shift of 𝜎 that is defined by 𝜎_𝑘(𝜉)=𝜎(𝜉−𝑘),𝜎₀(𝜉)=𝜎(𝜉). (27)

For each 𝑘∈ℤ^𝑛, we use 𝜎_𝑘(𝜉) as its symbol of a smooth projection 𝜋_𝑘 on the frequency space. Precisely, for any 𝑓∈𝒮^′(ℝ^𝑛), we have 𝜋_𝑘𝑓=𝜎_𝑘𝑓. (28)

Let 𝑋 be a Banach space of measurable functions on ℝ^𝑛 with quasi-norm ‖⋅‖_𝑋. We define the modulation space 𝑀^𝑠𝑋,ℓ^𝑞ℝ^𝑛=𝑓ismeasurable∶𝑓_{𝑀𝑠(𝑋,ℓ𝑞)}<∞, (29) where 𝑓_{𝑀𝑠(𝑋,ℓ𝑞)}=⟨𝑘⟩^𝑠𝜋_𝑘𝑓_𝑋ℓ𝑞,⟨𝑘⟩=1+|𝑘|. (30) By definition, we have the inclusion 𝑀^𝑠𝑋,ℓ^𝑞⊂𝑀^𝑟𝑋,ℓ^𝑞if𝑟≤𝑠. (31) It is known that the definition of the modulation space 𝑀^𝑠(𝑋,ℓ^𝑞) is independent of the choice of functions 𝜎. In this paper, we are particularly interested in the cases 𝑋=𝐿^𝑝 and 𝑋=𝐻^𝑝, where 𝐿^𝑝 is the Lebesgue space and 𝐻^𝑝 is the real Hardy space. For all 0<𝑝,𝑞≤∞, we call 𝑀^𝑠(𝐿^𝑝,ℓ^𝑞) the modulation spaces and 𝑀^𝑠(𝐻^𝑝,ℓ^𝑞) the modulation Hardy space. As a usual notation 𝑀^𝑠𝐿^𝑝,ℓ^𝑞=𝑀^𝑠_𝑝,𝑞=𝑀^𝑠_𝑝,𝑞ℝ^𝑛, (32) we similarly define 𝑀^𝑠𝐻^𝑝,ℓ^𝑞=𝜇^𝑠_𝑝,𝑞=𝜇^𝑠_𝑝,𝑞ℝ^𝑛. (33) By the definition and known properties of 𝐻^𝑝, we have that for all 1<𝑝<∞, 𝜇^𝑠_𝑝,𝑞ℝ^𝑛=𝑀^𝑠_𝑝,𝑞ℝ^𝑛, (34) and for all 0<𝑝≤1, 𝜇^𝑠_𝑝,𝑞ℝ^𝑛⊂𝑀^𝑠_𝑝,𝑞ℝ^𝑛. (35) For simplicity in notation, we denote 𝜇_𝑝,𝑞ℝ^𝑛=𝜇⁰_𝑝,𝑞ℝ^𝑛,𝑀_𝑝,𝑞ℝ^𝑛=𝑀⁰_𝑝,𝑞ℝ^𝑛. (36) The following imbedding relation can be found in Proposition 5.1.5 of [18]. Let 𝑠₁,𝑠₂∈ℝ, 0<𝑝₁,𝑝₂,𝑞₁,𝑞₂≤∞. If 𝑠₂≤𝑠₁,𝑝₁≤𝑝₂,𝑞₁≤𝑞₂ (37) then 𝑀^𝑠1_𝑝1,𝑞1⊂𝑀^𝑠2_𝑝2,𝑞2. (38)

2.2. 𝐻^𝑝 Spaces

It is well known that the Hardy space 𝐻^𝑝(ℝ^𝑛) coincides with the Lebesgue space 𝐿^𝑝(ℝ^𝑛) when 1<𝑝<∞. For 0<𝑝≤1, the space 𝐻^𝑝(ℝ^𝑛) has many characterizations. We will use its Riesz transform characterization in this paper. For an integer 𝐿≥0and multi-index 𝐽=(𝑗₁,…,𝑗_𝑁)∈{1,2,…,𝑛}^𝐿, let 𝑅_𝐽 denote the generalized Riesz transform 𝑅_𝐽𝑓=𝑅_𝑗1𝑅_𝑗2𝑅_𝑗𝐿𝑓, (39) where each 𝑅_𝑗(𝑓) is the 𝑗𝑡ℎ Riesz transform of 𝑓 if 𝑗≠0and 𝑅₀(𝑓)=𝑓. It is known that for 𝑝>(𝑛−1)/(𝑛−1+𝐿) and all 𝑓∈𝐻^𝑝∩𝐿², 𝑓_𝐻𝑝≃_𝐽𝑅_𝐽𝑓_𝐿𝑝, (40) where ∑_𝐽 is a sum of finite terms.

The operator 𝜋_𝑘𝑒^{𝑖𝑡|Δ|𝛼/2} is a convolution. We have 𝜋_𝑘𝑒^{𝑖𝑡|Δ|𝛼/2}𝑓_𝐻𝑝≃_𝐽𝜋_𝑘𝑒^{𝑖𝑡|Δ|𝛼/2}𝑅_𝐽𝑓_𝐿𝑝. (41) Also it is well known that 𝑅_𝐽 is bounded on 𝐻^𝑝 spaces for any 0<𝑝<∞.

2.3. Some Lemmas and Known Results

Lemma 8. Let 0<𝑝<∞and 𝑡≥1. Suppose that there is an integer 𝑁>0, such that for all test functions 𝑓𝜋_𝑘𝑒^{𝑖𝑡|Δ|𝛼/2}𝑓_𝐿𝑝⪯𝑡^𝑏1𝑓_𝐻𝑝 (42) for |𝑘|<𝑁 and 𝜋_𝑘𝑒^{𝑖𝑡|Δ|𝛼/2}𝑓_𝐿𝑝⪯𝑡^𝑏2|𝑘|^𝑑𝑓_𝐻𝑝 (43) for |𝑘|≥𝑁. Here 𝑏₁≥𝑏₂≥0 and 𝑑 is a real number. Then for 𝑓∈𝐻^𝑝∩𝐿², one has 𝑒^{𝑖𝑡|Δ|𝛼/2}𝑓_{𝑀𝑠𝑝,𝑞}⪯𝑡^𝑏1𝑓_{𝑀𝑠−𝜈𝑝,𝑞}+𝑡^𝑏2𝑓_{𝑀𝑠+𝑑𝑝,𝑞}, (44) where 𝜈 is an arbitrary positive number.

Proof. The case 𝑝≥1 is proved in [11]. It suffices to show the lemma for 0<𝑝≤1. By the Riesz transform characterization of 𝐻^𝑝, for |𝑘|≥𝑁, we have 𝜋_𝑘𝑒^{𝑖𝑡|Δ|𝛼/2}𝑓_𝐻𝑝≃_𝐽𝜋_𝑘𝑒^{𝑖𝑡|Δ|𝛼/2}𝑅_𝐽𝑓_𝐿𝑝⪯_𝐽𝑡^𝑏2|𝑘|^𝑑𝑅_𝐽𝑓_𝐿𝑝≃𝑡^𝑏2|𝑘|^𝑑𝑓_𝐻𝑝. (45) By checking the Fourier transform, we have the identity 𝜋_𝑘=_{|𝑗|∞≤1}𝜋_𝑘+𝑗𝜋_𝑘, (46) where 𝑗=𝑗₁,…,𝑗_𝑛,𝑗_∞=max𝑗₁,…,𝑗_𝑛. (47) So for |𝑘|≥𝑁−1, one has 𝜋_𝑘𝑒^{𝑖𝑡|Δ|𝛼/2}𝑓_𝐻𝑝=_{|𝑗|∞≤1}𝜋_𝑘+𝑗𝜋_𝑘𝑆_𝛼(𝑡)𝑓_𝐻𝑝⪯_{|𝑗|∞≤1}𝜋_𝑘+𝑗𝑒^{𝑖𝑡|Δ|𝛼/2}𝜋_𝑘𝑓_𝐻𝑝⪯𝑡^𝑏2|𝑘|^𝑑𝜋_𝑘𝑓_𝐻𝑝. (48) A similar argument shows that for |𝑘|<𝑁−1, 𝜋_𝑘𝑒^{𝑖𝑡|Δ|𝛼/2}𝑓_𝐻𝑝⪯𝑡^𝑏1𝜋_𝑘𝑓_𝐻𝑝 (49) for any 𝑓∈𝐻^𝑝∩𝐿². The rest of the lemma easily follows from the definition of the modulation spaces.

Lemma 9 (see [18, 19]). Let 𝐸⊂ℝ^𝑛 denote an open set and 𝜓∈𝐶^∞₀(𝐸). If 𝜙∈𝐶^∞(𝐸) and the rank of the matrix 𝐷_𝑥𝑗𝐷_𝑥𝑖𝜙(𝑥)^𝑛_𝑖,𝑗=1 (50) is at least 𝑚>0 for all 𝑥∈supp(𝜓), then _ℝ𝑛𝑒^{𝑖𝜆𝜙(𝑥)}𝜓(𝑥)𝑑𝑥⪯|𝜆|^−𝑚/2𝜓_𝐶2𝑛. (51)

Lemma 10. Let 2≥𝛼>0and 𝛼≠1. Suppose that 𝜎 is a 𝐶^∞ function with support in [−1,1]^𝑛. Then ℱ⁻¹𝜎(𝜉)𝑒^{𝑖𝑡|𝜉|𝛼}_{𝐿∞(ℝ𝑛)}⪯min1,𝑡^−𝑛/2. (52)

Proof. The case 𝛼=2 is known [20]. It then suffices to show that for 0<𝛼<2, ℱ⁻¹𝜎(𝜉)𝑒^{𝑖𝑡|𝜉|𝛼}_{𝐿∞(ℝ𝑛)}⪯𝑡^−𝑛/2 (53) for large 𝑡. Let Γ be a standard bump radial function supported in the set 1𝜉∶2<𝜉≤2 (54) and satisfying, for all 𝜉≠0, _𝑗Γ2^𝑗𝜉=1. (55) Noting the support condition of 𝜎, we write ℱ⁻¹𝜎(𝜉)𝑒^{𝑖𝑡|𝜉|𝛼}(𝑥≃_ℝ𝑛)𝜎(𝜉)𝑒^{𝑖𝑡|𝜉|𝛼}𝑒^{𝑖(𝑥⋅𝜉)}≃𝑑𝜉_{∞𝑗=−∞ℝ𝑛}Γ2^𝑗𝜉𝜎(𝜉)𝑒^{𝑖𝑡|𝜉|𝛼}𝑒^{𝑖(𝑥⋅𝜉)}⪯𝑑𝜉_{∞𝑗∈𝐴1∪𝐴2∪𝐴3}2^−𝑗𝑛_ℝ𝑛Γ𝜉𝜎2^−𝑗𝜉𝑒^{𝑖𝑡2−𝑗𝛼|𝜉|𝛼}𝑒^{𝑖2−𝑗(𝑥⋅𝜉)}𝑑𝜉, (56) where the sets 𝐴_𝑘,  𝑘=1,2,3 are defined by 𝐴₁=𝑗≥0∶2^{−𝑗(𝛼−1)−5}≥|𝑥|,𝐴𝛼𝑡₂=𝑗≥0∶𝑛2^{−𝑗(𝛼−1)+5}≤|𝑥|,𝐴𝛼𝑡₃=2𝑗≥0∶⁵|𝑥|𝛼𝑡>2^{𝑗(1−𝛼)}>2⁻⁵|𝑥|.𝛼𝑡𝑛 (57) For 𝑗∈𝐴₁, we use polar coordinates to write 2^−𝑗𝑛_ℝ𝑛Γ𝜉𝜎2^−𝑗𝜉𝑒^{𝑖𝑡2−𝑗𝛼|𝜉|𝛼}𝑒^{𝑖2−𝑗(𝑥⋅𝜉)}𝑑𝜉=2^−𝑗𝑛_𝑆𝑛−1^∞₀𝑟^𝑛−1Γ(𝑟)𝜎2^−𝑗𝑟𝜉𝜉×𝑒^{𝑖𝑡2−𝑗𝛼𝑟𝛼}𝑒^{𝑖2−𝑗𝑟(𝑥⋅(𝜉/|𝜉|))}𝑑𝑟𝑑𝜉^′, (58) where 𝑑𝜉^′ is the induced Lebesgue measure on the unit sphere 𝑆^𝑛−1. When 𝑛 is even, taking integration by parts for 𝑛/2 times on the inside integral, we obtain _𝑗∈𝐴12^−𝑗𝑛_ℝ𝑛Γ𝜉𝜎2^−𝑗𝜉𝑒^{𝑖𝑡2−𝑗𝛼|𝜉|𝛼}𝑒^{𝑖2−𝑗(𝑥⋅𝜉)}𝑑𝜉⪯𝑡^−𝑛/2_𝑗∈𝐴12^{−𝑗𝑛(1−𝛼/2)}⪯𝑡^−𝑛/2. (59) When 𝑛 is odd, we use integration by parts for 𝑛/2+1/2 times on the inside integral, _ℝ𝑛Γ𝜉𝜎2^−𝑗𝜉𝑒^{𝑖𝑡2−𝑗𝛼|𝜉|𝛼}𝑒^{𝑖2−𝑗(𝑥⋅𝜉)}⪯𝑑𝜉_ℝ𝑛Γ𝜉𝜎2^−𝑗𝜉𝑒^{𝑖𝑡2−𝑗𝛼|𝜉|𝛼}𝑒^{𝑖2−𝑗(𝑥⋅𝜉)}𝑑𝜉^{𝑛/(𝑛+1)}⪯𝑡^−𝑛/22^{𝑗𝛼𝑛/2}. (60) Again we obtain that for odd 𝑛, _𝑗∈𝐴12^−𝑗𝑛_ℝ𝑛Γ𝜉𝜎2^−𝑗𝜉𝑒^{𝑖𝑡2−𝑗𝛼|𝜉|𝛼}𝑒^{𝑖2−𝑗(𝑥⋅𝜉)}𝑑𝜉⪯𝑡^−𝑛/2. (61) For 𝑗∈𝐴₂, without loss of generality, we assume |𝑥₁|≥(|𝑥|/𝑛). Perform integration by parts on the 𝜉₁ variable for suitable amount of times. We similarly obtain _𝑗∈𝐴22^−𝑗𝑛_ℝ𝑛Γ𝜉𝜎2^−𝑗𝜉𝑒^{𝑖𝑡2−𝑗𝛼|𝜉|𝛼}𝑒^{𝑖2−𝑗(𝑥⋅𝜉)}𝑑𝜉⪯𝑡^−𝑛/2. (62) For 𝑗∈𝐴₃, invoking Lemma 9, we obtain _ℝ𝑛Γ𝜉𝜎2^−𝑗𝜉𝑒^{𝑖(𝑡2−𝑗𝛼|𝜉|𝛼+2−𝑗(𝑥⋅𝜉))}2𝑑𝜉⪯min1,^{𝑗𝛼𝑛/2}𝑡^𝑛/2. (63) Noting that 𝐴₃contains no more than (10/|1−𝛼|)+log₂𝑛 numbers of 𝑗, it is easy to check _𝑗∈𝐴32^−𝑗𝑛2min1,^{𝑗𝛼𝑛/2}𝑡^𝑛/2⪯𝑡^−𝑛/2. (64)
The lemma is proved.

Lemma 11 (see [21, pages 163–171]). Let 0<𝑝<2 and 𝑛1𝑁=𝑝−12+1. (65) Suppose that 𝑇_𝑚 is a Fourier multiplier with symbol 𝑚. If 𝑚 is a bounded function which is of class 𝐶^𝑁 in ℝ^𝑛⧵{0} and if 𝜕𝜕𝜉^𝐽𝑚(𝜉)≤𝐴𝜉^−1|𝐽|for|𝐽|≤𝑁 (66) with 𝐴≥1, then 𝑇_𝑚 is a bounded operator on 𝐻^𝑝 and 𝑇_{𝑚𝐻𝑝→𝐻𝑝}⪯𝐴^{𝑛(1/𝑝−1/2)}. (67)

Lemma 12. Let |𝑘|≥1 and 𝛼≠1. For all 1≤𝑝≤∞, one has 𝜋_𝑘𝑒^{𝑖𝑡|Δ|𝛼/2}𝑓_𝐿𝑝⪯𝜋_𝑘𝑓_𝐿𝑝if𝑡|𝑘|^𝛼−2𝜋≤1,_𝑘𝑒^{𝑖𝑡|Δ|𝛼/2}𝑓_𝐿𝑝⪯𝑡|𝑘|^{𝛼−2𝑛|(1/𝑝)−(1/2)|}𝜋_𝑘𝑓_𝐿𝑝if𝑡|𝑘|^𝛼−2≥1. (68)

This lemma can be found in Section 4.2 of [11].

Lemma 13. Let 𝐹 be a compact subset in ℝ^𝑛, and let 0<𝑝≤𝑞≤∞. There exists a constant 𝐶 depending only on the diameter of 𝐹 and 𝑝, such that 𝑓_𝐿𝑞≤𝐶𝑓_𝐿𝑝 (69) for all 𝑓∈𝐿^𝑝 satisfying supp𝑓⊂𝐹.

This lemma is the Nikol'skij-Triebel inequality, see Proposition 1.3.2 in [20] (also Lemma 2.5 in [22]).

Lemma 14. Let 0<𝑝≤1 and 𝐹,𝐹^′ be compact subsets of ℝ^𝑛. Then there exists a constant 𝐶 depending only on the diameters of 𝐹,𝐹^′ and 𝑝, such that 𝑓∗𝑔_𝐿𝑝≤𝐶𝑓_𝐿𝑝𝑔_𝐿𝑝 (70) for all 𝑓, 𝑔∈𝐿^𝑝 satisfying supp𝑓⊂𝐹 and supp𝑔⊂𝐹^′.

This is Lemma 2.6 in [22] (see also Proposition 1.5.3 in [20]).

Lemma 15 (Pitt's theorem). If 1<𝑝≤2 and 0≤𝛼<1/𝑝^′, then _ℝ𝑛𝑓(𝑥)^𝑝′|𝑥|^{−𝛼𝑛𝑝′}𝑑𝑥^1/𝑝′⪯_ℝ𝑛𝑓(𝑥)^𝑝|𝑥|^𝛼𝑛𝑝𝑑𝑥^1/𝑝. (71)

Lemma 16. Let 𝑠∈ℝ and 𝑝,𝑝_𝑗≥1(𝑗=1,2,…,𝑚) satisfy 1𝑝=1𝑝₁+1𝑝₂1+⋯+𝑝_𝑚. (72) Then one has ^𝑚_𝑗=1𝑢_{𝑗𝑀𝑠𝑝,1}⪯^𝑚_𝑗=1𝑢_{𝑗𝑀𝑠𝑝𝑗,1}. (73)

This result is a particular case of Lemma 2.5 in [8].

3. Proof of Theorem 1

The operator 𝜋_𝑘𝑒^{𝑖𝑡|Δ|𝛼/2} is a convolution operator with the symbol 𝜎(𝜉−𝑘)𝑒^{𝑖𝑡|𝜉|𝛼}. This symbol is a 𝐶^∞ function on ℝ^𝑛⧵{0} with compact support. Clearly for any 𝑁>0and 𝜉≠0, we have that for |𝑘|≤10, 𝜕𝜕𝜉^𝐽𝜎(𝜉−𝑘)𝑒^{𝑖𝑡|𝜉|𝛼}⪯(1+𝑡)𝜉^−1|𝐽|for|𝐽|≤𝑁. (74)

So Lemma 11 implies the following estimate.

Proposition 17. Let 0<𝑝<∞. For any 𝑘 with |𝑘|≤10, one has 𝜋_𝑘𝑒^{𝑖𝑡|Δ|𝛼/2}𝑓_𝐻𝑝⪯max1,𝑡^{𝑛|1/𝑝−1/2|}𝑓_𝐻𝑝. (75)

By the proof of Lemma 8 and Proposition 17, we have that for all |𝑘|≤9, 𝜋_𝑘𝑒^{𝑖𝑡|Δ|𝛼/2}𝑓_𝐻𝑝⪯max1,𝑡^{𝑛|1/𝑝−1/2|}𝜋_𝑘𝑓_𝐻𝑝. (76)

The following proposition extends Lemma 12 to all 𝑝∈(0,1].

Proposition 18. Let 0<𝑝≤1. For any 𝑘 with |𝑘|≥1, for any 𝛼∈ℝ⧵{0,1}, one has 𝜋_𝑘𝑒^{𝑖𝑡|Δ|𝛼/2}𝑓_𝐿𝑝⪯max1,𝑡|𝑘|^{𝛼−2𝑛(1/𝑝−1/2)}_{|𝑗|∞≤1}𝜋_𝑘+𝑗𝑓_𝐿𝑝. (77)

Proof. The proof uses the same idea used in proving the case 𝑝≥1 which was represented in [11]. For the convenience of the reader, we present its proof.
Let Ω_𝑘 be the kernel of 𝜋_𝑘𝑒^{𝑖𝑡|Δ|𝛼/2}. Then Ω_𝑘=ℱ⁻¹𝜎(𝜉−𝑘)𝑒^{𝑖𝑡|𝜉|𝛼}. (78) By Lemma 14 and (46), we have 𝜋_𝑘𝑒^{𝑖𝑡|Δ|𝛼/2}𝑓_𝐿𝑝⪯Ω_{𝑘𝐿𝑝|𝑗|∞≤1}𝜋_𝑘+𝑗𝑓_𝐿𝑝. (79) Thus to prove the proposition, it suffices to show ℱ⁻¹𝜎(𝜉−𝑘)𝑒^{𝑖𝑡|𝜉|𝛼}_𝐿𝑝⪯max1,𝑡|𝑘|^{𝛼−2𝑛(1/𝑝−1/2)}. (80)
For simplicity, we prove the case 𝑛=2. The proof for 𝑛≥3, is tedious but shares the same idea as that for 𝑛=2.
First we study the case 𝑡|𝑘|^𝛼−2>1. For 𝑖=1,2, and 𝑘=(𝑘₁,𝑘₂), if 𝛼≥2 we denote 𝐶_𝑖(𝑘)=𝛼𝑡𝑘_𝑖+1𝑘₁+1²+𝑘₂+1^{2(𝛼/2)−1},𝐷_𝑖(𝑘)=𝛼𝑡𝑘_𝑖−1𝑘₁−1²+𝑘₂−1^{2(𝛼/2)−1}. (81) If 𝛼<2, we denote 𝐶_𝑖(𝑘)=|𝛼|𝑡𝑘_𝑖+1𝑘₁−1²+𝑘₂−1^{2(𝛼/2)−1},𝐷_𝑖(𝑘)=|𝛼|𝑡𝑘_𝑖−1𝑘₁+1²+𝑘₂+1^{2(𝛼/2)−1}. (82)
Also, for 𝑖=1,2and 𝑗∈ℕ, we define sets 𝐹_𝑖=𝑥_𝑖∈ℝ∶𝐷_𝑖(𝑘)−𝑡|𝑘|^𝛼−2<𝑥_𝑖≤𝐶_𝑖(𝑘)+𝑡|𝑘|^𝛼−2,𝐺_𝑖,𝑗=𝑥_𝑖∈ℝ∶𝐶_𝑖(𝑘)+𝑡|𝑘|^𝛼−2<𝑥+𝑗−1_𝑖≤𝐶_𝑖(𝑘)+𝑡|𝑘|^𝛼−2,𝐻+𝑗_𝑖,𝑗=𝑥_𝑖∈ℝ∶𝐷_𝑖(𝑘)−𝑡|𝑘|^𝛼−2<𝑥−𝑗_𝑖≤𝐷_𝑖(𝑘)−𝑡|𝑘|^𝛼−2−𝑗+1. (83) It is easy to check Length𝐹_𝑖⪯𝑡|𝑘|^𝛼−2,Length𝐺_𝑖,𝑗=Length𝐻_𝑖,𝑗=1. (84) Let 𝐾_𝑖,𝑗=𝐺_𝑖,𝑗∪𝐻_𝑖,𝑗. (85) We have for 𝑖=1,2, 𝜒_𝐹𝑖𝑥_𝑖+^∞_𝑗=1𝜒_{𝐾𝑖,𝑗}𝑥_𝑖≡1. (86) Write _ℝ2ℱ⁻¹𝜎_𝑘(𝜉)𝑒^{𝑖𝑡|𝜉|𝛼}(𝑥)^𝑝=𝑑𝑥_ℝ2ℱ⁻¹𝜎(𝜉)𝑒^{𝑖𝑡|𝜉+𝑘|𝛼}(𝑥)^𝑝⪯𝑑𝑥_ℝ2𝜒_𝐹1𝑥₁𝜒_𝐹2𝑥₂ℱ⁻¹𝜎(𝜉)𝑒^{𝑖𝑡|𝜉+𝑘|𝛼}(𝑥)^𝑝+𝑑𝑥_{∞𝑗=1ℝ2}𝜒_𝐾1,𝑗𝑥₁𝜒_𝐹2𝑥₂×ℱ⁻¹𝜎(𝜉)𝑒^{𝑖𝑡|𝜉+𝑘|𝛼}(𝑥)^𝑝+𝑑𝑥_{∞𝑚=1ℝ2}𝜒_𝐾2,𝑚𝑥₂𝜒_𝐹1𝑥₁×ℱ⁻¹𝜎(𝜉)𝑒^{𝑖𝑡|𝜉+𝑘|𝛼}(𝑥)^𝑝+𝑑𝑥_{∞∞𝑗=1𝑚=1ℝ2}𝜒_𝐾1,𝑗𝑥₁𝜒_𝐾2,𝑚𝑥₂×ℱ⁻¹𝜎(𝜉)𝑒^{𝑖𝑡|𝜉+𝑘|𝛼}(𝑥)^𝑝𝑑𝑥=𝐼𝐼₁+𝐼𝐼₂+𝐼𝐼₃+𝐼𝐼₄, (87) where ℱ⁻¹𝜎(𝜉)𝑒^{𝑖𝑡|𝜉+𝑘|𝛼}(𝑥)≃_ℝ2𝑒^{𝑖(𝑡|𝜉+𝑘|𝛼+⟨𝑥,𝜉⟩)}𝜎(𝜉)𝑑𝜉. (88)
It is easy to check that if 𝑡|𝑘|^𝛼−2≥1 and 𝜉∈ supp(𝜎), the phase function Φ(𝑡,𝑘,𝑥,𝜉)=𝑡𝜉+𝑘^𝛼+(𝑥⋅𝜉), (89) satisfies det𝐷_𝜉𝑗𝐷_𝜉𝑖Φ≥𝑡|𝑘|^𝛼−2. (90) So by Lemma 9, we have 𝐼𝐼₁⪯𝑡⁻¹|𝑘|^2−𝛼𝑝_ℝ2𝜒_𝐹1𝑥₁𝜒_𝐹2𝑥₂⪯𝑡𝑑𝑥₋₁|𝑘|^2−𝛼𝑝𝑡|𝑘|^𝛼−22=𝑡^2−𝑝|𝑘|^{(𝛼−2)(2−𝑝)}. (91)
Observe the easy fact that if 𝑥∈𝐾_1,𝑗 and 𝜉∈ supp (𝜎), for any integer 𝐿, 𝜕^𝐿𝜕𝜉^𝐿_𝑙𝜎(𝜉)Φ_𝜉𝑖()1𝑡,𝑘,𝑥,𝜉=𝑂𝑗+𝑡|𝑘|^𝛼−2𝐿,𝑖,𝑙=1,2. (92) Perform integration by parts on 𝜉₁ and 𝜉₂ variables both for 𝐿 times such that 𝐿𝑝>1. An easy computation shows that 𝐼𝐼₄⪯^∞∞_{𝑗=1𝑚=1ℝ2}𝜒_𝐾1,𝑗𝑥₁𝜒_𝐾2,𝑚𝑥₂×1𝑗+𝑡|𝑘|^{𝛼−2𝐿𝑝}𝑚+𝑡|𝑘|^{𝛼−2𝐿𝑝}⪯^∞_{|𝑘|𝑗≥𝑡𝛼−2}^∞_{|𝑘|𝑚≥𝑡𝛼−2}1𝑗^𝐿𝑝𝑚^𝐿𝑝×_ℝ2𝜒_𝐾1,𝑗𝑥₁𝜒_𝐾2,𝑚𝑥₂𝑑𝑥⪯1. (93)
The estimates for 𝐼𝐼₂ and 𝐼𝐼₃ are exactly the same. We only estimate 𝐼𝐼₂. Take integration by parts on 𝜉₁ variable for 𝐿 times with 𝑝𝐿>1. Again, a simple computation shows that 𝐼𝐼₂⪯^∞_{|𝑘|𝑗≥𝑡𝛼−2}1𝑗^𝐿𝑝_ℝ²𝜒_𝐾1,𝑗𝑥₁𝜒_𝐹2𝑥₂⪯𝑡|𝑘|𝑑𝑥_{𝛼−2−𝐿𝑝+2}⪯1, (94) if we chose a suitably large 𝐿. These estimates on 𝐼𝐼_𝑗, 𝑗=1,2,3,4, indicate ℱ⁻¹𝜎(𝜉−𝑘)𝑒^{𝑖𝑡|𝜉|𝛼}_𝐿𝑝⪯𝑡^2/𝑝−1|𝑘|^{(𝛼−2)(2/𝑝−1)}=𝑡|𝑘|^{𝛼−22(1/𝑝−1/2)}, (95) provided 𝑡|𝑘|^𝛼−2≥1.
We now turn to show the case 𝑡|𝑘|^𝛼−2<1. For 𝑖=1,2, and 𝑘=(𝑘₁,𝑘₂), let 𝐶_𝑖(𝑘)𝐷_𝑖(𝑘) be the numbers defined above. For 𝑖=1,2and 𝑗∈ℕ, we define sets 𝔉_𝑖=𝑥_𝑖∶𝐷_𝑖(𝑘)−1<𝑥_𝑖≤𝐶_𝑖(𝑘),𝔊+1_𝑖,𝑗=𝑥_𝑖∶𝐶_𝑖(𝑘)+𝑗<𝑥_𝑖≤𝐶_𝑖(𝑘),ℌ+1+𝑗_𝑖,𝑗=𝑥_𝑖∶𝐷_𝑖(𝑘)−1−𝑗<𝑥_𝑖≤𝐷_𝑖(𝑘)−𝑗. (96) It is easy to check Length𝔉_𝑖𝔊⪯1,Length_𝑖,𝑗=Lengthℌ_𝑖,𝑗=1. (97) Let 𝔎_𝑖,𝑗=𝔊_𝑖,𝑗∪ℌ_𝑖,𝑗. (98) Thus, _ℝ2ℱ⁻¹𝜎_𝑘(𝜉)𝑒^{𝑖𝑡|𝜉|𝛼}(𝑥)^𝑝⪯𝑑𝑥_ℝ2𝜒_𝔉1𝑥₁𝜒_𝔉2𝑥₂ℱ⁻¹𝜎(𝜉)𝑒^{𝑖𝑡|𝜉+𝑘|𝛼}(𝑥)^𝑝+𝑑𝑥_{∞𝑗=1ℝ2}𝜒_𝔎1,𝑗𝑥₁𝜒_𝔉2𝑥₂×ℱ⁻¹𝜎(𝜉)𝑒^{𝑖𝑡|𝜉+𝑘|𝛼}(𝑥)^𝑝+𝑑𝑥_{∞𝑚=1ℝ2}𝜒_𝔎2,𝑚𝑥₂𝜒_𝔉1𝑥₁×ℱ⁻¹𝜎(𝜉)𝑒^{𝑖𝑡|𝜉+𝑘|𝛼}(𝑥)^𝑝+𝑑𝑥_{∞∞𝑗=1𝑚=1ℝ2}𝜒_𝔎1,𝑗𝑥₁𝜒_𝔎2,𝑚𝑥₂×ℱ⁻¹𝜎(𝜉)𝑒^{𝑖𝑡|𝜉+𝑘|𝛼}(𝑥)^𝑝𝑑𝑥. (99) Using the same argument as we used before, we can show ℱ⁻¹𝜎(𝜉−𝑘)𝑒^{𝑖𝑡|𝜉|𝛼}_𝐿𝑝⪯1. (100) We complete the proof of Proposition 18.

We are now in a position to prove Theorem 1.

Proof. By an argument involving interpolation and duality, it suffices to show the case 𝑝≤1. Using Proposition 18, the inequality in (76) and the definition of the modulation spaces, we easily obtain (ii) in Theorem 1.
To show (i) and (iii) in Theorem 1, by Proposition 18 and the definition of the modulation spaces, it suffices to show 𝜋₀𝑆_𝛼(𝑡)𝑓_𝐿𝑝⪯max1,𝑡^{𝑛(1/𝑝−1/2)}_{|𝑗|∞≤1}𝜋_𝑗𝑓_𝐿𝑝. (101) Again, by Lemma 14, the proof of the inequality in (101) can be reduced to show that for 𝑡≥1, _{ℝ𝑛ℝ𝑛}𝜎(𝜉)𝑒^{𝑖𝑡|𝜉|𝛼}𝑒^{𝑖(𝜉⋅𝑥)}𝑑𝜉^𝑝𝑑𝑥^1/𝑝⪯𝑡^{𝑛(1/𝑝−1/2)}. (102) We show (iii) first. The proof of 𝑛=1 may illustrate the method. When 𝑛=1_ℝℝ𝜎(𝜉)𝑒^{𝑖𝑡|𝜉|𝛼}𝑒^{𝑖(𝜉⋅𝑥)}𝑑𝜉^𝑝𝑑𝑥^1/𝑝⪯_{|𝑥|<𝑡ℝ}𝜎(𝜉)𝑒^{𝑖𝑡|𝜉|𝛼}𝑒^{𝑖(𝜉⋅𝑥)}𝑑𝜉^𝑝𝑑𝑥^1/𝑝+_{|𝑥|≥𝑡ℝ}𝜎(𝜉)𝑒^{𝑖𝑡|𝜉|𝛼}𝑒^{𝑖(𝜉⋅𝑥)}𝑑𝜉^𝑝𝑑𝑥^1/𝑝. (103) By Hölder's inequality and the Plancherel theorem, the first term above _{|𝑥|<𝑡ℝ}𝜎(𝜉)𝑒^{𝑖𝑡|𝜉|𝛼}𝑒^{𝑖(𝜉⋅𝑥)}𝑑𝜉^𝑝𝑑𝑥^1/𝑝⪯_ℝ𝜎(𝜉)𝑒^{𝑖𝑡|𝜉|𝛼2}𝑑𝜉^1/2𝑡^{(1/𝑝−1/2)}⪯𝑡^{(1/𝑝−1/2)}. (104) For the second term, performing integration by parts, we obtain _{|𝑥|≥𝑡ℝ}𝜎(𝜉)𝑒^{𝑖𝑡|𝜉|𝛼}𝑒^{𝑖(𝜉⋅𝑥)}𝑑𝜉^𝑝𝑑𝑥^1/𝑝=2_{|𝑥|≥𝑡}^∞₀𝜎(𝜉)𝑒^{𝑖𝑡𝜉𝛼}cos𝜉𝑥𝑑𝜉^𝑝𝑑𝑥^1/𝑝=2_|𝑥|>𝑡1|𝑥|^𝑝₀^+∞𝑖𝛼𝑡𝜎(𝜉)𝜉^𝛼−1+𝜎^′(𝜉)𝑒^{𝑖𝑡|𝜉|𝛼}×sin𝜉𝑥𝑑𝜉^𝑝𝑑𝑥^1/𝑝≤2_|𝑥|>𝑡1|𝑥|^{2𝑝/(2−𝑝)}𝑑𝑥^{(2−𝑝)/2𝑝}×_|𝑥|>𝑡0^+∞𝑖𝛼𝑡𝜎(𝜉)𝜉^𝛼−1+𝜎^′(𝜉)×𝑒^{𝑖𝑡|𝜉|𝛼}sin𝜉𝑥𝑑𝜉²𝑑𝑥^1/2⪯𝑡_|𝑥|>𝑡1|𝑥|^{2𝑝/(2−𝑝)}𝑑𝑥^{(2−𝑝)/2𝑝}×_𝑅1𝜎(𝜉)𝜉^𝛼−1+𝜎^′(𝜉)²𝑑𝜉^1/2⪯𝑡^1/𝑝−1/2, (105) since 2𝑝2−𝑝>1. (106)
Now we return to show (i) of Theorem 1. We will prove only the case 𝛼<1. Write _{ℝ𝑛ℝ𝑛}𝜎(𝜉)𝑒^{𝑖𝑡|𝜉|𝛼}𝑒^{𝑖(𝜉⋅𝑥)}𝑑𝜉^𝑝𝑑𝑥^1/𝑝⪯_{|𝑥|<𝑡ℝ𝑛}𝜎(𝜉)𝑒^{𝑖𝑡|𝜉|𝛼}𝑒^{𝑖(𝜉⋅𝑥)}𝑑𝜉^𝑝𝑑𝑥^1/𝑝_{|𝑥|≥𝑡ℝ𝑛}𝜎(𝜉)𝑒^{𝑖𝑡|𝜉|𝛼}𝑒^{𝑖(𝜉⋅𝑥)}𝑑𝜉^𝑝𝑑𝑥^1/𝑝=𝐽₁+𝐽₂. (107) Using Hölder's inequality and the Plancherel theorem, we obtain 𝐽₁⪯_ℝ𝑛𝜎(𝜉)𝑒^{𝑖𝑡|𝜉|𝛼2}𝑑𝜉^1/2𝑡^{𝑛(1/𝑝−1/2)}⪯𝑡^{𝑛(1/𝑝−1/2)}. (108)
For 𝑖,𝑗∈{1,2,…,𝑛}, we denote sets 𝐸_𝑡=𝑥∈ℝ^𝑛∶|𝑥|,𝐸≥𝑡_𝑡,𝑖=𝑥∈𝐸_𝑡∶𝑥_𝑖≥𝑥_𝑗∀𝑗≠𝑖. (109) We now write 𝐽₂⪯^𝑛_{𝑖=1𝐸𝑡,𝑖ℝ𝑛}𝜎(𝜉)𝑒^{𝑖𝑡|𝜉|𝛼}𝑒^{𝑖(𝜉⋅𝑥)}𝑑𝜉^𝑝𝑑𝑥^1/𝑝. (110) To show (102), it now suffices to show that for each 𝑖, 𝐼=_{𝐸𝑡,𝑖ℝ𝑛}𝜎(𝜉)𝑒^{𝑖𝑡|𝜉|𝛼}𝑒^{𝑖(𝜉⋅𝑥)}𝑑𝜉^𝑝𝑑𝑥^1/𝑝⪯𝑡^{𝑛(1/𝑝−1/2)}. (111)
Using the Leibniz rule, for any positive integer 𝐿, we have 𝜕_𝑖^(𝐿)𝜎(𝑥)𝑒^{𝑖𝑡|𝑥|𝛼}=^𝐿_𝑘=0𝐶^𝑘_𝐿𝜕_𝑖^{(𝐿−𝑘)}(𝜎(𝑥))𝜕_𝑖^(𝑘)𝑒^{𝑖𝑡|𝑥|𝛼}. (112) Here, an easy induction argument shows that, for 𝑘≥1, 𝜕_𝑖^(𝑘)𝑒^{𝑖𝑡|𝑥|𝛼}=𝑒^{𝑖𝑡|𝑥|𝛼𝑘}_𝑗=1𝑡^𝑗𝜑^(𝑘,𝑗)_{𝑗𝛼−𝑘}(𝑥), (113) where 𝜑^(𝑗.𝑘)_{𝑗𝛼−𝑘}∈𝐶^∞(𝑅^𝑛−{0}) is a homogeneous function of degree 𝑗𝛼−𝑘 for each 𝑗. We now write 𝜕_𝑖^(𝐿)𝜎(𝑥)𝑒^{𝑖𝑡|𝑥|𝛼}=𝑒^{𝑖𝑡|𝑥|𝛼}𝜕_𝑖^(𝐿)(𝜎(𝑥))+^𝐿_𝑗=1𝑡^𝑗|𝑥|^{𝑗𝛼−𝐿}Φ_{𝑗,𝐿,𝛼}(𝑥), (114) where Φ_{𝑗,𝐿,𝛼}(𝑥)=^𝐿_𝑘=𝑗𝐶^𝑘_𝐿𝜕_𝑖^{(𝐿−𝑘)}(𝜎(𝑥))𝜑^(𝑘,𝑗)_{𝑗𝛼−𝑘}𝑥|𝑥||𝑥|^𝐿−𝑘. (115) By the definition, it is easy to see that each Φ_{𝑗,𝐿,𝛼} is an 𝐿^∞ and 𝐶^∞(𝑅^𝑛−{0}) function with support in the cube [−1/2,1/2]^𝑛.
Let 𝐿=[(𝑛/2)+𝛼]+1. Performing integration by parts on 𝜉_𝑖 variables for 𝐿 times, we have 𝐼=_{𝐸𝑡,𝑖𝑅𝑛}𝜎(𝜉)𝑒^{𝑖𝑡|𝜉|𝛼}𝑒^{𝑖(𝑥⋅𝜉)}𝑑𝜉^𝑝𝑑𝑥^1/𝑝⪯_{𝐸𝑡,𝑖}|𝑥|^−𝐿𝑝_𝑅𝑛𝜕_𝑖^(𝐿)(𝜎(𝜉))𝑒^{𝑖𝑡|𝜉|𝛼}𝑒^{𝑖(𝑥⋅𝜉)}𝑑𝜉^𝑝𝑑𝑥^1/𝑝+^𝐿_𝑗=1𝑡^𝑗_{𝐸𝑡,𝑖}|𝑥|^−𝐿𝑝_𝑅𝑛𝜉^{𝑗𝛼−𝐿}Φ_{𝑗,𝐿,𝛼}×(𝜉)𝑒^{𝑖𝑡|𝜉|𝛼}𝑒^{𝑖(𝑥⋅𝜉)}𝑑𝜉^𝑝𝑥^1/𝑝=^𝐿_𝑗=0𝐼_𝑗. (116)
We first estimate each 𝐼_𝑗, 0≤𝑗≤𝐿−1. Recall that we assume 𝑝∈(0,2). Let 𝑞=2/𝑝, so 𝑝𝑞^′=2𝑝/(2−𝑝). By the choice of 𝐿 and the assumption 𝑛𝑝>,𝑛+𝛼 (117) it is easy to see 𝐿𝑝𝑞^′>𝑛. Therefore, by Hölder's inequality, we obtain 𝐼₀≤_{|𝑥|≥𝑡}|𝑥|^{−𝐿𝑝𝑞′}𝑑𝑥^{1/(𝑝𝑞′)}×_{𝑅𝑛𝑅𝑛}𝜕_𝑖^(𝐿)(𝜎(𝜉))𝑒^{𝑖𝑡|𝜉|𝛼}𝑒^{𝑖(𝜉⋅𝑥)}𝑑𝜉²𝑑𝑥^1/2⪯𝑡^{−𝐿+𝑛(1/𝑝−1/2)}𝜕_𝑖^(𝐿)𝜎₂⪯𝑡^{−𝐿+𝑛(1/𝑝−1/2)}. (118) For each 𝑗=1,2,…,𝐿−1, by the choice of 𝐿, the assumption on 𝑝, and an easy computation, it is not difficult to see that we may obtain a number 𝛼_𝑗 in the interval [0,𝑛/2) satisfying 𝐿−𝛼_𝑗1>𝑛𝑝−12,𝛼_𝑗𝑛+𝑗𝛼−𝐿>−2,𝛼_𝑗+𝑗−𝐿≤0. (119) By Hölder's inequality and Pitt's theorem, for each 𝑗, we obtain 𝐼_𝑗≤𝑡^𝑗_{𝐸𝑡,𝑖}|𝑥|^{−(𝐿−𝛼𝑗)𝑝}|𝑥|^−𝛼𝑗_𝑅𝑛𝜉^{𝑗𝛼−𝐿}Φ_{𝑗,𝐿,𝛼}(𝜉)𝑒^{𝑖𝑡|𝜉|𝛼}×𝑒^{𝑖(𝑥⋅𝜉)}𝑑𝜉^𝑝𝑑𝑥^1/𝑝≤_{|𝑥|≥𝑡}|𝑥|^{−(𝐿−𝛼𝑗)𝑝𝑞′}𝑑𝑥^{1/(𝑝𝑞′)}×_𝑅𝑛|𝑥|^−𝛼𝑗_𝑅𝑛𝜉^{𝑗𝛼−𝐿}Φ_{𝑗,𝐿,𝛼}(𝜉)𝑒^{𝑖𝑡|𝜉|𝛼}𝑒^{𝑖(𝑥⋅𝜉)}𝑑𝜉²𝑑𝑥^1/2⪯𝑡^{−𝐿+𝛼𝑗+𝑗+𝑛(1/𝑝−1/2)}_𝑅𝑛|𝑥|^𝛼𝑗|𝑥|^{𝑗𝛼−𝐿}Φ_{𝑗,𝐿,𝛼}(𝑥)²𝑑𝑥^1/2⪯𝑡^{−𝐿+𝛼𝑗+𝑗+𝑛(1/𝑝−1/2)}. (120) Combining all the estimates, we have _{𝐸𝑡,𝑖𝑅𝑛}𝜎(𝜉)𝑒^{𝑖𝑡|𝜉|𝛼}𝑒^{𝑖(𝑥⋅𝜉)}𝑑𝜉^𝑝𝑑𝑥^1/𝑝⪯𝐼_𝐿+^𝐿−1_𝑗=0𝑡^{−𝐿+𝛼𝑗+𝑗+𝑛(1/𝑝−1/2)}⪯𝐼_𝐿+𝑡^{𝑛(1/𝑝−1/2)}. (121) It remains to estimate 𝐼_𝐿.
It is easy to see that the choice of 𝐿 and the condition 𝛼>𝛼₀=23𝑛+2if𝑛iseven𝑛+3if𝑛isodd (122) in the theorem imply 𝐿𝛼−𝐿>−𝑛/2. So, by Hölder's inequality and Pitt's theorem again, we obtain 𝐼_𝐿=𝑡^𝐿_{𝐸𝑡,𝑖}|𝑥|^−𝐿𝑝_𝑅𝑛𝜉^{𝐿𝛼−𝐿}Φ_{𝐿,𝐿,𝛼}(𝜉)𝑒^{𝑖𝑡|𝜉|𝛼}𝑒^{𝑖(𝑥⋅𝜉)}𝑑𝜉^𝑝𝑑𝑥^1/𝑝⪯𝑡^𝐿_{|𝑥|≥𝑡}|𝑥|^{−𝐿𝑝𝑞/}𝑑𝑥^{1/(𝑝𝑞′)}×_{𝑅𝑛𝑅𝑛}𝜉^{𝐿𝛼−𝐿}Φ_{𝐿,𝐿,𝛼}(𝜉)𝑒^{𝑖𝑡|𝜉|𝛼}𝑒^{𝑖(𝑥⋅𝜉)}𝑑𝜉²𝑑𝑥^1/2⪯𝑡^{𝑛(1/𝑝−1/2)}𝜉^{𝐿𝛼−𝐿}Φ_{𝐿,𝐿,𝛼}(𝜉)₂⪯𝑡^{𝑛(1/𝑝−1/2)}. (123) This completes the proof of (102).
When 𝛼=2,4,…, we have _{𝐸𝑡,𝑖ℝ𝑛}𝜎(𝜉)𝑒^{𝑖𝑡|𝜉|𝛼}𝑒^{𝑖(𝜉⋅𝑥)}𝑑𝜉^𝑝𝑑𝑥^1/𝑝⪯𝑡^𝐿_{𝐸𝑡,𝑖}1|𝑥|^{𝑝𝐿|𝜉|≤2}𝜔(𝜉)𝑒^{𝑖𝑡|𝜉|𝛼}𝑒^{𝑖(𝜉⋅𝑥)}𝑑𝜉^𝑝𝑑𝑥^1/𝑝 (124) for any integer 𝐿, where 𝜔(𝜉) is a 𝐶^∞ function. Thus it is trivial to see that _{𝐸𝑡,𝑖ℝ𝑛}𝜎(𝜉)𝑒^{𝑖𝑡|𝜉|𝛼}𝑒^{𝑖(𝜉⋅𝑥)}𝑑𝜉^𝑝𝑑𝑥^1/𝑝⪯𝑡^{𝑛(1/𝑝−1/2)} (125) for all 𝑝>0. This proves (i) in Theorem 1.

4. Proof of Theorem 3

Recall that the function 𝜎 chosen in the definition of the modulation space is flexible. We may choose 𝜎=^𝑛_𝑗=1𝜃_𝑗, (126) where each 𝜃_𝑗 is a nonnegative valued function in 𝑆(ℝ) with support in [−4/5,4/5] and satisfies 𝜃_𝑗(𝜉_𝑗)=1 if 𝜉_𝑗∈[−2/5,2/5] and _{𝑘𝑗∈ℤ}𝜃_{𝑗,𝑘𝑗}𝜉_𝑗=1,𝜃_{𝑗,𝑘𝑗}𝜉_𝑗=𝜃_𝑗𝜉_𝑗+𝑘_𝑗. (127)

For simplicity, we work on the case 0<𝑝<2. Let 𝜙∈𝐶^∞(ℝ) be a nonnegative function with support in the set [1/16,1/8]. For 𝑥=𝑥₁,𝑥₂,…,𝑥_𝑛, (128) define a function Φ on ℝ^𝑛 by Φ(𝑥)=^𝑛_𝑗=1𝜙𝑥_𝑗. (129) and an 𝑓∈𝑆(ℝ^𝑛) by 𝑓=ℱ⁻¹(Φ). Let 𝑘∈ℤ^𝑛. It is easy to see that 𝜋_𝑘𝑓(𝑥)=𝜖(𝑘)𝑓(𝑥), (130) where 𝜖(𝑘)=1 if 𝑘=0 and 𝜖(𝑘)=0 if 𝑘≠0. Similarly we have 𝜋_𝑘𝑒^𝑖𝑡|Δ|𝑓(𝑥)=𝜖(𝑘)𝑒^𝑖𝑡|Δ|𝑓(𝑥). (131) Suppose that we have some 𝛿 such that 𝑒^𝑖𝑡|Δ|𝑓_{𝑀𝑠𝑝,𝑞(ℝ𝑛)}⪯𝑡^𝛿𝑓_{𝑀𝑠𝑝,𝑞(ℝ𝑛)}. (132) By the choice of 𝑓, we have 𝑒^𝑖𝑡|Δ|𝑓_{𝑀𝑠𝑝,𝑞(ℝ𝑛)}=𝜋₀𝑒^𝑖𝑡|Δ|𝑓_{𝐿𝑝(ℝ𝑛)}⪯𝑡^𝛿𝑓_{𝑀𝑠𝑝,𝑞(ℝ𝑛)}=𝑡^𝛿𝑓_{𝐿𝑝(ℝ𝑛)}. (133) On the other hand, 𝜋₀𝑒^𝑖𝑡|Δ|𝑓_{𝐿𝑝(ℝ𝑛)}=_ℝ𝑛^𝑛_𝑗=1ℝ𝜙(𝜉_𝑗)𝑒^{𝑖𝜓(𝜉𝑗,𝑥𝑗)}𝑑𝜉_𝑗^𝑝𝑑𝑥^1/𝑝, (134) where the phase function 𝜓(𝜉_𝑗,𝑥_𝑗) is defined by 𝜓𝜉_𝑗,𝑥_𝑗=𝑡𝜉²_𝑗+𝜉_𝑗𝑥_𝑗. (135) Since 𝜕𝜕𝜉_𝑗𝜓𝜉_𝑗,𝑥_𝑗=2𝑡𝜉_𝑗+𝑥_𝑗, (136) The critical point of 𝜓(𝜉_𝑗,𝑥_𝑗) is at 𝜉^∗_𝑗=−𝑥_𝑗.2𝑡 (137) Thus, by the stationary phase method (see [19, Proposition 3, page 334]), an easy computation gives that, as 𝑡→∞, _ℝ𝑛^𝑛_𝑗=1ℝ𝜙𝜉_𝑗𝑒^{𝑖𝜓(𝜉𝑗,𝑥𝑗)}𝑑𝜉_𝑗^𝑝𝑑𝑥^1/𝑝≥^𝑛_{𝑗=1|𝑥𝑗|≤2𝑡ℝ}𝜙𝜉_𝑗𝑒^{𝑖𝜓(𝜉𝑗,𝑥𝑗)}𝑑𝜉_𝑗^𝑝𝑑𝑥_𝑗^1/𝑝≃𝑡^{−𝑛/2𝑛}_{𝑗=1|𝑥𝑗|≤2𝑡}𝜙−𝑥_𝑗2𝑡^𝑝𝑑𝑥_𝑗^1/𝑝+𝑂𝑡^{𝑛(1/𝑝−1/2)−1/2}≃𝑡^{𝑛(1/𝑝−1/2)}. (138) Thus the inequality 𝑒^𝑖𝑡|Δ|𝑓_{𝑀𝑠𝑝,𝑞(ℝ𝑛)}⪯𝑡^𝛿𝑓_{𝑀𝑠𝑝,𝑞(ℝ𝑛)} (139) implies 𝑡^{𝑛(1/𝑝−1/2)}⪯𝑡^𝛿,as𝑡→∞. (140) This shows the conclusion.

5. Applications

5.1. Operators 𝑆_𝛼,𝛽(𝑡)

Let 𝛼≠1and 𝛽∈ℝ. In [23], to investigate the absolute convergence for multiple Fourier series, Wainger studied the oscillating multipliers 𝑆_𝛼,𝛽(t) with symbol 𝑚_𝛼,𝛽()=𝑒𝑡,𝜉^{𝑖𝑡|𝜉|𝛼}1+𝜉^2𝛽/2. (141) In [24], Miyachi proved that in the case 𝛼>0 and 𝛼≠1, for 0<𝑝<∞, 𝑆_𝛼,𝛽(1)𝑓_{𝐻𝑝(ℝ𝑛)}⪯𝑓_{𝐻𝑝(ℝ𝑛)} (142) if and only if 1𝛽≥𝑛𝛼𝑝−12. (143) By Theorem 1 and its proof, we not only obtain the boundedness of 𝑆_𝛼,𝛽(𝑡) on 𝜇^𝑠_𝑝,𝑞(ℝ^𝑛) for any 𝛽∈ℝ, but also gain a regularity of 𝛽 if 𝛽>0.

Theorem 19. Let 0<𝑝<∞ and 𝛼≠1. One has for 𝑡≥1𝑆_𝛼,𝛽(𝑡)𝑓_{𝜇𝑠𝑝,𝑞}⪯𝑡^{𝑛|1/𝑝−1/2|}𝑓_𝜇^{𝑠+𝛾−𝛽}_𝑝,𝑞+𝑓_{𝜇𝑠𝑝,𝑞}, (144) where (1𝛾=𝑛𝛼−2)𝑝−12. (145)

Particularly, if 𝛽=𝑛(𝛼−2)|1/𝑝−1/2|, we have 𝑆_𝛼,𝛽(𝑡)𝑓_{𝜇𝑠𝑝,𝑞}⪯𝑡^{𝑛|1/𝑝−1/2|}𝑓_{𝜇𝑠𝑝,𝑞}. (146)

Proof. The proof of Theorem 19 is the same as the proof of Theorem 1. We skip it.

5.2. 𝑀^𝑠_𝑝1,𝑞→𝑀^𝑠_𝑝2,𝑞 Boundedness

We will study the 𝑀^𝑠_𝑝1,𝑞→𝑀^𝑠_𝑝2,𝑞 boundedness for the operator 𝑒^{𝑖𝑡|Δ|𝛼/2} in the case 0<𝛼≤2and 𝛼≠1. The other cases of 𝛼will be addressed in another paper. First we estimate 𝜋_𝑘𝑒^{𝑖𝑡|Δ|𝛼/2}for |𝑘|≤10.

Proposition 20. Let 𝛼≠1, 0<𝛼≤2 and |𝑘|≤9. One has for all 0<𝑟≤1≤𝑝≤∞, 𝜋_𝑘𝑒^{𝑖𝑡|Δ|𝛼/2}𝑓_𝐿𝑝⪯(1+𝑡)^{−𝑛(1/2−1/𝑝)}𝜋_𝑘𝑓_𝐿𝑟. (147)

Proof. By Lemma 9, we know 𝜋_𝑘𝑒^{𝑖𝑡|Δ|𝛼/2}_{𝐿1→𝐿∞}⪯(1+𝑡)^−𝑛/2. (148) On the other hand, it is known from [11] 𝜋_𝑘𝑒^{𝑖𝑡|Δ|𝛼/2}_{𝐿𝑝→𝐿𝑝}⪯(1+𝑡)^{𝑛|1/𝑝−1/2|} (149) for all 1≤𝑝≤∞. Interpolating these two inequalities with the energy inequality 𝜋_𝑘𝑒^{𝑖𝑡|Δ|𝛼/2}_{𝐿2→𝐿2}⪯1, (150) we find that for any 1≤𝑝≤∞, 𝜋_𝑘𝑒^{𝑖𝑡|Δ|𝛼/2}_{𝐿𝑝→𝐿∞}⪯(1+𝑡)^{−𝑛(1/𝑝−1/2)}. (151) Let 1≤𝑝≤∞. For 𝑓∈𝐿¹, we have 𝜋_𝑘𝑒^{𝑖𝑡|Δ|𝛼/2}𝑓_𝐿𝑝=sup_{𝑔𝐿𝑝′=1}<𝜋_𝑘𝑆_𝛼(𝑡)𝑓,𝑔>=sup_{𝑔𝐿𝑝′=1}<𝑓,𝜋_𝑘𝑒^{𝑖𝑡|Δ|𝛼/2∗}𝑔>⪯sup_{𝑔𝐿𝑝′=1}𝜋_𝑘𝑒^{𝑖𝑡|Δ|𝛼/2∗}𝑔_𝐿∞𝑓_𝐿1⪯(1+𝑡)^{𝑛(1/𝑝−1/2)}𝑓_𝐿1. (152) This shows that for |𝑘|≤10, 𝜋_𝑘𝑒^{𝑖𝑡|Δ|𝛼/2}𝑓_𝐿𝑝⪯(1+𝑡)^{−𝑛(1/2−1/𝑝)}𝑓_𝐿1. (153) Now if 0<𝑟≤1, we obtain that for all 1≤𝑝≤∞, 𝜋_𝑘𝑒^{𝑖𝑡|Δ|𝛼/2}𝑓_𝐿𝑝⪯_|ℓ|≤1𝜋_𝑘+ℓ𝑆_𝛼(𝑡)𝜋_𝑘𝑓_𝐿𝑝⪯(1+𝑡)^{−𝑛(1/2−1/𝑝)}𝜋_𝑘𝑓_𝐿1⪯(1+𝑡)^{−𝑛(1/2−1/𝑝)}𝜋_𝑘𝑓_𝐿𝑟. (154) The last inequality is from Lemma 13.

As these discussions, we obtain the following corollary.

Corollary 21. Let 0<𝛼≤2, 𝛼≠1, |𝑘|≤10 and 0<𝑝₁≤1≤𝑝₂≤∞, one has 𝜋_𝑘𝑒^{𝑖𝑡|Δ|𝛼/2}𝑓_𝐻𝑝2⪯𝜋_𝑘𝑓_𝐻𝑝1𝜋if𝑡≤1,_𝑘𝑒^{𝑖𝑡|Δ|𝛼/2}𝑓_𝐻𝑝2⪯𝑡^{𝑛(1/𝑝2−1/2)}𝜋_𝑘𝑓_𝐻𝑝1if𝑡>1, (155) where for convenience, one denotes 𝐿^∞=𝐻^∞.

Proposition 22. Let 0<𝛼≤2 and 𝛼≠1. One has for |𝑘|≥9 and all 0<𝑟≤1≤𝑝≤∞𝜋_𝑘𝑒^{𝑖𝑡|Δ|𝛼/2}𝑓_𝐿𝑝⪯1+𝑡|𝑘|^{𝛼−2−𝑛(1/2−1/𝑝)}𝜋_𝑘𝑓_𝐿𝑟. (156)

Proof. By Lemma 9, it is trivial to see that 𝜋_𝑘𝑒^{𝑖𝑡|Δ|𝛼/2}_{𝐿1→𝐿∞}⪯min1,𝑡^−𝑛/2|𝑘|^{−𝑛(𝛼−2)/2}. (157) Interpolating this inequality with the energy inequality, we find that for any 𝑝≤2, 𝜋_𝑘𝑒^{𝑖𝑡|Δ|𝛼/2}_{𝐿𝑝→𝐿∞}⪯1+𝑡|𝑘|^{𝛼−2−𝑛(1/𝑝−1/2)}. (158) So by duality, for |𝑘|≥9, 𝜋_𝑘𝑒^{𝑖𝑡|Δ|𝛼/2}𝑓_𝐿𝑝⪯1+𝑡|𝑘|^{𝛼−2𝑛(1/𝑝−1/2)}𝑓_𝐿1. (159) Now for any 0<𝑟≤1, we obtain that for all 1≤𝑝≤∞, 𝜋_𝑘𝑒^{𝑖𝑡|Δ|𝛼/2}𝑓_𝐿𝑝⪯_|ℓ|≤1𝜋_𝑘+ℓ𝑆_𝛼(𝑡)𝜋_𝑘𝑓_𝐿𝑝⪯1+𝑡|𝑘|^{𝛼−2−𝑛(1/𝑝−1/2)}𝜋_𝑘𝑓_𝐿1⪯1+𝑡|𝑘|^{𝛼−2𝑛(1/𝑝−1/2)}𝜋_𝑘𝑓_𝐿𝑟. (160)

As a consequence the proposition, we have the following.

Corollary 23. Let 0<𝛼≤2, 𝛼≠1, |𝑘|≥9 and 0<𝑝₁≤1≤𝑝₂≤∞, one has 𝜋_𝑘𝑒^{𝑖𝑡|Δ|𝛼/2}𝑓_𝐻𝑝2⪯𝜋_𝑘𝑓_𝐻𝑝1if𝑡|𝑘|^𝛼−2𝜋≤1,_𝑘𝑒^{𝑖𝑡|Δ|𝛼/2}𝑓_𝐻𝑝2⪯|𝑘|^{𝑛(𝛼−2)(1/𝑝2−1/2)}𝑡^{𝑛(1/𝑝2−1/2)}𝜋_𝑘𝑓_𝐻𝑝1if𝑡|𝑘|^𝛼−2≥1. (161)

By Corollaries 21 and 23 and the definition of the modulation spaces, we now obtain the 𝜇^𝑠_𝑝1,𝑞→𝑀^𝑠_𝑝2,𝑞 boundedness of 𝑒^{𝑖𝑡|Δ|𝛼/2} on the modulation spaces.

Theorem 24. Let 0<𝛼≤2, 𝛼≠1, and 0<𝑝₁≤1≤𝑝₂≤∞, one has that 𝑒^{𝑖𝑡|Δ|𝛼/2}𝑓_{𝑀𝑠𝑝2,𝑞}⪯(1+𝑡)^{𝑛(1/𝑝2−1/2)}𝑓_𝜇𝑝1^𝑠+𝛾_,𝑞, (162) where ()1𝛾=𝑛𝛼−2𝑝₂−12. (163)

Next, we give the proof of Theorem 4.

Proof. By the definition of the modulation space, we know 𝑒^{𝑖𝑡|Δ|𝛼/2}𝑓_{𝑀𝑠𝑝2,𝑞(ℝ𝑛)𝐿𝑟(0,∞)}=^∞_0𝑘(1+|𝑘|)^𝑠𝑞𝜋_𝑘𝑒^{𝑖𝑡|Δ|𝛼/2}𝑓^𝑞_𝐿𝑝2^𝑟/𝑞𝑑𝑡^1/𝑟. (164) If 𝑟/𝑞≥1, write (1+|𝑘|)^𝑠𝑞𝜋_𝑘𝑆_𝛼(𝑡)𝑓^𝑞_𝐿𝑝2=𝑔_𝑘(𝑡). (165) By the Minkowski inequality, we have 𝑒^{𝑖𝑡|Δ|𝛼/2}𝑓_{𝑀𝑠𝑝2,𝑞(ℝ𝑛)𝐿𝑟(0,∞)}⪯_𝑘^∞₀𝑔_𝑘(𝑡)^𝑟/𝑞𝑑𝑡^{𝑞/𝑟1/𝑞}=_𝑘^∞₀(1+|𝑘|)^𝑠𝑟𝜋_𝑘𝑒^{𝑖𝑡|Δ|𝛼/2}𝑓^𝑟_𝐿𝑝2𝑑𝑡^{𝑞/𝑟1/𝑞}=_𝑘(1+|𝑘|)^𝑠𝑞∞₀𝜋_𝑘𝑒^{𝑖𝑡|Δ|𝛼/2}𝑓^𝑟_𝐿𝑝2𝑑𝑡^{𝑞/𝑟1/𝑞}. (166) When |𝑘|≤10, ^∞₀𝜋_𝑘𝑒^{𝑖𝑡|Δ|𝛼/2}𝑓^𝑟_𝐿𝑝2𝑑𝑡^𝑞/𝑟⪯𝜋_𝑘𝑓^𝑟_𝐻𝑝1^∞₀(1+𝑡)^{−𝑟𝑛(1/2−1/𝑝2)}𝑑𝑡^𝑞/𝑟⪯𝜋_𝑘𝑓^𝑞_𝐻𝑝1. (167) When |𝑘|≥9, ^∞₀𝜋_𝑘𝑒^{𝑖𝑡|Δ|𝛼/2}𝑓^𝑟_𝐿𝑝2𝑑𝑡^𝑞/𝑟⪯^{|𝑘|2−𝛼}₀𝜋_𝑘𝑒^{𝑖𝑡|Δ|𝛼/2}𝑓^𝑟_𝐿𝑝2𝑑𝑡^𝑞/𝑟+^∞_{|𝑘|2−𝛼}𝜋_𝑘𝑒^{𝑖𝑡|Δ|𝛼/2}𝑓^𝑟_𝐿𝑝2𝑑𝑡^𝑞/𝑟⪯𝜋_𝑘𝑓^𝑟_𝐿𝑝1^{|𝑘|2−𝛼}₀𝑑𝑡^𝑞/𝑟+𝜋_𝑘𝑓^𝑞_𝐿𝑝1^∞_{|𝑘|2−𝛼}𝑡|𝑘|^{𝛼−2𝑟𝑛(1/2−1/𝑝2)}𝑑𝑡^𝑞/𝑟⪯𝜋_𝑘𝑓^𝑞_𝐿𝑝1|𝑘|^{(2−𝛼)𝑞/𝑟}+𝜋_𝑘𝑓^𝑞_𝐿𝑝1|𝑘|^{(𝛼−2)𝑟𝑛(1/𝑝2−1/2)∞}_|k|2−𝛼𝑡^{−𝑟𝑛(1/2−1/𝑝2)}𝑑𝑡^𝑞/𝑟⪯𝜋_𝑘𝑓^𝑞_𝐿𝑝1|𝑘|^{(2−𝛼)𝑞/𝑟}. (168) This shows that if 𝑟≥𝑞, then 𝑒^{𝑖𝑡|Δ|𝛼/2}𝑓_{𝑀𝑠𝑝2,𝑞(ℝ𝑛)𝐿𝑟(0,∞)}⪯_𝑘(1+|𝑘|)^𝑠𝑞⟨𝑘⟩^{(2−𝛼)𝑞/𝑟}𝜋_𝑘𝑓^𝑞_𝐿𝑝1^1/𝑞=𝑓_𝜇𝑝1^{𝑠+(2−𝛼)/𝑟}_{,𝑞(ℝ𝑛)}. (169)
If 𝑟/𝑞≤1, then 𝑒^{𝑖𝑡|Δ|𝛼/2}𝑓_{𝑀𝑠𝑝2,𝑞(ℝ𝑛)𝐿𝑟(0,∞)}⪯^∞_0𝑘(1+|𝑘|)^𝑠𝑟𝜋_𝑘𝑒^{𝑖𝑡|Δ|𝛼/2}𝑓^𝑟_𝐿𝑝2𝑑𝑡^1/𝑟. (170) As in the previous case, ^∞₀𝜋_𝑘𝑒^{𝑖𝑡|Δ|𝛼/2}𝑓^𝑟_𝐿𝑝2𝑑𝑡⪯𝜋_𝑘𝑓_𝐻𝑝1 (171) for |𝑘|≤10, and (1+|𝑘|)^𝑠𝑟∞₀𝜋_𝑘𝑒^{𝑖𝑡|Δ|𝛼/2}𝑓^𝑟_𝐿𝑝2⪯(|𝑘𝑑𝑡1+|)^𝑠𝑟𝜋_𝑘𝑓^𝑟_𝐿𝑝1|𝑘|^{(𝛼−2)𝑟𝑛(1/2−1/𝑝2)}×^∞_{|𝑘|2−𝛼}𝑡^{−𝑟𝑛(1/2−1/𝑝2)}+(|𝑘𝑑𝑡1+|)^𝑠𝑟𝜋_𝑘𝑓^𝑟_𝐿𝑝1^∞_{|𝑘|2−𝛼}𝑡^{−𝑟𝑛(1/2−1/𝑝2)}⪯(|𝑘𝑑𝑡1+|)^𝑠𝑟𝜋_𝑘𝑓^𝑟_𝐿𝑝1⟨𝑘⟩^(𝛼−2). (172) This shows that 𝑒^{𝑖t|Δ|𝛼/2}𝑓_{𝑀𝑠𝑝2,𝑞(ℝ𝑛)𝐿𝑟(0,∞)}⪯_𝑘(1+|𝑘|)^𝑠𝑟𝜋_𝑘𝑓^𝑟_𝐿𝑝1⟨𝑘⟩^{(𝛼−2)1/𝑟}=𝑓_𝜇𝑝1^{𝑠+(2−𝛼)/𝑟}_{,𝑟(ℝ𝑛)}. (173) The theorem is proved.

5.3. Schrödinger Equation

Consider the Cauchy problem of the linear free Schrödinger equation 𝑖𝜕_𝑡𝑢=Δ_𝑥𝑢,(𝑡,𝑥)∈ℝ₊×ℝ^𝑛𝑢(0,𝑥)=𝑢₀(𝑥). (174)

The formal solution to this equation is given by 𝑢()=1𝑡,𝑥(2𝜋)^𝑛_ℝ^𝑛𝑒^{𝑖𝑡|𝜉|2}𝑢₀(𝜉)𝑒^{2𝜋𝑖𝜉⋅𝑥}𝑑𝜉=𝑒^{−𝑖𝑡Δ}𝑢₀(𝑥). (175) By Theorem 1, we obtain the growth rate as 𝑡→∞ for the solution to the linear free Schrödinger equation.

Theorem 25. Let 𝑢(𝑡,𝑥) be the solution of the above Cauchy problem of the Schrödinger equation. For 0<𝑝≤∞, one has ‖𝑢(𝑡,⋅)‖_𝑀^𝑠_{𝑝,𝑞(ℝ}^𝑛₎⪯1+𝑡^{𝑛(1/𝑝−1/2)}𝑢_{0𝑀𝑠𝑝,𝑞(ℝ𝑛)}, (176) where the asymptotic factor 𝑡^{𝑛(1/𝑝−1/2)} is sharp as 𝑡→∞.

5.4. Linear Cauchy Problem with Negative Power

We start with the following linear Cauchy problem with negative power: 𝑖𝜕_𝑡𝑢+Δ_𝑥^−𝛼/2𝑢=0,(𝑡,𝑥)∈ℝ₊×ℝ^𝑛𝑢(),𝛼>0,0,𝑥=𝑢₀(𝑥). (177)

The formal solution to this equation is given by 𝑢()=1𝑡,𝑥(2𝜋)^𝑛_ℝ^𝑛𝑒^{𝑖𝑡|𝜉|𝛼}𝑢₀(𝜉)𝑒^{2𝜋𝑖𝜉⋅𝑥}𝑑𝜉=𝑒^{−𝑖𝑡|Δ|−𝛼/2}𝑢₀(𝑥). (178)

Proposition 26. Let 𝛼>0 and 1≤𝑝≤2. One has 𝜋₀𝑒^{−𝑖𝑡|Δ|−𝛼/2}𝑓_{𝐿𝑝(ℝ𝑛)}⪯(1+𝑡)^{([𝑛/2]+1)|2/𝑝−1|}_{|𝑗|∞≤1}𝜋_𝑗𝑓_{𝐻𝑟(ℝ𝑛)}, (179) where 1𝑟>𝑛+𝛼([𝑛/2]+1)2−𝑝.𝑛𝑝 (180)

Proof. We only prove the case of odd 𝑛, since the proof for even 𝑛 is similar. For any fixed 𝑛𝛽>𝛼2+1, (181) we write 𝜋₀𝑒^{−𝑖𝑡|Δ|−𝛼/2}𝑓(𝑥)=1(2𝜋)^𝑛_ℝ^𝑛𝜎(𝜉)𝜉^𝛽𝑒^{𝑖𝑡|𝜉|−𝛼}𝜉^−𝛽𝑓(𝜉)𝑒^{2𝜋𝑖𝜉⋅𝑥}𝑑𝜉≃𝜋₀|Δ|^𝛽/2𝑒^{−𝑖𝑡|Δ|−𝛼/2}𝐼_𝛽𝑓, (182) where 𝐼_𝛽 is the Riesz potential of order 𝛽. The kernel of 𝜋₀|Δ|^𝛽/2𝑒^{−𝑖𝑡|Δ|−𝛼/2} is Ω_𝛼,𝛽(𝑥)=_ℝ𝑛𝜉^𝛽𝑒^{𝑖𝑡|𝜉|−𝛼}𝜎(𝜉)𝑒^{2𝜋𝑖𝜉⋅𝑥}𝑑𝜉. (183) We first show Ω_𝛼,𝛽∗𝑓_𝐿1⪯(1+𝑡)^[𝑛2^]+1𝑓_𝐿1. (184) To this end, by Young's inequality, we need to show Ω_{𝛼,𝛽𝐿1(ℝ𝑛)}⪯(1+𝑡)^[𝑛/2]+1. (185) It suffices to show the case 𝑡≥1. As the same argument in the proof of Theorem 1, with the Schwarz inequality we have Ω_{𝛼,𝛽𝐿1(ℝ𝑛)}⪯𝑡^𝑛/2+_{|𝑥|>𝑡ℝ𝑛}𝜉^𝛽𝑒^{𝑖𝑡|𝜉|−𝛼}𝜎(𝜉)𝑒^{2𝜋𝑖𝜉⋅𝑥}𝑑𝜉𝑑𝑥. (186) Performing integration by parts (𝑛+1)/2 times on the second term, without loss of generality, we may write _{|𝑥|>𝑡ℝ𝑛}𝜉^𝛽𝑒^{𝑖𝑡|𝜉|−𝛼}𝜎(𝜉)𝑒^{2𝜋𝑖𝜉⋅𝑥}𝑑𝜉𝑑𝑥⪯𝑡^(𝑛+1)/2_|𝑥|>𝑡1|𝑥|^{(𝑛+𝜀)/2}×1|𝑥|^{(1−𝜀)/2ℝ𝑛}𝜓(𝜉)𝑒^{𝑖𝑡|𝜉|−𝛼}𝜎(𝜉)𝑒^{2𝜋𝑖𝜉⋅𝑥}𝑑𝜉𝑑𝑥, (187) where 𝜎(𝜉) is a 𝐶^∞ function supported in [−1,1]^𝑛 and 𝜓(𝜉) is a function satisfying 𝜓(𝜉)⪯𝜉^{𝛽−((𝑛+1)𝛼/2)−((𝑛+1)/2)}. (188) Choose a small 𝜀>0 such 2𝛽−𝛼(𝑛+1)−𝜀>0. (189) Thus by Schwarz's inequality and the Pitt's theorem, we obtain _{|𝑥|>𝑡ℝ𝑛}𝜉^𝛽𝑒^{𝑖𝑡|𝜉|−𝛼}𝜎(𝜉)𝑒^{2𝜋𝑖𝜉⋅𝑥}𝑑𝜉𝑑𝑥⪯𝑡^(𝑛+1)/2_ℝ𝑛1|𝑥|^{(1−𝜀)/2ℝ𝑛}𝜓(𝜉)𝑒^{𝑖𝑡|𝜉|−𝛼}𝜎(𝜉)×𝑒^{2𝜋𝑖𝜉⋅𝑥}𝑑𝜉²𝑑𝑥^1/2,𝑡^(𝑛+1)/2_ℝ𝑛𝜎(𝜉)²𝜉^{−𝑛−𝜀}𝜉^{2𝛽−𝛼(𝑛+1)}𝑑𝜉^1/2⪯𝑡^(𝑛+1)/2. (190) Combining these estimates, we have 𝜋₀𝑒^{−𝑖𝑡|Δ|−𝛼/2}𝑓_𝐿1⪯𝑡^(𝑛+1)/2𝐼_𝛽𝑓_𝐿1⪯𝑡^(𝑛+1)/2𝑓_𝐿1−𝛽, (191) where 𝐿¹_−𝛽 denotes the 𝐿¹ Sobolev space of order −𝛽. On the other hand, we have the easy energy estimate 𝜋₀𝑒^{−𝑖𝑡|Δ|−𝛼/2}𝑓_𝐿2⪯𝑓_𝐿2. (192) An interpolation yields that for all 1≤𝑝≤2, 𝜋₀𝑒^{−𝑖𝑡|Δ|−𝛼/2}𝑓_𝐿𝑝⪯𝑡^{(𝑛+1)(1/𝑝−1/2)}𝑓_𝐿^𝑝_−𝛾, (193) where 2𝛾=𝛽𝑝−1. (194) We now use the Sobolev imbedding theorem and the almost orthogonality of {𝜋_𝑘} to obtain 𝜋₀𝑒^{−𝑖𝑡|Δ|−𝛼/2}𝑓_{𝐿𝑝(ℝ𝑛)}⪯𝑡^{(𝑛+1)(1/𝑝−1/2)}_{|𝑗|∞≤1}𝜋_𝑗𝑓_{𝐻𝑟(ℝ𝑛)} (195) with 1𝑟=1𝑝+𝛽2/𝑝−1𝑛=𝑛+𝛽2−𝑝.𝑛𝑝 (196) Since 𝛽 is an arbitrary number larger than 𝛼([𝑛/2]+1), the proposition now follows from Lemma 13.

Lemma 27. Let 𝛼>0 and 2≤𝑝≤∞. One has 𝜋₀𝑒^{−𝑖𝑡|Δ|−𝛼/2}𝑓_{𝐿2(ℝ𝑛)}⪯_{|𝑗|∞≤1}𝜋_𝑗𝑓_{𝐻𝑟(ℝ𝑛)}, (197) for any 𝑟≤2.

Proof. The almost orthogonality (Identity (46)) and the energy estimate give 𝜋₀𝑒^{−𝑖𝑡|Δ|−𝛼/2}𝑓_{𝐿2(ℝ𝑛)}⪯_{|𝑗|∞≤1}𝜋_𝑗𝑓_{𝐿2(ℝ𝑛)}. (198) Thus the lemma follows from Lemma 13.

Now we are ready to give the proof of Theorem 5.

Proof. The proof of (i) can be obtained from Propositions 18 and 26, and the definition and the modulation spaces. Similarly, the proof of (ii) follows by Proposition 18, Lemma 27 and the definition of the modulation spaces.

5.5. Nonlinear Cauchy Problem with Negative Power

Now, we study the following Cauchy problem of the nonlinear dispersive equations (NDE): 𝑖𝜕𝑢−|Δ|𝜕𝑡^−𝛼/2(𝑢)𝑢()𝑢+𝐹=0,0,𝑥=𝑢₀(𝑥), (199) where 𝐹(𝑢)=|𝑢|^2𝑘𝑢 for some positive integer 𝑘. For 𝑇>0, the space 𝑌=𝐶([0,𝑇],𝑀^𝑠_𝑝,1) is defined by 𝐶[0,𝑇],𝑀^𝑠_𝑝,1=𝑢(𝑡,𝑥)∶‖𝑢‖_{𝐶([0,𝑇],𝑀}^𝑠_𝑝,1)=sup_{0≤𝑡≤𝑇}‖𝑢(𝑡,⋅)‖_𝑀^𝑠_𝑝,1<∞. (200)

Our proof will follow the same method used in [8], or, more precisely, the idea introduced in an earlier paper [13].

Now we give the proof of Theorem 6.

Proof. In the proof, the letters 𝐶_𝑗, 𝑗=1,2,3 denote some positive constants that are independent of all essential variables. We write the Cauchy problem in the equivalent form 𝑢(𝑡,⋅)=𝑒^{i𝑡|Δ|−𝛼/2}𝑢₀−𝑖^𝑡₀𝑒^{𝑖(𝑡−𝜏)|Δ|−𝛼/2}𝐹(𝑢(𝜏,⋅))𝑑𝜏, (201) and consider the mapping ℑ𝑢=𝑒^{𝑖𝑡|Δ|−𝛼/2}𝑢₀−𝑖^𝑡₀𝑒^{𝑖(𝑡−𝜏)|Δ|−𝛼/2}𝐹(𝑢(𝜏,⋅))𝑑𝜏. (202) We want to show that ℑ is a contraction.
By Theorem 5 and Lemma 12, 𝑒^{𝑖𝑡|Δ|−𝛼/2}𝑢_0𝑌≤𝐶₁(1+𝑇)^{([𝑛/2]+1)|2/𝑝−1|}𝑢_0𝑀^𝑠−𝛾_{𝑟,1(ℝ𝑛)}+𝐶₁𝑢_{0𝑀𝑠𝑝,1(ℝ𝑛)}. (203) By Lemma 16, there is a constant 𝐴_2𝑘+1>0 for which we have |𝑢(𝑡,⋅)|^2𝑘𝑢(𝑡,⋅)_{𝑀𝑠𝑟,1}≤𝐴_2𝑘+1‖𝑢(𝑡,⋅)‖_𝑀^2𝑘+1𝑠_{(2𝑘+1)𝑟,1}. (204) Thus, by Theorem 5 and Lemma 12, ^𝑡₀𝑒^{𝑖(𝑡−𝜏)|Δ|−𝛼/2}𝐹(𝑢(𝜏,⋅))𝑑𝜏_{𝐶([0,𝑇],𝑀𝑠𝑝,1)}⪯sup_{0<𝑡≤𝑇}^𝑡₀𝑒^{𝑖(𝑡−𝜏)|Δ|−𝛼/2}𝐹(𝑢(𝜏,⋅))_{𝑀𝑠𝑝,1}𝑑𝜏≤𝐶₁sup_{0<𝑡≤𝑇}^𝑡₀(1+(𝑡−𝜏))^{([𝑛/2]+1)|2/𝑝−1|}×|𝑢(𝜏,⋅)|^2𝑘𝑢(𝜏,⋅)_{𝑀𝑠𝑟,1}𝑑𝜏≤𝐶₂𝑀_𝑘𝑇(1+𝑇)^{([𝑛/2]+1)|2/𝑝−1|}sup_{0≤𝑡≤𝑇}‖𝑢(𝑡,⋅)‖_𝑀^2𝑘+1𝑠_{𝑟(2𝑘+1),1}⪯𝐶₂𝑀_𝑘𝑇(1+𝑇)^{([𝑛/2]+1)|2/𝑝−1|}sup_{0≤𝑡≤𝑇}‖𝑢(𝑡,⋅)‖_𝑀^2𝑘+1𝑠_𝑝,1. (205) The last inequality is because 𝑟(2𝑘+1)≥𝑝 (206) and the imbedding relation 𝑀^𝑠_𝑝,1⊂𝑀^𝑠_{𝑟(2𝑘+1),1}.
As a consequence, ‖ℑ𝑢‖_{𝐶([0,𝑇],𝑀}^𝑠_𝑝,1)=sup_{0≤𝑡≤𝑇}‖ℑ𝑢(𝑡,⋅)‖_𝑀^𝑠_𝑝,1⪯𝐶₃(1+𝑇)^{([𝑛/2]+1)|2/𝑝−1|}×𝑢_0𝑀^𝑠−𝛾_{𝑟,1(ℝ𝑛)}+𝑇sup_{0≤𝑡≤𝑇}‖𝑢(𝑡,⋅)‖_𝑀^2𝑘+1𝑠_𝑝,1+𝐶₁𝑢_{0𝑀𝑠𝑝,1(ℝ𝑛)}. (207) Fix an 𝐿>0 such that 𝐶₃𝑢_0𝑀^𝑠−𝛾_{𝑟,1(ℝ𝑛)}+𝐶₁𝑢_{0𝑀𝑠𝑝,1(ℝ𝑛)}<𝐿4. (208) Let 𝐵_𝐿 be the closed ball of radial 𝐿 centered at the origin in the space 𝐶([0,𝑇],𝑀^𝑠_𝑝,1). Now we choose 𝑇₀ such that for any 𝑢∈𝐵_𝐿and 𝑇∈(0,𝑇₀], ‖ℑ𝑢‖_{𝐶([0,𝑇],𝑀}^𝑠_𝑝,1)≤𝐶₃(1+𝑇)^{([𝑛/2]+1)|2/𝑝−1|}𝑢_0𝑀^𝑠−𝛾_{𝑟,1(ℝ𝑛)}+𝑇𝐿^2𝑘+1+𝐶₁𝑢_{0𝑀𝑠𝑝,1(ℝ𝑛)}≤𝐿. (209) By this choice, ℑ is a mapping form 𝐵_𝐿into 𝐵_𝐿.
Furthermore, an easy computation gives that ‖ℑ𝑢−ℑ𝑣‖_{𝐶([0,𝑇],𝑀}^𝑠_𝑝,1)≤𝐶₃(1+𝑇)^{([𝑛/2]+1)|2/𝑝−1|}×𝑇sup_{0≤𝑡≤𝑇}‖𝑢(𝑡,⋅)−𝑣(𝑡,⋅)‖_𝑀^𝑠_𝑝,1(2𝑘+1)𝐿^2𝑘≤12‖𝑢−𝑣‖_{𝐶([0,𝑇],𝑀}^𝑠_𝑝,1), (210) for all 𝑇∈(0,𝑇₀] if 𝑇₀ is suitably chosen. Finally, using the Banach contraction mapping theorem, we obtain a fixed point in 𝐵_𝐿. This is the solution of the Cauchy problem. We prove the theorem.

Using a similar argument with the help of (ii) in Theorem 5, we can prove Theorem 7.

Acknowledgment

This work is partially supported by the NSF of China (Grants nos. 10931001, 10871173, 11201103).