The Initial Value Problem for the Quadratic Nonlinear Klein-Gordon Equation

Abstract

We study the initial value problem for the quadratic nonlinear Klein-Gordon equation ℒ𝑢=⟨𝑖𝜕_𝑥⟩⁻¹𝑢², (𝑡,𝑥)∈𝐑×𝐑, 𝑢(0,𝑥)=𝑢₀(𝑥), 𝑥∈𝐑, where ℒ=𝜕_𝑡+𝑖⟨𝑖𝜕_𝑥⟩ and ⟨𝑖𝜕_𝑥⟩=1−𝜕²_𝑥. Using the Shatah normal forms method, we obtain a sharp asymptotic behavior of small solutions without the condition of a compact support on the initial data which was assumed in the previous works.

1. Introduction

Let us consider the Cauchy problem for the nonlinear Klein-Gordon equation with a quadratic nonlinearity in one dimensional case

ℒ𝑢=𝜆⟨𝑖𝜕_𝑥⟩⁻¹𝑢²,(𝑡,𝑥)∈𝐑×𝐑,𝑢(0,𝑥)=𝑢₀(𝑥),𝑥∈𝐑, (1.1) where 𝜆∈𝐂,ℒ=𝜕_𝑡+𝑖⟨𝑖𝜕_𝑥⟩, and ⟨𝑖𝜕_𝑥⟩=1−𝜕²_𝑥.

Our purpose is to obtain the large time asymptotic profile of small solutions to the Cauchy problem (1.1) without the restriction of a compact support on the initial data which was assumed in the previous work [1]. One of the important tools of paper [1] was based on the transformation of the equation by virtue of the hyperbolic polar coordinates following to paper [2]. The application of the hyperbolic polar coordinates implies the restriction to the interior of the light cone, and therefore, requires the compactness of the initial data. Problem (1.1) is related to the Cauchy problem

𝑣_𝑡𝑡+𝑣−𝑣_𝑥𝑥=𝜇𝑣²,(𝑡,𝑥)∈𝐑×𝐑,𝑣(0,𝑥)=𝑣₀(𝑥),𝑣_𝑡(0,𝑥)=𝑣₁(𝑥),𝑥∈𝐑, (1.2) where 𝑣₀ and 𝑣₁ are the real-valued functions, and 𝜇∈𝐑. Indeed we can put 𝑢=(1/2)(𝑣+𝑖⟨𝑖𝜕_𝑥⟩⁻¹𝑣_𝑡), then 𝑢 satisfies

𝑖ℒ𝑢=2𝜇⟨𝑖𝜕_𝑥⟩⁻¹(𝑢+𝑢)²,(𝑡,𝑥)∈𝐑×𝐑,𝑢(0,𝑥)=𝑢₀(𝑥),𝑥∈𝐑, (1.3) where 𝑢₀=(1/2)(𝑣₀+𝑖⟨𝑖𝜕_𝑥⟩⁻¹𝑣₁).

There are a lot of works devoted to the study of the cubic nonlinear Klein-Gordon equation

𝑣_𝑡𝑡+𝑣−𝑣_𝑥𝑥=𝜇𝑣³,(𝑡,𝑥)∈𝐑×𝐑,𝑣(0,𝑥)=𝑣₀(𝑥),𝑣_𝑡(0,𝑥)=𝑣₁(𝑥),𝑥∈𝐑 (1.4) with 𝜇∈𝐑. When 𝜇<0, the global existence of solutions to (1.4) can be easily obtained in the energy space, which is, however, insufficient for determining the large-time asymptotic behavior of solutions. The sharp 𝐋^∞- time decay estimates of solutions and nonexistence of the usual scattering states for (1.4) were shown in [3] by using hyperbolic polar coordinates under the conditions that the initial data are sufficiently regular and have a compact support.

The initial value problem for the nonlinear Klein-Gordon equation with various cubic nonlinearities depending on 𝑣,𝑣_𝑡,𝑣_𝑥,𝑣_𝑥𝑥,𝑣_𝑡𝑥 and having a suitable nonresonance structure was studied in [4–6], where small solutions were found in the neighborhood of the free solutions when the initial data are small and regular and decay rapidly at infinity. Hence the cubic nonlinearities are not necessarily critical; however the resonant nonlinear term 𝑣³ was excluded in these works. In paper [4], the nonresonant nonlinearities were classified into two types, one of them can be treated by the nonlinear transformation which is different from the method of normal forms [7] and the other reveals an additional time decay rate via the operator 𝑥𝜕_𝑡+𝑡𝜕_𝑥 which was used in [2]. This nonlinear transformation was refined in [8] and applied to a system of nonlinear Klein-Gordon equations in one or two space dimensions with nonresonant nonlinearities. It seems that the method of normal forms is very useful in the case of a single equation; however it does not work well in the case of a system of nonlinear Klein-Gordon equations. Some sufficient conditions on quadratic or cubic nonlinearities were given in [1], which allow us to prove global existence and to find sharp asymptotics of small solutions to the Cauchy problem including (1.2) with small and regular initial data having a compact support. Moreover it was proved that the asymptotic profile differs from that of the linear Klein-Gordon equation. See also [9, 10] in which asymptotic behavior of solutions to (1.4) was studied as in [1] by using hyperbolic polar coordinates. Compactness condition on the data was removed in [11] in the case of the cubic nonlinearity 𝑣³ and a real-valued solution. Final value problem with the cubic nonlinearity was studied in [12] for a real-valued solution. As far as we know the problem of finding the large-time asymptotics is still open for the case of the cubic nonlinearity 𝑣³ and the complex valued initial data. When the initial data are complex-valued, global existence and 𝐋^∞-time decay estimates of small solutions to the Klein-Gordon equation with cubic nonlinearity |𝑣|²𝑣 were obtained in paper [13] under the conditions that the initial data are smooth and have a compact support.

The scattering problem and the time decay rates of small solutions to (1.4) with super-critical nonlinearities |𝑣|^𝑝−1𝑣 and |𝑣|^𝑝 with 𝑝>3 were studied in papers of [14, 15]. Finally, we note that the Klein-Gordon equation (1.4) with quadratic nonlinearities in two space dimensions was studied in [16], where combining the method of the normal forms of [7] and the time decay estimate through the operator 𝑥𝜕_𝑡+𝑡𝜕_𝑥 of [17], it was shown that every quadratic nonlinearity is nonresonant.

We denote the Lebesgue space by 𝐋^𝑝={𝜙∈𝒮^′;‖𝜙‖_𝐋𝑝<∞}, with the norm ‖𝜙‖_𝐋𝑝=(∫_𝐑|𝜙(𝑥)|^𝑝𝑑𝑥)^1/𝑝 if 1≤𝑝<∞ and ‖𝜙‖_𝐋∞=sup_𝑥∈𝐑|𝜙(𝑥)| if 𝑝=∞. The weighted Sobolev space is

𝐇_𝑝^𝑚,𝑠={𝜙∈𝐋^𝑝;‖⟨𝑥⟩^𝑠⟨𝑖𝜕_𝑥⟩^𝑚𝜙‖_𝐋𝑝<∞}, (1.5) for 𝑚,𝑠∈𝐑, 1≤𝑝≤∞, where ⟨𝑥⟩=√1+𝑥². For simplicity we write 𝐇^𝑚,𝑠=𝐇₂^𝑚,𝑠. The index 0 we usually omit if it does not cause a confusion. We denote by ℱ𝜙=ℱ_𝑥→𝜉𝜙≡𝜙=1/√2𝜋∫_𝐑𝑒^{−𝑖𝑥𝜉}𝜙(𝑥)𝑑𝑥 the Fourier transform of the function 𝜙. Then the inverse Fourier transformation is ℱ⁻¹𝜙=ℱ⁻¹_𝜉→𝑥𝜙=(1/√2𝜋)∫_𝐑𝑒^𝑖𝑥𝜉𝜙(𝜉)𝑑𝜉.

Our main result of this paper is the following.

Theorem 1.1. Let 𝑢₀∈𝐇^3,1 and the norm ‖𝑢₀‖_𝐇^3,1=𝜀. Then there exists 𝜀₀>0 such that for all 0<𝜀<𝜀₀ the Cauchy problem (1.1) has a unique global solution 𝑢(𝑡)∈𝐂([0,∞);𝐇^3,1) (1.6) satisfying the time decay estimate ‖𝑢(𝑡)‖_𝐇∞^1,0≤𝐶𝜀(1+𝑡)^−1/2. (1.7) Furthermore there exists a unique final state 𝑊₊∈𝐇_∞^0,1∩𝐇^0,1 such that ‖𝑢(𝑡)−𝑒^{−𝑖⟨𝑖𝜕𝑥⟩𝑡}ℱ⁻¹𝑊₊𝑒^{−2𝑖|𝜆|2Ω|}𝑊₊^|2log𝑡‖_𝐇^1,0≤𝐶𝜀^3/2𝑡^𝛾−1/4,‖ℱ𝑒^{𝑖⟨𝑖𝜕𝑥⟩𝑡}𝑢(𝑡)−𝑊₊𝑒^{−2𝑖|𝜆|2Ω|}𝑊_+|²_log𝑡‖_𝐇∞^0,1≤𝐶𝜀^3/2𝑡^𝛾−1/4, (1.8) where 𝛾∈(0,1/4), Ω(𝜉)≡⟨𝜉⟩²/⟨2𝜉⟩(2⟨𝜉⟩+⟨2𝜉⟩).

An important tool for obtaining the time decay estimates of solutions to the nonlinear Klein-Gordon equation is implementation of the operator

𝒥=⟨𝑖𝜕_𝑥⟩𝑒^{−𝑖⟨𝑖𝜕𝑥⟩𝑡}𝑥𝑒^{𝑖⟨𝑖𝜕𝑥⟩𝑡}=ℱ⁻¹⟨𝜉⟩𝑒^{−𝑖⟨𝜉⟩𝑡}𝑖𝜕_𝜉𝑒^{𝑖⟨𝜉⟩𝑡}ℱ=⟨𝑖𝜕_𝑥⟩𝑥+𝑖𝑡𝜕_𝑥, (1.9) which is analogous to the operator 𝑥+𝑖𝑡𝜕_𝑥=𝑒^{(−𝑖𝑡/2)𝜕2𝑥}𝑥𝑒^{(𝑖𝑡/2)𝜕2𝑥} in the case of the nonlinear Schrödinger equations used in [18]. The operator 𝒥 was used previously in paper of [15] for constructing the scattering operator for nonlinear Klein-Gordon equations with a supercritical nonlinearity. We have [𝑥,⟨𝑖𝜕_𝑥⟩^𝛼]=𝛼⟨𝑖𝜕_𝑥⟩^𝛼−2𝜕_𝑥; therefore the commutator [ℒ,𝒥]=ℒ𝒥−𝒥ℒ=0, where ℒ=𝜕_𝑡+𝑖⟨𝑖𝜕_𝑥⟩ is a linear part of (1.1). Since 𝒥 is not a purely differential operator, it is apparently difficult to calculate the action of 𝒥 on the nonlinearity in (1.1). So, instead we use the first-order differential operator

𝒫=𝑡𝜕_𝑥+𝑥𝜕_𝑡 (1.10) which is closely related to 𝒥 by the identity 𝒫=ℒ𝑥−𝑖𝒥 and acts easily on the nonlinearity. Moreover, it almost commutes with ℒ, since [ℒ,𝒫]=−𝑖⟨𝑖𝜕_𝑥⟩⁻¹𝜕_𝑥ℒ.

Also we use the method of normal forms of [7] by which we transform the quadratic nonlinearity into a cubic one with a nonlocal operator. We multiply both sides of equation (1.1) by the free Klein-Gordon evolution group ℱ𝒰(−𝑡)=ℱ𝑒^{𝑖𝑡⟨𝑖𝜕𝑥⟩}=𝑒^{𝑖𝑡⟨𝜉⟩}ℱ and put 𝑣(𝑡,𝜉)=𝑒^{𝑖𝑡⟨𝜉⟩}𝑢 to get

𝑣_𝑡(𝑡,𝜉)=𝜆𝑒^{𝑖𝑡⟨𝜉⟩}ℱ(⟨𝑖𝜕_𝑥⟩⁻¹𝑢²𝜆)=√2𝜋⟨𝜉⟩^𝐑𝑒^𝑖𝑡𝐴𝑣(𝑡,𝜂−𝜉)𝑣(𝑡,−𝜂)𝑑𝜂, (1.11) where 𝐴(𝜉,𝜂)=⟨𝜉⟩+⟨𝜉−𝜂⟩+⟨𝜂⟩. Integrating (1.11) with respect to time, we find

𝜆𝑣(𝑡,𝜉)=𝑣(0,𝜉)+√2𝜋⟨𝜉⟩^𝑡0𝑑𝜏_𝐑𝑒^𝑖𝜏𝐴𝑣(𝜏,𝜂−𝜉)𝑣(𝜏,−𝜂)𝑑𝜂. (1.12) Then we integrate by parts with respect to 𝜏, taking into account (1.11),

𝑣(𝑡,𝜉)+𝑖𝜆_𝐑𝑒^𝑖𝑡𝐴𝑣(𝑡,𝜂−𝜉)𝑣(𝑡,−𝜂)𝑑𝜂√2𝜋⟨𝜉⟩𝐴(𝜉,𝜂)=𝑣(0,𝜉)+𝑖𝜆_𝐑𝑣(0,𝜂−𝜉)𝑣(0,−𝜂)𝑑𝜂√2𝜋⟨𝜉⟩𝐴(𝜉,𝜂)−2𝑖|𝜆|^2𝑡₀𝑑𝜏_𝐑𝑑𝜂√𝑒2𝜋⟨𝜉⟩𝐴(𝜉,𝜂)^{𝑖𝜏(𝐴−⟨𝜂⟩)}𝑣(𝜏,𝜂−𝜉)ℱ_𝑥→𝜂(⟨𝑖𝜕_𝑥⟩⁻¹𝑢²). (1.13) Returning to the function 𝑢(𝑡,𝑥)=𝒰(𝑡)ℱ⁻¹_𝜉→𝑥𝑣=ℱ⁻¹_𝜉→𝑥(𝑒^{−𝑖𝑡⟨𝜉⟩}𝑣(𝑡,𝜉)), we obtain the following equation:

ℒ(𝑢+𝑖𝜆𝒢(𝑢,𝑢))=−2𝑖|𝜆|²𝒢(𝑢,⟨𝑖𝜕_𝑥⟩⁻¹𝑢²), (1.14) with the symmetric bilinear operator

𝒢(𝜙,𝜓)=ℱ⁻¹_{𝜉→𝑥𝐑}𝑔(𝜉−𝜂,𝜂)𝜙(𝜉−𝜂)𝜓(𝜂)𝑑𝜂, (1.15) where

1𝑔(𝜁,𝜂)=√,2𝜋⟨𝜁+𝜂⟩(⟨𝜁+𝜂⟩+⟨𝜁⟩+⟨𝜂⟩) (1.16) and 𝜁=𝜉−𝜂. Our main point in this paper is to show that the right-hand side of (1.14) can be decomposed into two terms; one of them is a cubic nonlinearity

−2𝑖|𝜆|²1𝑡𝒰(𝑡)ℱ⁻¹Ω(𝜉)|ℱ𝒰(−𝑡)𝑢(𝑡,𝜉)|²ℱ𝒰(−𝑡)𝑢(𝑡,𝜉), (1.17) and the other one is a remainder term with an estimate like 𝑂(𝑡^−5/4‖ℱ𝒰(−𝑡)𝑢(𝑡)‖³_𝐇1,3).

Remark 1.2. We believe that all quadratic nonlinear terms 𝑢²,𝑢²,|𝑢|² of problem (1.3) also could be considered by this approach. In the same way as in the derivation of (1.14) we get from (1.3) 𝑖ℒ(𝑢+2𝜇(𝒢(𝑢,𝑢)+𝒢₁(𝑢,𝑢)+2𝒢₂(𝑢,𝑢)))=𝑖𝜇²𝒢(𝑢,⟨𝑖𝜕_𝑥⟩⁻¹(𝑢+𝑢)²)+𝑖𝜇²𝒢₁(𝑢,⟨𝑖𝜕_𝑥⟩⁻¹(𝑢+𝑢)²)+𝑖𝜇²𝒢₂(𝑢+𝑢,⟨𝑖𝜕_𝑥⟩⁻¹(𝑢+𝑢)²), (1.18) where 𝒢_𝑗(𝜙,𝜓)=ℱ⁻¹_{𝜉→𝑥𝐑}𝑔_𝑗𝑔(𝜉−𝜂,𝜂)𝜙(𝜉−𝜂)𝜓(𝜂)𝑑𝜂,_𝑗1(𝜁,𝜂)=√2𝜋𝐴_𝑗(𝜁+𝜂,𝜂)⟨𝜁+𝜂⟩ (1.19) with 𝐴₁(𝜉,𝜂)=⟨𝜉⟩−⟨𝜉−𝜂⟩−⟨𝜂⟩,𝐴₂(𝜉,𝜂)=⟨𝜉⟩−⟨𝜉−𝜂⟩+⟨𝜂⟩. Some more regularity conditions are necessary to treat the bilinear operators 𝒢_𝑗. Also we have to show that 𝒢(𝑢,⟨𝑖𝜕_𝑥⟩⁻¹(𝑢²+2|𝑢|²)),𝒢₁(𝑢,⟨𝑖𝜕_𝑥⟩⁻¹(𝑢²+𝑢²𝒢)),₂(𝑢,⟨𝑖𝜕_𝑥⟩⁻¹(𝑢²+𝑢²)),𝒢₂(𝑢,⟨𝑖𝜕_𝑥⟩⁻¹(𝑢²+2|𝑢|²)) (1.20) are the nonresonant terms (i.e., remainders) and to remove the resonant terms 𝒢(𝑢,⟨𝑖𝜕_𝑥⟩⁻¹𝑢²),𝒢₁(𝑢,⟨𝑖𝜕_𝑥⟩⁻¹|𝑢|²),𝒢₂(𝑢,⟨𝑖𝜕_𝑥⟩⁻¹|𝑢|²),𝒢₂(𝑢,⟨𝑖𝜕_𝑥⟩⁻¹𝑢²) (1.21) by an appropriate phase function. We will dedicate a separate paper to this problem.

We prove our main result in Section 3. In the next section we prove several lemmas used in the proof of the main result.

2. Preliminaries

First we give some estimates for the symmetric bilinear operator

𝒢(𝜙,𝜓)=ℱ⁻¹_{𝜉→𝑥𝐑}𝑔(𝜉−𝜂,𝜂)𝜙(𝜉−𝜂)𝜓(𝜂)𝑑𝜂, (2.1) where

1𝑔(𝜁,𝜂)=√.2𝜋⟨𝜁+𝜂⟩(⟨𝜁+𝜂⟩+⟨𝜁⟩+⟨𝜂⟩) (2.2) Denote the kernel as follows:

1𝑔(𝑦,𝑧)=4𝜋²_𝐑²𝑒^{𝑖𝑦𝜁+𝑖𝑧𝜂}(⟨𝜁+𝜂⟩+⟨𝜁⟩+⟨𝜂⟩)⟨𝜁+𝜂⟩𝑑𝜂𝑑𝜁. (2.3)

Lemma 2.1. The representation is true 𝒢(𝜙,𝜓)=_𝐑2𝑔(𝑦,𝑧)𝜙(𝑥−𝑦)𝜓(𝑥−𝑧)𝑑𝑦𝑑𝑧, (2.4) where the kernel 𝑔(𝑦,𝑧) obeys the following estimate: |𝑔(𝑦,𝑧)|≤𝐶⟨𝑦⟩⁻³⟨𝑧⟩⁻³ln²1(1+)|𝑧−𝑦| (2.5) for all 𝑦,𝑧∈𝐑,𝑦≠𝑧. Moreover the following estimates are valid: ‖𝒢(𝜙,𝜓)‖_𝐋𝑝≤𝐶‖𝜙‖_𝐋𝑞‖𝜓‖_𝐋𝑟,‖𝒫𝒢(𝜙,𝜓)‖_𝐋𝑝≤𝐶‖𝒫𝜙‖_𝐋𝑞‖𝜓‖_𝐋𝑟+𝐶‖𝒫𝜓‖_𝐋𝑞‖𝜙‖_𝐋𝑟+𝐶‖𝜕_𝑡𝜙‖_𝐋𝑞‖𝜓‖_𝐋𝑟+𝐶‖𝜕_𝑡𝜓‖_𝐋𝑞‖𝜙‖_𝐋𝑟 (2.6) for 1≤𝑝≤∞,1<𝑞≤𝑝/𝛼,1<𝑟≤𝑝/(1−𝛼),𝛼∈[0,1], provided that the right-hand sides are bounded.

Proof. To prove representation (2.4), we substitute the direct Fourier transforms 1𝜙(𝜁)=√2𝜋^𝐑𝑒^{−𝑖(𝑥−𝑦)𝜁}1𝜙(𝑥−𝑦)𝑑𝑦,𝜓(𝜂)=√2𝜋^𝐑𝑒^{−𝑖(𝑥−𝑧)𝜂}𝜓(𝑥−𝑧)𝑑𝑧 (2.7) into the definition of the operator 𝒢. Then changing 𝜁=𝜉−𝜂, we find 𝒢(𝜙,𝜓)=(2𝜋)^−3/2_𝐑2𝜙(𝑥−𝑦)𝜓(𝑥−𝑧)(_𝐑2𝑔(𝜁,𝜂)𝑒^{𝑖𝑦𝜁+𝑖𝑧𝜂}=𝑑𝜂𝑑𝜁)𝑑𝑦𝑑𝑧_𝐑2𝑔(𝑦,𝑧)𝜙(𝑥−𝑦)𝜓(𝑥−𝑧)𝑑𝑦𝑑𝑧, (2.8) where the kernel 𝑔(𝑦,𝑧) is 𝑔(𝑦,𝑧)=(2𝜋)^−3/2_𝐑2𝑔(𝜁,𝜂)𝑒^{𝑖𝑦𝜁+𝑖𝑧𝜂}=1𝑑𝜂𝑑𝜁4𝜋²_𝐑²𝑒^{𝑖𝑦𝜁+𝑖𝑧𝜂}(⟨𝜁+𝜂⟩+⟨𝜁⟩+⟨𝜂⟩)⟨𝜁+𝜂⟩𝑑𝜂𝑑𝜁. (2.9) Changing the variables of integration 𝜁=𝜉/2−𝜂^′ and 𝜂=𝜉/2+𝜂^′ (the prime we will omit), we get 1𝑔(𝑦,𝑧)=4𝜋²_𝐑²𝑒^{𝑖𝑦𝜁+𝑖𝑧𝜂}=1(⟨𝜁+𝜂⟩+⟨𝜁⟩+⟨𝜂⟩)⟨𝜁+𝜂⟩𝑑𝜂𝑑𝜁4𝜋²_𝐑𝑑𝜉𝑒⟨𝜉⟩^{𝑖((𝑦+𝑧)/2)𝜉}_𝐑𝐵(𝜉,𝜂)𝑒^{𝑖(𝑧−𝑦)𝜂}𝑑𝜂, (2.10) where 𝐵(𝜉,𝜂)=1/(⟨𝜉⟩+⟨𝜉/2−𝜂⟩+⟨𝜉/2+𝜂⟩). We change 𝑠=𝑦+𝑧/2 and 𝜌=(𝑧−𝑦) and denote 1𝑔(𝑠,𝜌)=4𝜋²_𝐑𝑑𝜉𝑒⟨𝜉⟩^𝑖𝑠𝜉_𝐑𝐵(𝜉,𝜂)𝑒^𝑖𝜌𝜂𝑑𝜂. (2.11) For the case of |𝜌|≤1,|𝑠|≤1 we integrate by parts using the identity 𝑒^𝑖𝜌𝜂=𝐴(𝜂)𝜕_𝜂(𝜂𝑒^𝑖𝜌𝜂), where 𝐴(𝜂)=1/(1+𝑖𝜌𝜂). Then we get 1𝑔(𝑠,𝜌)=−4𝜋²_𝐑𝑑𝜉𝑒⟨𝜉⟩^𝑖𝑠𝜉_𝐑𝑒^𝑖𝜌𝜂𝜂𝜕_𝜂(𝐴(𝜂)𝐵(𝜉,𝜂))𝑑𝜂. (2.12) Note that 𝜂𝜕_𝜂𝜂𝐵(𝜉,𝜂)=(⟨𝜉⟩+⟨𝜉/2−𝜂⟩+⟨𝜉/2+𝜂⟩)²(𝜂−𝜉/2+⟨𝜂−𝜉/2⟩𝜂+𝜉/2⟨𝜉/2+𝜂⟩). (2.13) Then |⟨𝜉⟩⁻¹𝜂𝜕_𝜂𝐶(𝐴(𝜂)𝐵(𝜉,𝜂))|≤.⟨𝜉⟩(1+|𝜌||𝜂|)(⟨𝜉⟩+⟨𝜂⟩) (2.14) Hence we can estimate the kernel 𝑔(𝑠,𝜌) as follows: |𝑔(𝑠,𝜌)|≤𝐶_𝐑𝑑𝜉⟨𝜉⟩_𝐑𝑑𝜂(1+|𝜌||𝜂|)(⟨𝜉⟩+⟨𝜂⟩)≤𝐶^∞₁𝑑𝜂1+|𝜌|𝜂^∞₁𝑑𝜉𝜉(𝜂+𝜉)=𝐶^∞₁𝑑𝜂11+|𝜌|𝜂𝜂𝜉(ln𝜂+𝜉)|^∞₁𝑑𝜉≤𝐶^∞₁ln𝜂(1+|𝜌|𝜂)𝜂𝑑𝜂≤𝐶₁^1/|𝜌|ln𝜂𝑑𝜂𝜂+𝐶|𝜌|^∞_1/|𝜌|ln𝜂𝑑𝜂𝜂²≤𝐶ln²1(1+)|𝜌| (2.15) for the case of |𝜌|≤1,|𝑠|≤1. For the case of |𝜌|≤1,|𝑠|≥1 we integrate three times by parts with respect to 𝜉𝑔(𝑠,𝜌)=𝐶|𝑠|⁻³_𝐑𝑑𝜉𝑒^𝑖𝑠𝜉_𝐑𝑒^𝑖𝜌𝜂𝜕³_𝜉(⟨𝜉⟩⁻¹𝜂𝜕_𝜂(𝐴(𝜂)𝐵(𝜉,𝜂)))𝑑𝜂. (2.16) Note that |𝜕³_𝜉(⟨𝜉⟩⁻¹𝜂𝜕_𝜂𝐶(𝐴(𝜂)𝐵(𝜉,𝜂)))|≤.⟨𝜉⟩(1+|𝜌||𝜂|)(⟨𝜉⟩+⟨𝜂⟩) (2.17) Hence we can estimate the kernel 𝑔(𝑦,𝑧) as follows: |𝑔(𝑠,𝜌)|≤𝐶|𝑠|⁻³_𝐑𝑑𝜉⟨𝜉⟩_𝐑𝑑𝜂(1+|𝜌||𝜂|)(⟨𝜉⟩+⟨𝜂⟩)≤𝐶|𝑠|⁻³ln²1(1+)|𝜌| (2.18) for all |𝜌|≤1,|𝑠|≥1. For the case |𝜌|≥1,|𝑠|≤1 we integrate by parts three times with respect to 𝜂𝑔(𝑠,𝜌)=𝐶|𝜌|⁻³_𝐑𝑑𝜉𝑒⟨𝜉⟩^𝑖𝑠𝜉_𝐑𝑒^𝑖𝜌𝜂𝜕³_𝜂𝐵(𝜉,𝜂)𝑑𝜂. (2.19) Note that |𝜕³_𝜂𝐵(𝜉,𝜂)|≤𝐶(⟨𝜉⟩+⟨𝜂⟩)⁻³. Hence |𝑔(𝑠,𝜌)|≤𝐶|𝜌|⁻³_𝐑𝑑𝜉⟨𝜉⟩^2𝐑𝑑𝜂⟨𝜂⟩²≤𝐶|𝜌|⁻³. (2.20) Finally for the case of |𝜌|≥1,|𝑠|≥1 we integrate by parts three times with respect to 𝜉 and 𝜂𝑔(𝑠,𝜌)=𝐶|𝑠𝜌|⁻³_𝐑𝑑𝜉𝑒^𝑖𝑠𝜉_𝐑𝑒^𝑖𝜌𝜂𝜕³_𝜉𝜕³_𝜂(⟨𝜉⟩⁻¹𝐵(𝜉,𝜂))𝑑𝜂. (2.21) Since |𝜕³_𝜉𝜕³_𝜂(⟨𝜉⟩⁻¹𝐶𝐵(𝜉,𝜂))|≤⟨𝜉⟩²(⟨𝜉⟩+⟨𝜂⟩)², (2.22) then we can estimate the kernel 𝑔(𝑠,𝜌) as follows: |𝑔(𝑠,𝜌)|≤𝐶|𝑠𝜌|⁻³_𝐑𝑑𝜉⟨𝜉⟩^2𝐑𝑑𝜂⟨𝜂⟩²≤𝐶|𝑠𝜌|⁻³ (2.23) for all |𝜌|≥1,|𝑠|≥1. Hence estimate (2.5) is true.
By virtue of estimate (2.5) applying the Hölder inequality with 1/𝑝=1/𝑝₁+1/𝑝₂ and the Young inequality with 1/𝑝₁+1=1/𝑞+1/𝑞₁ and 1/𝑝₂+1=1/𝑟+1/𝑟₁, we find ‖𝒢(𝜙,𝜓)‖_𝐋𝑝≤𝐶‖_𝐑|𝜙(𝑥−𝑦)|𝑑𝑦⟨𝑦⟩³‖_{𝐋𝑝1𝑥}‖_𝐑ln²1(1+|𝑧−𝑦|)|𝜓(𝑥−𝑧)|𝑑𝑧⟨𝑧⟩³‖_{𝐋𝑝2𝑥𝐋∞𝑦}≤𝐶‖𝜙‖_𝐋𝑞‖𝜓‖_𝐋𝑟, (2.24) where 1≤𝑝≤∞,1<𝑞≤𝑝/𝛼,1<𝑟≤𝑝/(1−𝛼),𝛼∈[0,1].
We now estimate the operator 𝒫=𝑡𝜕_𝑥+𝑥𝜕_𝑡 as follows: 𝒫𝒢(𝜙,𝜓)=𝒢(𝒫𝜙,𝜓)+_𝐑2𝑦𝑔(𝑦,𝑧)𝜓(𝑥−𝑧)𝜕_𝑡𝜙(𝑥−𝑦)𝑑𝑦𝑑𝑧+𝒢(𝜙,𝒫𝜓)+_𝐑2𝑧𝑔(𝑦,𝑧)𝜙(𝑥−𝑦)𝜕_𝑡𝜓(𝑥−𝑧)𝑑𝑦𝑑𝑧. (2.25) Then by virtue of estimate (2.4) applying the Hölder and Young inequalities, we get ‖𝒫𝒢(𝜙,𝜓)‖_𝐋𝑝≤‖𝒢(𝒫𝜙,𝜓)‖_𝐋𝑝+‖𝒢(𝜙,𝒫𝜓)‖_𝐋𝑝+‖_𝐑2𝑦𝑔(𝑦,𝑧)𝜓(𝑥−𝑧)𝜕_𝑡𝜙(𝑥−𝑦)𝑑𝑦𝑑𝑧‖_𝐋𝑝+‖_𝐑2𝑧𝑔(𝑦,𝑧)𝜙(𝑥−𝑦)𝜕_𝑡𝜓(𝑥−𝑧)𝑑𝑦𝑑𝑧‖_𝐋𝑝≤𝐶‖𝒫𝜙‖_𝐋𝑞‖𝜓‖_𝐋𝑟+𝐶‖𝒫𝜓‖_𝐋𝑞‖𝜙‖_𝐋𝑟+𝐶‖𝜕_𝑡𝜙‖_𝐋𝑞‖𝜓‖_𝐋𝑟+𝐶‖𝜕_𝑡𝜓‖_𝐋𝑞‖𝜙‖_𝐋𝑟. (2.26) Lemma 2.1 is proved.

We now decompose the free Klein-Gordon evolution group 𝒰(𝑡)=𝑒^{−𝑖⟨𝑖𝜕𝑥⟩𝑡}=ℱ⁻¹𝐸(𝑡)ℱ, where 𝐸(𝑡)=𝑒^{−𝑖𝑡⟨𝜉⟩} similarly to the factorization of the free Schrödinger evolution group. We denote the dilation operator by

𝒟_𝜔1𝜙=√𝑥𝑖𝜔𝜙(𝜔),(𝒟_𝜔)⁻¹=𝑖𝒟_1/𝜔. (2.27) Define the multiplication factor 𝑀(𝑡)=𝑒^{−𝑖𝑡⟨𝑖𝑥⟩𝜃(𝑥)}, where 𝜃(𝑥)=1 for |𝑥|<1 and 𝜃(𝑥)=0 for |𝑥|≥1. We introduce the operator

ℬ𝜙=𝜃(𝑥)⟨𝑖𝑥⟩^3/2𝑥𝜙(⟨𝑖𝑥⟩). (2.28) The inverse operator ℬ⁻¹ acts on the functions 𝜙(𝑥) defined on (−1,1) as follows:

ℬ⁻¹1𝜙=⟨𝜉⟩^3/2𝜉𝜙()⟨𝜉⟩ (2.29) for all 𝜉∈𝐑, since 𝜉=𝑥/⟨𝑖𝑥⟩∈𝐑 and 𝑥=𝜉/⟨𝜉⟩∈(−1,1). We now introduce the operators

𝒱(𝑡)=ℬ⁻¹𝑀(𝑡)𝒟_𝑡⁻¹ℱ⁻¹𝑒^{−𝑖𝑡⟨𝜉⟩},𝒲(𝑡)=𝑀(𝑡)(1−𝜃)𝒟_𝑡⁻¹ℱ⁻¹𝑒^{−𝑖𝑡⟨𝜉⟩}, (2.30) so that we have the representation for the free Klein-Gordon evolution group

𝒰(𝑡)ℱ⁻¹=𝑒^{−𝑖𝑡⟨𝑖𝜕𝑥⟩}ℱ⁻¹=ℱ⁻¹𝑒^{−𝑖𝑡⟨𝜉⟩}=𝒟_𝑡𝑀(𝑡)(ℬ𝒱(𝑡)+𝒲(𝑡))=𝒟_𝑡𝑀(𝑡)ℬ+𝒟_𝑡𝑀(𝑡)ℬ(𝒱(𝑡)−1)+𝒟_𝑡𝑀(𝑡)𝒲(𝑡). (2.31) The first term 𝒟_𝑡𝑀(𝑡)ℬ𝜙 of the right-hand side of (2.31) describes inside the light cone the well-known leading term of the large-time asymptotics of solutions of the linear Klein-Gordon equation ℒ𝑢=0 with initial data 𝜙. The second term of the right-hand side of (2.31) is a remainder inside of the light cone, whereas the last term represents the large time asymptotics outside of the light cone which decays more rapidly in time. We also have

ℱ𝒰(−𝑡)=ℱ𝑒^{𝑖𝑡⟨𝑖𝜕𝑥⟩}=𝑒^{𝑖𝑡⟨𝜉⟩}ℱ=𝒱⁻¹(𝑡)ℬ⁻¹𝑀(𝑡)𝒟_𝑡⁻¹+𝒲⁻¹(𝑡)𝒟_𝑡⁻¹=ℬ⁻¹𝑀(𝑡)𝒟_𝑡⁻¹+(𝒱⁻¹(𝑡)−1)ℬ⁻¹𝑀(𝑡)𝒟_𝑡⁻¹+𝒲⁻¹(𝑡)𝒟_𝑡⁻¹, (2.32) where the right-inverse operators are

𝒱⁻¹(𝑡)=𝐸(𝑡)ℱ𝒟_𝑡𝒲𝑀(𝑡)ℬ,₋₁(𝑡)=𝐸(𝑡)ℱ𝒟_𝑡(1−𝜃), (2.33) where 𝐸(𝑡)=𝑒^{−𝑖𝑡⟨𝜉⟩}.

In the next lemma we state the estimates of the operators 𝒱(𝑡)=ℬ⁻¹𝑀(𝑡)𝒟_𝑡⁻¹ℱ⁻¹𝑒^{−𝑖𝑡⟨𝜉⟩}.

Lemma 2.2. The estimates hold as follows: ‖𝒱(𝑡)𝜙‖_𝐋2≤𝐶‖𝜙‖_𝐋2,‖⟨𝜉⟩^𝜌𝜕_𝜉𝒱(𝑡)𝜙‖_𝐋2≤𝐶‖⟨𝜉⟩^𝛼𝜕_𝜉𝜙‖_𝐋2+𝐶‖⟨𝜉⟩^𝛼−1𝜙‖_𝐋2, (2.34) where 𝜌<𝛼,𝛼∈[1,2], and ‖⟨𝜉⟩^𝜌𝜕_𝜉𝒱⁻¹(𝑡)𝜙‖_𝐋2≤𝐶‖⟨𝜉⟩^1/4+𝛼𝜕_𝜉𝜙‖_𝐋2+𝐶‖⟨𝜉⟩^𝛼−3/4𝜙‖_𝐋2, (2.35) where 𝜌<𝛼,𝛼∈[0,1], provided the right-hand sides are finite.

Proof. Changing the variable of integration 𝑥=𝜉⟨𝜉⟩⁻¹, we see that ‖ℬ⁻¹𝜙‖²_𝐋2=_𝐑𝜉|𝜙(⟨𝜉⟩)|²𝑑𝜉⟨𝜉⟩³=¹₋₁|𝜙(𝑥)|²𝑑𝑥=‖𝜙‖²_{𝐋2(−1,1)}. (2.36) Hence ‖𝒱(𝑡)𝜙‖_𝐋2=‖𝜙‖_𝐋2.
Consider the estimate for the derivative 𝜕_𝜉𝒱(𝑡)𝜙. Define 𝑆(𝜉,𝜂)=⟨𝜂⟩−(𝜉𝜂+1)/⟨𝜉⟩. Note that 𝜕𝑆/𝜕𝜂=𝜂/⟨𝜂⟩−𝜉/⟨𝜉⟩=((1+⟨𝜂⟩⟨𝜉⟩−𝜂𝜉)/⟨𝜉⟩⟨𝜂⟩(⟨𝜂⟩+⟨𝜉⟩))(𝜂−𝜉) and 𝜕𝑆/𝜕𝜉=(𝜉−𝜂)/⟨𝜉⟩³. Hence integrating by parts one time yields 𝜕_𝜉𝐑𝑒^{−𝑖𝑡𝑆}𝜙(𝜂)𝑑𝜂=−𝑖𝑡⟨𝜉⟩⁻³_𝐑𝑒^{−𝑖𝑡𝑆}𝜙(𝜂)(𝜉−𝜂)𝑑𝜂=⟨𝜉⟩⁻³_𝐑𝑒^{−𝑖𝑡𝑆}(𝜙^′(𝜂)𝑔(𝜉,𝜂)+𝜙(𝜂)𝑔_𝜂(𝜉,𝜂))𝑑𝜂, (2.37) where 𝑔(𝜉,𝜂)=(𝜉−𝜂)/(𝜉/⟨𝜉⟩−𝜂/⟨𝜂⟩)=⟨𝜉⟩⟨𝜂⟩(⟨𝜂⟩+⟨𝜉⟩)/(1+⟨𝜂⟩⟨𝜉⟩−𝜂𝜉). Also we have (1+⟨𝜂⟩⟨𝜉⟩−𝜂𝜉)⁻¹≤𝐶⟨𝜉⟩⟨𝜂⟩(⟨𝜂⟩+⟨𝜉⟩)². (2.38) Hence the estimate is true 𝑔(𝜉,𝜂)=⟨𝜉⟩⟨𝜂⟩(⟨𝜂⟩+⟨𝜉⟩)1+⟨𝜂⟩⟨𝜉⟩−𝜂𝜉≤𝐶⟨𝜉⟩²⟨𝜂⟩².⟨𝜂⟩+⟨𝜉⟩ (2.39) Since 𝜕_𝜉⟨𝜉⟩^𝑎=𝑎𝜉⟨𝜉⟩^𝑎−2, we get ⟨𝜉⟩^𝜌𝜕_𝜉𝒱(𝑡)𝜙=⟨𝜉⟩^𝜌𝑖𝑡𝜕2𝜋^𝜉⟨𝜉⟩^−3/2_𝐑𝑒^{−𝑖𝑡𝑆}3𝜙(𝜂)𝑑𝜂=−2𝜉⟨𝜉⟩^𝜌−2𝒱(𝑡)𝜙+⟨𝜉⟩^𝜌−9/2𝑖𝑡2𝜋_𝐑𝑒^{−𝑖𝑡𝑆}𝜙^′(𝜂)𝑔(𝜉,𝜂)𝑑𝜂+⟨𝜉⟩^𝜌−9/2𝑖𝑡2𝜋_𝐑𝑒^{−𝑖𝑡𝑆}𝜙(𝜂)𝑔_𝜂(𝜉,𝜂)𝑑𝜂. (2.40) Consider the 𝐋²-estimate of the integral √𝑡⟨𝜉⟩^𝜌−9/2_𝐑𝑒^{𝑖𝑡(𝜉𝜂/⟨𝜉⟩)}𝜓(𝜂)𝑔(𝜉,𝜂)𝑑𝜂. (2.41) We have ‖√𝑡_𝐑𝑒^{𝑖𝑡(𝜉𝜂/⟨𝜉⟩)}𝜓(𝜂)𝑔(𝜉,𝜂)⟨𝜉⟩^𝜌−9/2𝑑𝜂‖²_𝐋2=𝑡_𝐑𝑑𝜂𝜓(𝜂)_𝐑𝑑𝜁𝜓(𝜁)_𝐑𝑒^{𝑖𝑡(𝜉/⟨𝜉⟩)(𝜂−𝜁)}𝑔(𝜉,𝜂)𝑔(𝜉,𝜁)⟨𝜉⟩^2𝜌−9=𝑑𝜉_𝐑𝑑𝜂𝜓(𝜂)_𝐑𝑑𝜁𝜓(𝜁)𝐺(𝜂,𝜁), (2.42) where the kernel 𝐺(𝜂,𝜁)=𝑡_𝐑𝑒^{𝑖𝑡(𝜉/⟨𝜉⟩)(𝜂−𝜁)}𝑔(𝜉,𝜂)𝑔(𝜉,𝜁)⟨𝜉⟩^2𝜌−9𝑑𝜉=𝑡⟨𝜂⟩⟨𝜁⟩_𝐑𝑒^{𝑖𝑡(𝜉/⟨𝜉⟩)(𝜂−𝜁)}⟨𝜂⟩+⟨𝜉⟩(1+⟨𝜂⟩⟨𝜉⟩−𝜂𝜉)⟨𝜁⟩+⟨𝜉⟩(1+⟨𝜁⟩⟨𝜉⟩−𝜁𝜉)⟨𝜉⟩^2𝜌−7𝑑𝜉. (2.43) After a change 𝜉=𝑥/⟨𝑖𝑥⟩, we find 𝐺(𝜂,𝜁)=𝑡⟨𝜂⟩⟨𝜁⟩¹₋₁𝑒^{𝑖𝑡𝑥(𝜂−𝜁)}⟨𝜂⟩+1/⟨𝑖𝑥⟩(1+⟨𝜂⟩/⟨𝑖𝑥⟩−𝜂𝑥/⟨𝑖𝑥⟩)⟨𝜁⟩+1/⟨𝑖𝑥⟩(1+⟨𝜁⟩/⟨𝑖𝑥⟩−𝜁𝑥/⟨𝑖𝑥⟩)⟨𝑖𝑥⟩^4−2𝜌𝑑𝑥. (2.44) We now change the contour 𝑧=𝑥+𝑖𝑦∈Γ of integration in 𝐺 as 𝑦=sign(𝜂−𝜁)⟨𝑖𝑥⟩²; then 𝐺(𝜂,𝜁)=𝑡⟨𝜂⟩⟨𝜁⟩_Γ𝑒^{𝑖𝑡𝑧(𝜂−𝜁)}⟨𝜂⟩+1/⟨𝑖𝑧⟩(1+⟨𝜂⟩/⟨𝑖𝑧⟩−𝜂𝑧/⟨𝑖𝑧⟩)⟨𝜁⟩+1/⟨𝑖𝑧⟩(1+⟨𝜁⟩/⟨𝑖𝑧⟩−𝜁𝑧/⟨𝑖𝑧⟩)⟨𝑖𝑧⟩^4−2𝜌𝑑𝑧. (2.45) Since |⟨𝑖𝑧⟩|=((⟨𝑖𝑥⟩²+𝑦²)²+4𝑥²𝑦²)^1/4∼⟨𝑖𝑥⟩, using (2.38) and the inequality (⟨𝜂⟩+⟨𝜉⟩)≥⟨𝜂⟩^𝛼−1⟨𝜉⟩^2−𝛼 for 𝛼∈[1,2], we get |(⟨𝜂⟩+1/⟨𝑖𝑧⟩)/(1+⟨𝜂⟩/⟨𝑖𝑧⟩−𝜂𝑧/⟨𝑖𝑧⟩)|≤𝐶⟨𝜂⟩^𝛼−1⟨𝑖𝑥⟩^𝛼−2; also we have |𝑒^{𝑖𝑡𝑧(𝜂−𝜁)}|=𝑒^{−⟨𝑖𝑥⟩2𝑡|𝜂−𝜁|}≤𝐶⟨𝑖𝑥⟩^−2𝛿(1+𝑡|𝜂−𝜁|)^𝛿 (2.46) for 𝛿>1. Therefore we obtain the estimate |𝐺(𝜂,𝜁)|≤𝐶𝑡⟨𝜂⟩^𝛼⟨𝜁⟩^𝛼(1+𝑡|𝜂−𝜁|)^𝛿1−1⟨𝑖𝑥⟩^{2(𝛼−𝜌−𝛿)}𝑑𝑥≤𝐶𝑡⟨𝜂⟩^𝛼⟨𝜁⟩^𝛼(1+𝑡|𝜂−𝜁|)^𝛿, (2.47) if we choose 𝜌<𝛼,𝛼∈[1,2] and 1<𝛿<1+𝛼−𝜌. Thus we get ‖√𝑡_𝐑𝑒^{𝑖𝑡(𝜉𝜂/⟨𝜉⟩)}𝜓(𝜂)𝑔(𝜉,𝜂)⟨𝜉⟩^𝜌−9/2𝑑𝜂‖²_𝐋2=_𝐑𝑑𝜂𝜓(𝜂)_𝐑𝑑𝜁𝜓(𝜁)𝐺(𝜂,𝜁)≤𝐶‖⟨𝜂⟩^𝛼𝜓(𝜂)‖_𝐋2‖_𝐑𝑑𝜁𝜓(𝜁)⟨𝜁⟩^𝛼𝑡(1+𝑡|𝜂−𝜁|)^𝛿‖_𝐋2≤𝐶𝑡‖⟨⋅⟩^𝛼𝜓‖²_𝐋2𝐑𝑑𝜂(1+𝑡|𝜂|)^𝛿≤𝐶‖⟨⋅⟩^𝛼𝜓‖²_𝐋2 (2.48) since 𝛿>1.
In the same manner we consider the estimate of the integral ‖√𝑡_𝐑𝑒^{𝑖𝑡(𝜉𝜂/⟨𝜉⟩)}𝜓(𝜂)𝑔_𝜂(𝜉,𝜂)⟨𝜉⟩^𝜌−9/2𝑑𝜂‖²_𝐋2=𝑡_𝐑𝑑𝜂𝜓(𝜂)_𝐑𝑑𝜁𝜓(𝜁)_𝐑𝑒^{𝑖𝑡(𝜉/⟨𝜉⟩)(𝜂−𝜁)}𝑔_𝜂(𝜉,𝜂)𝑔_𝜁(𝜉,𝜁)⟨𝜉⟩^2𝜌−9=𝑑𝜉_𝐑𝑑𝜂𝜓(𝜂)_𝐑𝑑𝜁𝜓(𝜁)𝐺(𝜂,𝜁), (2.49) where the kernel 𝐺(𝜂,𝜁)=𝑡_𝐑𝑒^{𝑖𝑡(𝜉/⟨𝜉⟩)(𝜂−𝜁)}𝑔_𝜂(𝜉,𝜂)𝑔_𝜁(𝜉,𝜁)⟨𝜉⟩^2𝜌−9𝑑𝜉=𝑡_𝐑𝑒^{𝑖𝑡(𝜉/⟨𝜉⟩)(𝜂−𝜁)}(𝜂+𝜉+(𝜂/⟨𝜂⟩)⟨𝜉⟩)(1+⟨𝜂⟩⟨𝜉⟩−𝜂𝜉)(𝜁+𝜉+(𝜁/⟨𝜁⟩)⟨𝜉⟩)(1+⟨𝜁⟩⟨𝜉⟩−𝜁𝜉)⟨𝜉⟩^2𝜌−7𝑑𝜉, (2.50) since by a direct calculation 𝜕_𝜂⟨𝜂⟩(⟨𝜂⟩+⟨𝜉⟩)=(1+⟨𝜂⟩⟨𝜉⟩−𝜂𝜉)𝜂+𝜉+(𝜂/⟨𝜂⟩)⟨𝜉⟩.(1+⟨𝜂⟩⟨𝜉⟩−𝜂𝜉) (2.51) In the same way as in (2.47) we have |𝐺(𝜂,𝜁)|≤𝐶𝑡⟨𝜂⟩^𝛼−1⟨𝜁⟩^𝛼−1(1+𝑡|𝜂−𝜁|)^𝛿 (2.52) if we choose 𝜌<𝛼,𝛼∈[1,2], and 1<𝛿<1+𝛼−𝜌. Therefore we get ‖√𝑡_𝐑𝑒^{𝑖𝑡(𝜉𝜂/⟨𝜉⟩)}𝜓(𝜂)𝑔_𝜂(𝜉,𝜂)⟨𝜉⟩^𝜌−9/2𝑑𝜂‖²_𝐋2≤𝐶‖⟨𝜂⟩^𝛼−1𝜓(𝜂)‖²_𝐋2𝐑𝑡𝑑𝜂(1+𝑡|𝜂|)^𝛿≤𝐶‖⟨⋅⟩^𝛼−1𝜓‖²_𝐋2. (2.53) Hence ‖⟨𝜉⟩^𝜌𝜕_𝜉𝒱(𝑡)𝜙‖_𝐋2≤𝐶‖⟨𝜉⟩^𝛼𝜕_𝜉𝜙‖_𝐋2+𝐶‖⟨𝜉⟩^𝛼−1𝜙‖_𝐋2. Thus we have the second estimate of the lemma.
Consider the estimate for the derivative 𝜕_𝜉𝒱⁻¹(𝑡)𝜙. Note that 𝜕𝑆(𝜂,𝜉)/𝜕𝜉=𝜉/⟨𝜉⟩−𝜂/⟨𝜂⟩=((1+⟨𝜂⟩⟨𝜉⟩−𝜂𝜉)/⟨𝜉⟩⟨𝜂⟩(⟨𝜂⟩+⟨𝜉⟩))(𝜉−𝜂) and 𝜕𝑆(𝜂,𝜉)/𝜕𝜂=(𝜂−𝜉)/⟨𝜂⟩³. Hence integrating by parts one time yields ⟨𝜉⟩^𝜌𝜕_𝜉𝒱⁻¹√(𝑡)𝜙=−𝑡√2𝑖𝜋⟨𝜉⟩^𝜌−1_𝐑1+⟨𝜂⟩⟨𝜉⟩−𝜂𝜉(⟨𝜂⟩+⟨𝜉⟩)⟨𝜂⟩^1/2𝜙(𝜂)𝑑𝑒^{𝑖𝑡𝑆(𝜂,𝜉)}=√𝑡√2𝑖𝜋⟨𝜉⟩^𝜌−1_𝐑𝑒^{𝑖𝑡𝑆(𝜂,𝜉)}(𝜙^′(𝜂)𝑔₁(𝜉,𝜂)+𝜙(𝜂)𝑔_1𝜂(𝜉,𝜂))𝑑𝜂, (2.54) where 𝑆(𝜂,𝜉)=⟨𝜉⟩−(𝜉𝜂+1)/⟨𝜂⟩ and 𝑔₁(𝜉,𝜂)=(((1+⟨𝜂⟩⟨𝜉⟩−𝜂𝜉))/(⟨𝜂⟩+⟨𝜉⟩))⟨𝜂⟩^1/2. The estimate is true 𝑔₁(𝜉,𝜂)=1+⟨𝜂⟩⟨𝜉⟩−𝜂𝜉⟨𝜂⟩+⟨𝜉⟩⟨𝜂⟩^1/2≤𝐶⟨𝜂⟩^3/2⟨𝜉⟩.⟨𝜂⟩+⟨𝜉⟩ (2.55) Consider the 𝐋²-estimate of the integral ⟨𝜉⟩^𝜌−1√𝑡_𝐑𝑒^{−𝑖𝑡(𝜉𝜂/⟨𝜂⟩)}𝜓(𝜂)𝑔₁(𝜉,𝜂)𝑑𝜂. (2.56) We have, changing 𝜂/⟨𝜂⟩=𝑦 and 𝜁/⟨𝜁⟩=𝑧, ‖⟨𝜉⟩^𝜌−1√𝑡_𝐑𝑒^{−𝑖𝑡(𝜉𝜂/⟨𝜂⟩)}𝜓(𝜂)𝑔₁(𝜉,𝜂)𝑑𝜂‖²_𝐋2=𝑡_𝐑𝑑𝜂𝜓(𝜂)_𝐑𝑑𝜁𝜓(𝜁)_𝐑𝑒^{−𝑖𝑡𝜉(𝜂/⟨𝜂⟩−𝜁/⟨𝜁⟩)}𝑔₁(𝜉,𝜂)𝑔₁(𝜉,𝜁)⟨𝜉⟩^2𝜌−2𝑑𝜉=𝑡¹₋₁𝑑𝑦⟨𝑖𝑦⟩³𝑦𝜓()⟨𝑖𝑦⟩¹₋₁𝑑𝑧⟨𝑖𝑧⟩³𝑧𝜓()⟨𝑖𝑧⟩×_𝐑𝑒^{−𝑖𝑡𝜉(𝑦−𝑧)}𝑔₁𝑦(𝜉,⟨𝑖𝑦⟩)𝑔₁𝑧(𝜉,⟨𝑖𝑧⟩)⟨𝜉⟩^2𝜌−2=𝑑𝜉₁₋₁𝑑𝑦⟨𝑖𝑦⟩³𝑦𝜓()⟨𝑖𝑦⟩¹₋₁𝑑𝑧⟨𝑖𝑧⟩³𝑧𝜓(⟨𝑖𝑧⟩)𝐺₁(𝑦,𝑧), (2.57) where the kernel 𝐺₁(𝑦,𝑧)=𝑡_𝐑𝑒^{−𝑖𝑡𝜉(𝑦−𝑧)}𝑔₁𝑦(𝜉,⟨𝑖𝑦⟩)𝑔₁𝑧(𝜉,⟨𝑖𝑧⟩)⟨𝜉⟩^2𝜌−2𝑑𝜉=𝑡⟨𝑖𝑦⟩^−1/2⟨𝑖𝑧⟩^−1/2_𝐑𝑒^{−𝑖𝑡𝜉(𝑦−𝑧)}1+⟨𝜉⟩/⟨𝑖𝑦⟩−𝑦𝜉/⟨𝑖𝑦⟩1/⟨𝑖𝑦⟩+⟨𝜉⟩1+⟨𝜉⟩/⟨𝑖𝑧⟩−𝑧𝜉/⟨𝑖𝑧⟩1/⟨𝑖𝑧⟩+⟨𝜉⟩⟨𝜉⟩^2𝜌−2𝑑𝜉. (2.58) We now change the contour 𝜉=𝑥+𝑖𝑦∈Γ of integration in 𝐺 as 𝑦=sign(𝑦−𝑧)⟨𝑥⟩²; then 𝐺₁(𝑦,𝑧)=𝑡⟨𝑖𝑦⟩^−1/2⟨𝑖𝑧⟩^−1/2_Γ𝑒^{−𝑖𝑡𝜉(𝑦−𝑧)}1+⟨𝜉⟩/⟨𝑖𝑦⟩−𝑦𝜉/⟨𝑖𝑦⟩1/⟨𝑖𝑦⟩+⟨𝜉⟩1+⟨𝜉⟩/⟨𝑖𝑧⟩−𝑧𝜉/⟨𝑖𝑧⟩1/⟨𝑖𝑧⟩+⟨𝜉⟩⟨𝜉⟩^2𝜌−2𝑑𝜉. (2.59) Since |⟨𝜉⟩|=((⟨𝑥⟩²−𝑦²)²+4𝑥²𝑦²)^1/2∼⟨𝑥⟩ and by (2.55) |(1+⟨𝜉⟩/⟨𝑖𝑦⟩−𝑦𝜉/⟨𝑖𝑦⟩)/(1/⟨𝑖𝑦⟩+⟨𝜉⟩)|≤𝐶⟨𝜉⟩^1−𝛼⟨𝑖𝑦⟩^−𝛼 for 𝛼∈[0,1], and also |𝑒^{−𝑖𝑡𝜉(𝑦−𝑧)}𝐶|≤(1+𝑡⟨𝑥⟩|𝑦−𝑧|)²≤𝐶𝑡⟨𝑥⟩^𝛿|𝑦−𝑧|^𝛿⟨𝑦−𝑧⟩ (2.60) for 𝛿<1,we obtain the estimate |𝐺₁(𝑦,𝑧)|≤𝐶⟨𝑖𝑦⟩^{−1/2−𝛼}⟨𝑖𝑧⟩^{−1/2−𝛼}|𝑦−𝑧|^𝛿⟨𝑦−𝑧⟩^𝐑⟨𝑥⟩^{−2𝛼+2𝜌−𝛿}𝑑𝑥≤𝐶⟨𝑖𝑦⟩^{−1/2−𝛼}⟨𝑖𝑧⟩^{−1/2−𝛼}|𝑦−𝑧|^𝛿,⟨𝑦−𝑧⟩ (2.61) if we choose 𝜌<𝛼, and 1>𝛿>1−2𝛼+2𝜌. Therefore we get ‖√𝑡_𝐑𝑒^{𝑖𝑡(𝜉𝜂/⟨𝜉⟩)}𝜓(𝜂)𝑔(𝜉,𝜂)⟨𝜉⟩^𝜌−9/2𝑑𝜂‖²_𝐋2=¹₋₁𝑑𝑦⟨𝑖𝑦⟩^5/2−𝛼𝑦𝜓()⟨𝑖𝑦⟩¹₋₁𝑑𝑧⟨𝑖𝑧⟩^5/2−𝛼𝑧𝜓()𝐶⟨𝑖𝑧⟩|𝑦−𝑧|^𝛿⟨𝑦−𝑧⟩≤𝐶¹₋₁𝑑𝑦⟨𝑖𝑦⟩^5/2−2𝛼𝑦|𝜓(⟨𝑖𝑦⟩)|²≤𝐶¹₋₁𝑑𝜉⟨𝜉⟩^1/2+2𝛼|𝜓(𝜉)|²≤𝐶‖⟨⋅⟩^1/4+𝛼𝜓‖²_𝐋2. (2.62)
In the same manner we consider the estimate of the integral with 𝜙(𝜂)𝑔_1𝜂(𝜉,𝜂). Hence ‖⟨𝜉⟩^𝜌𝜕_𝜉𝒱⁻¹(𝑡)𝜙‖_𝐋2≤𝐶‖⟨𝜉⟩^1/4+𝛼𝜕_𝜉𝜙‖_𝐋2+𝐶‖⟨𝜉⟩^𝛼−3/4𝜙‖_𝐋2, (2.63) where 𝛼>𝜌. Thus we have the second estimate of the lemma. Lemma 2.2 is proved.

In the next lemma we prove an auxiliary asymptotics for the integral ∫^∞₀𝑒^{−𝑖𝑡𝑧2}Φ(𝜉,𝑧)𝑑𝑧.

Lemma 2.3. The estimate holds as follows: |⟨𝜉⟩^𝛼(^∞₀𝑒^{−𝑖𝑡𝑧2}Φ(𝜉,𝑧)𝑑𝑧−Φ(𝜉,0)𝜋4𝑖𝑡)|≤𝐶𝑡^−3/4‖⟨𝜉⟩^𝛼Φ_𝑧(𝜉,𝑧)‖_𝐋^∞_𝜉𝐋²_𝑧 (2.64) provided the right-hand side is finite, where 𝛼∈𝐑.

Proof. We represent the integral ^∞₀𝑒^{−𝑖𝑡𝑧2}Φ(𝜉,𝑧)𝑑𝑧=Φ(𝜉,0)^∞₀𝑒^{−𝑖𝑡𝑧2}𝑑𝑧+𝑅=Φ(𝜉,0)𝜋4𝑖𝑡+𝑅, (2.65) where the remainder term is 𝑅=^∞₀𝑒^{−𝑖𝑡𝑧2}(Φ(𝜉,𝑧)−Φ(𝜉,0))𝑑𝑧. (2.66) In the remainder term we integrate by parts via identity 𝑒^{−𝑖𝑡𝑧2}=(1/(1−2𝑖𝑡𝑧²))𝜕_𝑧(𝑧𝑒^{−𝑖𝑡𝑧2})‖⟨𝜉⟩^𝛼𝑅‖_𝐋^∞_𝜉≤𝐶‖⟨𝜉⟩^𝛼∞₀11+𝑡𝑧²(Φ(𝜉,𝑧)−Φ(𝜉,0))𝑑𝑧‖_𝐋∞𝜉+𝐶‖⟨𝜉⟩^𝛼∞₀𝑧1+𝑡𝑧²Φ_𝑧(𝜉,𝑧)𝑑𝑧‖_𝐋∞𝜉≤𝐶‖⟨𝜉⟩^𝛼Φ_𝑧(𝜉,𝑧)‖_𝐋^∞_𝜉𝐋²_𝑧‖√𝑧(1+𝑡𝑧²)⁻¹‖_𝐋¹_𝑧+𝐶‖⟨𝜉⟩^𝛼Φ_𝑧(𝜉,𝑧)‖_𝐋^∞_𝜉𝐋²_𝑧‖𝑧(1+𝑡𝑧²)⁻¹‖_𝐋²_𝑧≤𝐶𝑡^−3/4‖⟨𝜉⟩^𝛼Φ_𝑧(𝜉,𝑧)‖_𝐋^∞_𝜉𝐋²_𝑧, (2.67) since |Φ(𝜉,𝑧)−Φ(𝜉,0)|≤^𝑧₀|Φ_𝑧(𝜉,𝑧)|𝑑𝑧≤𝐶|𝑧|‖Φ_𝑧(𝜉,𝑧)‖_𝐋^∞_𝜉𝐋²_𝑧. (2.68) Thus we have the estimate for the remainder in the asymptotic formula. Lemma 2.3 is proved.

In the next lemma we obtain the asymptotics for the operator 𝒱(𝑡)=ℬ⁻¹𝑀(𝑡)𝒟_𝑡⁻¹ℱ⁻¹𝑒^{−𝑖𝑡⟨𝜉⟩} and the right-inverse operator 𝒱⁻¹(𝑡)=𝐸(𝑡)ℱ𝒟_𝑡𝑀(𝑡)ℬ.

Lemma 2.4. The estimates hold as follows: ‖⟨𝜉⟩^𝛽(𝒱(𝑡)−1)𝜙‖_𝐋∞≤𝐶𝑡^−1/4(‖⟨𝜉⟩^𝛽+3/4𝜕_𝜉𝜙‖_𝐋2+‖⟨𝜉⟩^𝛽−1/4𝜙‖_𝐋2),‖⟨𝜉⟩^𝛽−3/4(𝒱⁻¹(𝑡)−1)𝜙‖_𝐋∞≤𝐶𝑡^−1/4(‖⟨𝜉⟩^3/2+𝛽𝜕_𝜉𝜙‖_𝐋2+‖⟨𝜉⟩^1/2+𝛽𝜙‖_𝐋2), (2.69) for all 𝑡≥1, where 0≤𝛽≤3/2, provided the right-hand sides are finite.

Proof. We have the identities 𝜉𝑆(𝜉,𝜂)=⟨𝜂⟩−⟨𝜉⟩−⟨𝜉⟩(𝜂−𝜉)=1+⟨𝜂⟩⟨𝜉⟩−𝜂𝜉(⟨𝜂⟩+⟨𝜉⟩)²⟨𝜉⟩(𝜂−𝜉)²,𝑆_𝜂𝜂(𝜉,𝜂)=−𝜉⟨𝜂⟩=⟨𝜉⟩1+⟨𝜂⟩⟨𝜉⟩−𝜂𝜉⟨𝜉⟩⟨𝜂⟩(⟨𝜂⟩+⟨𝜉⟩)(𝜂−𝜉), (2.70) and 𝑆_𝜂𝜂(𝜉,𝜂)=⟨𝜂⟩⁻³. We now change 𝑧=𝑆(𝜉,𝜂). We denote The inverse functions by 𝜂_𝑗(𝑧), so that 𝜂₁(𝑧)∶(0,∞)→(𝜉,∞) and 𝜂₂(𝑧)∶(0,∞)→(−∞,𝜉).
Thus the stationary point 𝑧=0 transforms into 𝜂=𝜉. Hence we can write the representation 𝒱(𝑡)𝜙=⟨𝜉⟩^−3/2𝑖𝑡2𝜋_𝐑𝑒^{−𝑖𝑡𝑆(𝜉,𝜂)}𝜙(𝜂)𝑑𝜂=2⟨𝜉⟩^−3/2𝑖𝑡2𝜋²_𝑗=1^∞₀𝑒^{−𝑖𝑡𝑧2}𝜙(𝜂_𝑗(𝑧))𝑆(𝜉,𝜂_𝑗(𝑧))|𝑆_𝜂(𝜉,𝜂_𝑗(𝑧))|𝑑𝑧. (2.71) By a direct calculation we find 𝑑𝑧=𝑑𝜂|𝑆_𝜂(𝜉,𝜂)|=𝑆(𝜉,𝜂)1+⟨𝜂⟩⟨𝜉⟩−𝜂𝜉⟨𝜂⟩⟨𝜉⟩^1/2. (2.72) Therefore we obtain 𝒱(𝑡)𝜙=^∞₀𝑒^{−𝑖𝑡𝑧2}Φ(𝜉,𝑧)𝑑𝑧, (2.73) where Φ(𝜉,𝑧)=2⟨𝜉⟩⁻¹𝑖𝑡2𝜋²_𝑗=1⟨𝜂⟩𝜙(𝜂)|1+⟨𝜉⟩⟨𝜂⟩−𝜉𝜂_{𝜂=𝜂𝑗(𝑧)}. (2.74) Application of Lemma 2.3 yields 𝒱(𝑡)𝜙=𝜙(𝜉)+𝑂(𝑡^−3/4‖Φ_𝑧(𝜉,𝑧)‖_𝐋^∞_𝜉𝐋²_𝑧). (2.75) By a direct calculation we have Φ_𝑧(𝜉,𝑧)𝑑𝑧=𝑑𝜂𝑖𝑡2𝜋²_𝑗=12⟨𝜂⟩^3/2𝜙^′(𝜂)⟨𝜉⟩^3/4(1+⟨𝜂⟩⟨𝜉⟩−𝜂𝜉)^3/4+𝑖𝑡2𝜋²_𝑗=1𝜂+(𝜂⟨𝜉⟩+𝜉⟨𝜂⟩)/(⟨𝜂⟩+⟨𝜉⟩)⟨𝜉⟩^3/4⟨𝜂⟩^1/2(1+⟨𝜂⟩⟨𝜉⟩−𝜂𝜉)^3/4𝜙(𝜂). (2.76) By (2.38) we get the estimate ‖⟨𝜉⟩^𝛽Φ_𝑧(𝜉,𝑧)‖_𝐋^∞_𝜉𝐋²_𝑧=‖⟨𝜉⟩^𝛽Φ_𝑧(𝜉,𝑧)𝑑𝑧‖𝑑𝜂_{𝐋∞𝜉𝐋2𝜂}≤𝐶√𝑡‖⟨𝜉⟩^𝛽⟨𝜂⟩^9/4(⟨𝜂⟩+⟨𝜉⟩)^3/2𝜙^′(𝜂)‖_{𝐋∞𝜉𝐋2𝜂}+𝐶√𝑡‖⟨𝜉⟩^𝛽⟨𝜂⟩^5/4(⟨𝜂⟩+⟨𝜉⟩)^3/2𝜙(𝜂)‖_{𝐋∞𝜉𝐋2𝜂}≤𝐶√𝑡‖⟨𝜂⟩^3/4+𝛽𝜙^′‖_𝐋2+𝐶√𝑡‖⟨𝜂⟩^𝛽−1/4𝜙‖_𝐋2 (2.77) for 0≤𝛽≤3/2 since (⟨𝜂⟩+⟨𝜉⟩)≥𝐶⟨𝜉⟩^(2/3)𝛽⟨𝜂⟩^{1−(2/3)𝛽}. This yields the first estimate of the lemma.
We now prove the second estimate. We have 𝒱⁻¹√(𝑡)𝜙=𝑡√2𝑖𝜋¹⁻¹𝑒^{𝑖𝑡(⟨𝜉⟩−𝜉𝑦−⟨𝑖𝑦⟩)}⟨𝑖𝑦⟩^−3/2𝑦𝜙(⟨𝑖𝑦⟩)𝑑𝑦, (2.78) then changing 𝑦=𝜂⟨𝜂⟩⁻¹, we find 𝒱⁻¹√(𝑡)𝜙(𝑥)=𝑡√2𝑖𝜋^R𝑒^{𝑖𝑡𝑆(𝜂,𝜉)}𝜙(𝜂)⟨𝜂⟩^−3/2𝑑𝜂, (2.79) where 𝑆(𝜂,𝜉)=⟨𝜉⟩−𝜉𝜂+1=⟨𝜂⟩1+⟨𝜂⟩⟨𝜉⟩−𝜂𝜉(⟨𝜂⟩+⟨𝜉⟩)²⟨𝜂⟩(𝜂−𝜉)². (2.80) As above we change 𝑧=𝑆(𝜂,𝜉) and represent 𝒱⁻¹√(𝑡)𝜙(𝑥)=𝑡√2𝑖𝜋^𝐑𝑒^{𝑖𝑡𝑆(𝜂,𝜉)}𝜙(𝜂)⟨𝜂⟩^−3/2=√𝑑𝜂2𝑡√𝑖𝜋²_𝑗=1^∞0𝑒^{𝑖𝑡𝑧2}𝜙(𝜂_𝑗(𝑧))⟨𝜂_𝑗(𝑧)⟩^−3/2𝑆(𝜂_𝑗(𝑧),𝜉)|𝑆_𝜂(𝜂_𝑗(𝑧),𝜉)|𝑑𝑧. (2.81) By a direct calculation we find 𝑆_𝜂(𝜂,𝜉)=(𝜂−𝜉)/⟨𝜂⟩³ and 𝑑𝑧=𝑑𝜂|𝑆_𝜂(𝜂,𝜉)|2=𝑆(𝜂,𝜉)⟨𝜂⟩+⟨𝜉⟩2⟨𝜂⟩^5/2.1+⟨𝜂⟩⟨𝜉⟩−𝜂𝜉 (2.82) Therefore we obtain 𝒱⁻¹(𝑡)𝜙(𝑥)=^∞₀𝑒^{𝑖𝑡𝑧2}Φ(𝜉,𝑧)𝑑𝑧, (2.83) where 2√Φ(𝜉,𝑧)=2𝑡√𝑖𝜋²_𝑗=1⟨𝜂⟩1+⟨𝜂⟩⟨𝜉⟩−𝜂𝜉⟨𝜂⟩+⟨𝜉⟩𝜙(𝜂)|_{𝜂=𝜂𝑗(𝑧)}. (2.84) Application of Lemma 2.3 yields 𝒱⁻¹(𝑡)𝜙(𝑥)=𝜙(𝜉)+𝑂(𝑡^−3/4‖Φ_𝑧(𝜉,𝑧)‖_𝐋^∞_𝜉𝐋²_𝑧). (2.85) By a direct calculation we have Φ_𝑧(𝜉,𝑧)𝑑𝑧=2√𝑑𝜂2𝑡√𝑖𝜋²_𝑗=1(1+⟨𝜂⟩⟨𝜉⟩−𝜂𝜉)^3/4(⟨𝜂⟩+⟨𝜉⟩)^3/2⟨𝜂⟩^9/4𝜙^′+2√(𝜂)2𝑡√𝑖𝜋²_𝑗=1⟨𝜂⟩^5/4(1+⟨𝜂⟩⟨𝜉⟩−𝜂𝜉)^1/4(⟨𝜂⟩+⟨𝜉⟩⟨𝜂⟩1+⟨𝜂⟩⟨𝜉⟩−𝜂𝜉)⟨𝜂⟩+⟨𝜉⟩_𝜂𝜙(𝜂), (2.86) Since |(⟨𝜂⟩1+⟨𝜂⟩⟨𝜉⟩−𝜂𝜉)⟨𝜂⟩+⟨𝜉⟩_𝜂|⟨𝜂⟩^5/4(1+⟨𝜂⟩⟨𝜉⟩−𝜂𝜉)^1/4⟨𝜂⟩+⟨𝜉⟩=|⟨𝜉⟩𝜂⟨𝜂⟩^1/4(1+⟨𝜂⟩⟨𝜉⟩−𝜂𝜉)^3/4(⟨𝜂⟩+⟨𝜉⟩)^5/2+⟨𝜂⟩^5/4(𝜂⟨𝜉⟩−𝜉⟨𝜂⟩)2(⟨𝜂⟩+⟨𝜉⟩)^3/2(1+⟨𝜂⟩⟨𝜉⟩−𝜂𝜉)^1/4|≤𝐶⟨𝜉⟩^3/4⟨𝜂⟩²(⟨𝜉⟩+⟨𝜂⟩)^3/2, (2.87) therefore we get the estimate ‖⟨𝜉⟩^𝛽−3/4Φ_𝑧(𝜉,𝑧)‖_𝐋^∞_𝜉𝐋²_𝑧=‖⟨𝜉⟩^𝛽−3/4Φ_𝑧(𝜉,𝑧)𝑑𝑧‖𝑑𝜂_{𝐋∞𝜉𝐋2𝜂}≤𝐶√𝑡(‖⟨𝜉⟩^𝛽⟨𝜂⟩³(⟨𝜂⟩+⟨𝜉⟩)^3/2𝜙^′(𝜂)‖_{𝐋∞𝜉𝐋2𝜂}+‖⟨𝜉⟩^𝛽⟨𝜂⟩²(⟨𝜉⟩+⟨𝜂⟩)^3/2𝜙(𝜂)‖_{𝐋∞𝜉𝐋2𝜂})≤𝐶√𝑡(‖⟨𝜂⟩^3/2+𝛽𝜙^′‖_𝐋2+‖⟨𝜂⟩^1/2+𝛽𝜙‖_𝐋2) (2.88) for 0≤𝛽≤2/3, since (⟨𝜂⟩+⟨𝜉⟩)≥𝐶⟨𝜉⟩^2/3𝛽⟨𝜂⟩^1−2/3𝛽. This yields the second estimate of the lemma. Lemma 2.4 is proved.

In the next lemma we find the estimates for the operator 𝒲(𝑡)=(1−𝜃)𝑀(𝑡)𝒟_𝑡⁻¹ℱ⁻¹𝑒^{−𝑖𝑡⟨𝜉⟩}.

Lemma 2.5. The estimates hold as follows: ‖𝒲(𝑡)𝜙‖_𝐋𝑟≤𝐶𝑡^{−1/2+(1−𝛿)(1−1/𝑟)}(‖⟨𝜉⟩^{(1+2𝛿)(1−1/𝑟)}𝜕_𝜉𝜙‖_𝐋1+‖⟨𝜉⟩^{(1+2𝛿)(1−1/𝑟)−1}𝜙‖_𝐋1) (2.89) for all 𝑡≥1, where 1<𝑟≤∞,𝛿∈[0,1], provided the right-hand side is finite.

Proof. Note that 𝒲(𝑡)𝜙=0 for |𝑥|≤1. To prove the estimate, we integrate by parts via the identity 𝑒^{𝑖𝑡(𝑥𝜉−⟨𝜉⟩)}=(1/𝑖𝑡(𝑥−𝜉/⟨𝜉⟩))𝜕_𝜉(𝑒^{𝑖𝑡(𝑥𝜉−⟨𝜉⟩)})√𝒲(𝑡)𝜙=𝑖𝑡√2𝜋^𝐑𝑒^{𝑖𝑡(𝑥𝜉−⟨𝜉⟩)}1𝜙(𝜉)𝑑𝜉=−√2𝜋𝑖𝑡^𝐑𝑒^{𝑖𝑡(𝑥𝜉−⟨𝜉⟩)}1𝜙𝑥−𝜉/⟨𝜉⟩^′−1(𝜉)𝑑𝜉√2𝜋𝑖𝑡^𝐑𝑒^{𝑖𝑡(𝑥𝜉−⟨𝜉⟩)}⟨𝜉⟩⁻³(𝑥−𝜉/⟨𝜉⟩)²𝜙(𝜉)𝑑𝜉 (2.90) for all |𝑥|≥2. Since 1/|𝑥−𝜉/⟨𝜉⟩|≤𝐶⟨𝑥⟩⁻¹ for all |𝑥|≥2, then we get |𝒲(𝑡)𝜙|≤𝐶𝑡^−1/2⟨𝑥⟩⁻¹_𝐑(|𝜙^′(𝜉)|+⟨𝜉⟩⁻³|𝜙(𝜉)|)𝑑𝜉≤𝐶𝑡^−1/2⟨𝑥⟩⁻¹(‖𝜕_𝜉𝜙‖_𝐋1+𝐶‖⟨𝜉⟩⁻³𝜙‖_𝐋1). (2.91) For the case of 1<|𝑥|≤2 we integrate by parts via the identity 𝑒^{𝑖𝑡(𝑥𝜉−⟨𝜉⟩)}=(1/(𝜉/⟨𝜉⟩+𝑖𝑡⟨𝜉⟩(𝑥−𝜉/⟨𝜉⟩)))𝜕_𝜉(⟨𝜉⟩𝑒^{𝑖𝑡(𝑥𝜉−⟨𝜉⟩)})√𝒲(𝑡)𝜙=−𝑖𝑡√2𝜋^𝐑𝑒^{𝑖𝑡(𝑥𝜉−⟨𝜉⟩)}⟨𝜉⟩𝜙𝜉/⟨𝜉⟩+𝑖𝑡⟨𝜉⟩(𝑥−𝜉/⟨𝜉⟩)^′+√(𝜉)𝑑𝜉𝑖𝑡√2𝜋^𝐑𝑒^{𝑖𝑡(𝑥𝜉−⟨𝜉⟩)}1/⟨𝜉⟩²+𝑖𝑡⟨𝜉⟩(𝜉𝑥/⟨𝜉⟩−1)(𝜉/⟨𝜉⟩+𝑖𝑡⟨𝜉⟩(𝑥−𝜉/⟨𝜉⟩))²𝜙(𝜉)𝑑𝜉. (2.92) Since |𝑥−𝜉/⟨𝜉⟩|≥||𝑥|−1|+|1−|𝜉|/⟨𝜉⟩| and |𝜉𝑥/⟨𝜉⟩−1|≤|𝜉|/⟨𝜉⟩||𝑥|−1|+|1−|𝜉|/⟨𝜉⟩|≤|𝑥−𝜉/⟨𝜉⟩| for 1<|𝑥|, therefore |𝒲(𝑡)𝜙|≤𝐶√𝑡_𝐑⟨𝜉⟩|𝜙^′(𝜉)|+|𝜙(𝜉)|√|𝜉|/⟨𝜉⟩+𝑡⟨𝜉⟩(||𝑥|−1|+|1−|𝜉|/⟨𝜉⟩|)𝑑𝜉≤𝐶𝑡_𝐑⟨𝜉⟩|𝜙^′(𝜉)|+|𝜙(𝜉)|1+𝑡⟨𝜉⟩⁻²+𝑡⟨𝜉⟩||𝑥|−1|𝑑𝜉 (2.93) for 1<|𝑥|≤2. Hence we get ‖𝒲(𝑡)𝜙‖_𝐋𝑟≤𝐶𝑡^−1/2‖⟨𝑥⟩⁻¹‖_𝐋^𝑟_(|𝑥|≥2)(‖𝜕_𝜉𝜙‖_𝐋1+𝐶‖⟨𝜉⟩⁻³𝜙‖_𝐋1)+𝐶√𝑡_𝐑𝑑𝜉⟨𝜉⟩|𝜙^′(𝜉)|+|𝜙(𝜉)|1+𝑡⟨𝜉⟩⁻²‖11+(𝑡⟨𝜉⟩/((1+𝑡⟨𝜉⟩⁽‖−2))))||𝑥|−1|_𝐋^𝑟_{(1<|𝑥|≤2)}≤𝐶𝑡^−1/2(‖𝜕_𝜉𝜙‖_𝐋1+𝐶‖⟨𝜉⟩⁻³𝜙‖_𝐋1)+𝐶𝑡^1/2−1/𝑟_𝐑(⟨𝜉⟩^1−1/𝑟|𝜙^′(𝜉)|+⟨𝜉⟩^−1/𝑟|𝜙(𝜉)|)(1+𝑡⟨𝜉⟩⁻²)^1/𝑟−1𝑑𝜉≤𝐶𝑡^{−1/2+(1−𝛿)(1−1/𝑟)}(‖⟨𝜉⟩^{(1+2𝛿)(1−1/𝑟)}𝜕_𝜉𝜙‖_𝐋1+‖⟨𝜉⟩^{(1+2𝛿)(1−1/𝑟)−1}𝜙‖_𝐋1) (2.94) for 𝛿∈[0,1]. This yields the estimate of the lemma. Lemma 2.5 is proved.

We next prove the time decay estimate in terms of the operator 𝒥.

Lemma 2.6. The estimate is valid ‖𝑢‖_𝐋^∞≤𝐶⟨𝑡⟩^−1/2‖𝑢‖_𝐇^1/2_3/2(‖𝑢‖_𝐇^1/2_3/2+‖𝒥𝑢‖_𝐇^1/2_1/2) (2.95) for all 𝑡≥0, provided that the right-hand side is finite.

Proof. Since ‖𝑢‖_𝐋¹≤𝐶‖𝑥𝑢‖_𝐋^1/2₂‖𝑢‖_𝐋^1/2₂, by applying the 𝐋^∞-𝐋¹ time decay estimate of the free evolution group 𝑒^{−𝑖⟨𝑖𝜕𝑥⟩𝑡} (see paper of [19, Lemma 1]), we get ‖𝑢‖_𝐋^∞=‖𝑒^{−𝑖⟨𝑖𝜕𝑥⟩𝑡}𝑒^{𝑖⟨𝑖𝜕𝑥⟩𝑡}𝑢‖_𝐋^∞≤𝐶𝑡^−1/2‖⟨𝑖𝜕_𝑥⟩^3/2𝑒^{𝑖⟨𝑖𝜕𝑥⟩𝑡}𝑢‖_𝐋¹≤𝐶𝑡^−1/2‖𝑥⟨𝑖𝜕_𝑥⟩^3/2𝑒^{𝑖⟨𝑖𝜕𝑥⟩𝑡}𝑢‖_𝐋^1/22‖⟨𝑖𝜕_𝑥⟩^3/2𝑒^{𝑖⟨𝑖𝜕𝑥⟩𝑡}𝑢‖_𝐋^1/22≤𝐶𝑡^−1/2‖𝒥𝑢‖_𝐇^1/2_1/2‖𝑢‖_𝐇^1/2_3/2 (2.96) for all 𝑡>0. Then by the Sobolev inequality we have ‖𝑢‖_𝐋^∞≤𝐶‖𝑢‖_𝐇¹. Thus the desired estimate follows. Lemma 2.6 is proved.

Next we obtain the asymptotics for the integral

_𝐑𝑒^{𝑖𝑡𝐴(𝜉,𝑦)}𝜙(𝜉,𝑦)𝑑𝑦, (2.97) where 𝐴(𝜉,𝑦)=⟨𝜉⟩+⟨𝜁⟩−2⟨𝜂⟩, and 𝜁=𝜉+2𝑦,𝜂=𝜉+𝑦.

Lemma 2.7. The following asymptotics is true: _𝐑𝑒^{𝑖𝑡𝐴(𝜉,𝑦)}𝜙(𝜉,𝑦)𝑑𝑦=𝑖𝜋𝑡⟨𝜉⟩^3/2𝜙(𝜉,0)+𝑂(𝑡^−3/4‖⟨𝜁⟩^3/2⟨𝜂⟩^3/2(⟨𝜁⟩+⟨𝜂⟩)^3/4𝜙_𝑦(𝜉,𝑦)‖_{𝐋∞𝜉𝐋2𝑦})+𝑂(𝑡^−3/4‖⟨𝜁⟩^1/2⟨𝜂⟩^1/2(⟨𝜁⟩+⟨𝜂⟩)^1/4𝜙(𝜉,𝑦)‖_{𝐋∞𝜉𝐋2𝑦}) (2.98) for 𝑡≥1 uniformly with respect to 𝜉∈𝐑, provided the right-hand side is finite.

Proof. Denote 𝜁=𝜉+2𝑦,𝜂=𝜉+𝑦, so that 𝑦=𝜁−𝜂=𝜂−𝜉. We have the identities 𝐴(𝜉,𝑦)=21+⟨𝜂⟩⟨𝜁⟩−𝜂𝜁+⟨𝜉⟩⟨𝜁⟩−𝜉𝜁+⟨𝜉⟩⟨𝜂⟩−𝜉𝜂𝑦(⟨𝜉⟩+⟨𝜁⟩)(⟨𝜉⟩+⟨𝜂⟩)(⟨𝜁⟩+⟨𝜂⟩)²,𝐴_𝑦𝜁(𝜉,𝑦)=2(−𝜂⟨𝜁⟩⟨𝜂⟩)=21+⟨𝜂⟩⟨𝜁⟩−𝜂𝜁⟨𝜁⟩⟨𝜂⟩(⟨𝜁⟩+⟨𝜂⟩)𝑦, (2.99) and 𝐴_𝑦𝑦(𝜉,𝑦)=2(2/⟨𝜁⟩³−1/⟨𝜂⟩³). We now change 𝑧=𝐴(𝜉,𝑦). We denote the inverse functions by 𝑦_𝑗(𝑧), so that 𝑦₁(𝑧)∶(0,∞)→(0,∞) and 𝑦₂(𝑧)∶(0,∞)→(−∞,0). Thus the stationary point 𝑧=0 transforms into 𝑦=0. Hence we can write the representation _𝐑𝑒^{𝑖𝑡𝐴(𝜉,𝑦)}𝜙(𝜉,𝑦)𝑑𝑦=2²_𝑗=1^∞₀𝑒^{−𝑖𝑡𝑧2}𝜙(𝜉,𝑦_𝑗(𝑧))𝐴(𝜉,𝑦_𝑗(𝑧))|𝐴_𝑦(𝜉,𝑦_𝑗(𝑧))|𝑑𝑧. (2.100) By a direct calculation we find 𝑑𝑧=𝑑𝑦|𝐴_𝑦(𝜉,𝑦)|2=1𝐴(𝜉,𝑦),𝑄(𝜉,𝜁,𝜂) (2.101) where √𝑄(𝜉,𝜁,𝜂)≡2⟨𝜁⟩⟨𝜂⟩⟨𝜁⟩+⟨𝜂⟩1+⟨𝜂⟩⟨𝜁⟩−𝜂𝜁+⟨𝜉⟩⟨𝜁⟩−𝜉𝜁+⟨𝜉⟩⟨𝜂⟩−𝜉𝜂(1+⟨𝜂⟩⟨𝜁⟩−𝜂𝜁)(⟨𝜉⟩+⟨𝜁⟩)(⟨𝜉⟩+⟨𝜂⟩). (2.102) Therefore we obtain _𝐑𝑒^{𝑖𝑡𝐴(𝜉,𝑦)}𝜙(𝜉,𝑦)𝑑𝑦=^∞₀𝑒^{−𝑖𝑡𝑧2}Φ(𝜉,𝑧)𝑑𝑧, (2.103) where Φ(𝜉,𝑧)=√2²_𝑗=1𝜙(𝜉,𝑦)𝑄(𝜉,𝜁,𝜂)|_{𝑦=𝑦𝑗(𝑧)}. (2.104) Since 𝑄(𝜉,𝜉,𝜉)=⟨𝜉⟩^3/2, then the application of Lemma 2.3 yields _𝐑𝑒^{𝑖𝑡𝐴(𝜉,𝑦)}𝜙(𝜉,𝑦)𝑑𝑦=𝑖𝜋𝑡⟨𝜉⟩^3/2𝜙(𝜉,0)+𝑂(𝑡^−3/4‖Φ_𝑧(𝜉,𝑧)‖_𝐋^∞_𝜉𝐋²_𝑧). (2.105) By a direct calculation we have Φ_𝑧(𝜉,𝑧)𝑑𝑧=𝑑𝑦²_𝑗=1𝜙_𝑦(𝜉,𝑦)(𝑄(𝜉,𝜁,𝜂))^3/2|_{𝑦=𝑦𝑗(𝑧)}+²_𝑗=1𝜙(𝜉,𝑦)𝑄(𝜉,𝜁,𝜂)(2𝑄_𝜁(𝜉,𝜁,𝜂)+𝑄_𝜂(𝜉,𝜁,𝜂))|_{𝑦=𝑦𝑗(𝑧)}. (2.106) By (2.38) 𝑄(𝜉,𝜁,𝜂)≤𝐶⟨𝜁⟩⟨𝜂⟩(⟨𝜁⟩+⟨𝜂⟩)^1/2,|𝑄_𝜁(𝜉,𝜁,𝜂)|≤𝐶⟨𝜁⟩⁻¹𝑄(𝜉,𝜁,𝜂)(1+|𝜂−𝜉|+⟨𝜂⟩+⟨𝜉⟩|𝜁−𝜉|⟨𝜁⟩+⟨𝜉⟩)≤𝐶⟨𝜂⟩(⟨𝜁⟩+⟨𝜂⟩)^1/2,|𝑄_𝜂(𝜉,𝜁,𝜂)|≤𝐶⟨𝜁⟩(⟨𝜁⟩+⟨𝜂⟩)^1/2. (2.107) Therefore we get the estimate ‖Φ_𝑧(𝜉,𝑧)‖_𝐋^∞_𝜉𝐋²_𝑧=‖Φ_𝑧(𝜉,𝑧)𝑑𝑧‖𝑑𝑦_{𝐋∞𝜉𝐋2𝑦}≤𝐶‖⟨𝜁⟩^3/2⟨𝜂⟩^3/2(⟨𝜁⟩+⟨𝜂⟩)^3/4𝜙_𝑦(𝜉,𝑦)‖_{𝐋∞𝜉𝐋2𝑦}+𝐶‖⟨𝜁⟩^1/2⟨𝜂⟩^1/2(⟨𝜁⟩+⟨𝜂⟩)^1/4𝜙(𝜉,𝑦)‖_{𝐋∞𝜉𝐋2𝑦}. (2.108) This yields the estimate of the lemma. Lemma 2.7 is proved.

In the next lemma we obtain the asymptotics for the nonlinear term ℱ𝒰(−𝑡)𝒢(𝑢,⟨𝑖𝜕_𝑥⟩⁻¹𝑢²) in (1.14).

Lemma 2.8. Let 𝑣∈𝐇^1,3. Then the asymptotic formula holds as follows: |⟨𝜉⟩(ℱ𝒰(−𝑡)𝒢(𝑢,⟨𝑖𝜕_𝑥⟩⁻¹𝑢²1)−𝑡Ω(𝜉)|𝑣(𝑡,𝜉)|²𝑣(𝑡,𝜉))|≤𝐶𝑡^−5/4‖𝑣‖³_𝐇1,3 (2.109) for 𝑡≥1 uniformly with respect to 𝜉∈𝐑, where 𝑣(𝑡,𝜉)=𝑒^{𝑖𝑡⟨𝜉⟩}𝑢(𝑡,𝜉),Ω(𝜉)≡⟨𝜉⟩²/⟨2𝜉⟩(2⟨𝜉⟩+⟨2𝜉⟩).

Proof. We first find the representation for ℱ𝒰(−𝑡)(⟨𝑖𝜕_𝑥⟩⁻¹𝑢²(𝑡)). We introduce the operator 𝒬(𝑡)=ℬ𝒱(𝑡)+𝒲(𝑡)=𝑀𝒟_𝑡⁻¹ℱ⁻¹𝑒^{−𝑖𝑡⟨𝜉⟩}, so that the representation for the free evolution group is true 𝒰(𝑡)ℱ⁻¹=𝑒^{−𝑖𝑡⟨𝑖𝜕𝑥⟩}ℱ⁻¹=ℱ⁻¹𝑒^{−𝑖𝑡⟨𝜉⟩}=𝒟_𝑡𝑀𝒬(𝑡). We also have ℱ𝑒^{𝑖𝑡⟨𝑖𝜕𝑥⟩}=𝒬⁻¹(𝑡)𝑀𝒟_𝑡⁻¹, where the right-inverse operator 𝒬⁻¹(𝑡)=𝒱⁻¹(𝑡)ℬ⁻¹+𝒲⁻¹(𝑡)=𝑒^{𝑖𝑡⟨𝜉⟩}ℱ𝒟_𝑡𝑀.
Since 𝒟_𝜔𝑡=√𝑖𝒟_𝜔𝒟_𝑡,𝒟_𝜔⁻¹=𝑖𝒟_1/𝜔, and ℱ𝒟_1/𝜔=𝒟_𝜔ℱ, we find 𝒬⁻¹(𝑡)𝑀^𝜔−1=𝑒^{𝑖𝑡⟨𝜉⟩}ℱ𝒟_𝑡𝑀^𝜔=√𝑖𝑒^{𝑖𝑡⟨𝜉⟩}𝒟_𝜔ℱ𝒟_𝜔𝑡𝑀^𝜔=√𝑖𝑒^{𝑖𝑡⟨𝜉⟩}𝒟_𝜔𝑒^{−𝑖𝜔𝑡⟨𝜉⟩}𝒬⁻¹(𝜔𝑡) (2.110) with 𝜔>0. Applying this formula with 𝜔=2 and putting 𝑢(𝑡)=𝒰(𝑡)ℱ⁻¹𝑣=𝒟_𝑡𝑀𝒬(𝑡)𝑣, we get ℱ𝒰(−𝑡)(⟨𝑖𝜕_𝑥⟩⁻¹𝑢²(𝑡))=⟨𝜉⟩⁻¹𝒬⁻¹(𝑡)𝑀𝒟_𝑡⁻¹(𝒟_𝑡𝑀𝒬(𝑡)𝑣)²=1⟨𝜉⟩√𝒬𝑖𝑡⁻¹(𝑡)𝑀(𝒬(𝑡)𝑣)²=𝑒^{𝑖𝑡⟨𝜉⟩}√𝑡𝒟₂𝑒^{−2𝑖𝑡⟨𝜉⟩}𝒬⟨2𝜉⟩⁻¹(2𝑡)(𝒬(𝑡)𝑣)². (2.111) Since ℱ𝒰(−𝑡)=ℱ𝑒^{𝑖⟨𝑖𝜕𝑥⟩𝑡}=𝑒^{𝑖𝑡⟨𝜉⟩}ℱ, we can write ℱ_𝑥→𝜂(⟨𝑖𝜕_𝑥⟩⁻¹𝑢²1(𝑡))=√𝑡𝒟₂𝑒^{−2𝑖𝑡⟨𝜂⟩}𝒬⟨2𝜂⟩⁻¹(2𝑡)(𝒬(𝑡)𝑣)²=1√𝑒2𝑖𝑡^{−2𝑖𝑡⟨𝜂/2⟩}𝜂Φ(𝑡,2), (2.112) where Φ(𝑡,𝜂)=⟨2𝜂⟩⁻¹𝒬⁻¹(2𝑡)(𝒬(𝑡)𝑣)². Hence we obtain ℱ𝒰(−𝑡)𝒢(𝑢,⟨𝑖𝜕_𝑥⟩⁻¹𝑢²)=𝑒^{𝑖𝑡⟨𝜉⟩}ℱ𝒢(𝑢,⟨𝑖𝜕_𝑥⟩⁻¹𝑢²)=𝑒^{𝑖𝑡⟨𝜉⟩}√2𝜋⟨𝜉⟩^𝐑𝑢(𝑡,𝜂−𝜉)ℱ_𝑥→𝜂(⟨𝑖𝜕_𝑥⟩⁻¹𝑢²(𝑡))𝑑𝜂.⟨𝜉⟩+⟨𝜉−𝜂⟩+⟨𝜂⟩ (2.113) Since 𝑢(𝑡,𝜂−𝜉)=𝑒^{𝑖𝑡⟨𝜂−𝜉⟩}𝑣(𝑡,𝜂−𝜉), we get ℱ𝒰(−𝑡)𝒢(𝑢,⟨𝑖𝜕_𝑥⟩⁻¹𝑢²1)=2√𝜋𝑖𝑡⟨𝜉⟩^𝐑𝑒^𝑖𝑡𝐴𝑣(𝑡,𝜂−𝜉)Φ(𝑡,𝜂/2)⟨𝜉⟩+⟨𝜉−𝜂⟩+⟨𝜂⟩𝑑𝜂 (2.114) with 𝐴=⟨𝜉⟩+⟨𝜂−𝜉⟩−2⟨𝜂/2⟩.
Since 𝒬(𝑡)=ℬ𝒱(𝑡) for |𝑥|≤1 and 𝒬(𝑡)=𝒲(𝑡) for |𝑥|≥1, we then have (𝒬(𝑡)𝑣)²=(ℬ𝒱(𝑡)𝑣)²+(𝒲(𝑡)𝑣)² (2.115) for all 𝑥∈𝐑. In the same manner applying the identity ℬ⁻¹(ℬ𝑣)²=⟨𝜉⟩^3/2𝑣²(𝜉), we obtain Φ(𝑡,𝜂)=⟨2𝜂⟩⁻¹𝒱⁻¹(2𝑡)ℬ⁻¹(ℬ𝒱(𝑡)𝑣)²+⟨2𝜂⟩⁻¹𝒲⁻¹(2𝑡)(𝒲(𝑡)𝑣)²=⟨2𝜂⟩⁻¹𝒱⁻¹(2𝑡)⟨𝜉⟩^3/2(𝒱(𝑡)𝑣)²+⟨2𝜂⟩⁻¹𝒲⁻¹(2𝑡)(𝒲(𝑡)𝑣)²=Φ₁(𝑡,𝜂)+Φ₂(𝑡,𝜂). (2.116) After a change 𝜂=2𝜉+2𝑦, we find ℱ𝒰(−𝑡)𝒢(𝑢,⟨𝑖𝜕_𝑥⟩⁻¹𝑢²)=_𝐑𝑒^{𝑖𝑡𝐴(𝜉,y)}𝜙₁(𝜉,𝑦)𝑑𝑦+_𝐑𝑒^{𝑖𝑡𝐴(𝜉,𝑦)}𝜙₂(𝜉,𝑦)𝑑𝑦, (2.117) where 𝜙₁(𝜉,𝑦)≡𝑣(𝑡,𝜉+2𝑦)Φ₁(𝑡,𝜉+𝑦)√,𝜙𝜋𝑖𝑡⟨𝜉⟩(⟨𝜉⟩+⟨𝜉+2𝑦⟩+⟨2(𝜉+𝑦)⟩)₂(𝜉,𝑦)≡𝑣(𝑡,𝜉+2𝑦)Φ₂(𝑡,𝜉+𝑦)√,𝜋𝑖𝑡⟨𝜉⟩(⟨𝜉⟩+⟨𝜉+2𝑦⟩+⟨2(𝜉+𝑦)⟩) (2.118) and 𝐴(𝜉,𝑦)=⟨𝜉⟩+⟨𝜉+2𝑦⟩−2⟨𝜉+𝑦⟩. Note that ‖𝒲⁻¹(2𝑡)𝜙‖_𝐋2=‖𝐸(2𝑡)ℱ𝒟_2𝑡(1−𝜃)𝜙‖_𝐋2=‖𝒟_2𝑡(1−𝜃)𝜙‖_𝐋2=‖(1−𝜃)𝜙‖_𝐋2≤‖𝜙‖_𝐋2. (2.119) Hence by Lemma 2.5 with 𝛿=5/6,𝑟=4‖𝒲(𝑡)𝜙‖²_𝐋4≤𝐶𝑡^−3/4(‖⟨𝜉⟩²𝜕_𝜉𝜙‖_𝐋1+‖⟨𝜉⟩𝜙‖_𝐋1) (2.120) for all 𝑡≥1, and ‖𝒲⁻¹(2𝑡)(𝒲(𝑡)𝑣)²‖_𝐋²≤‖𝒲(𝑡)𝑣‖²_𝐋4≤𝐶𝑡^−3/4(‖⟨𝜉⟩²𝜕_𝜉𝑣‖_𝐋1+‖⟨𝜉⟩𝑣‖_𝐋¹)². (2.121) Thus ‖⟨𝜉⟩_𝐑𝑒^{𝑖𝑡𝐴(𝜉,𝑦)}𝜙₂(𝜉,𝑦)𝑑𝑦‖_𝐋∞≤𝐶𝑡^−1/2‖𝑣(𝑡)‖_𝐋²‖⟨𝜉⟩⁻²𝒲⁻¹(2𝑡)(𝒲(𝑡)𝑣)²‖_𝐋²≤𝐶𝑡^−5/4‖𝑣‖³_𝐇1,3. (2.122)
We apply Lemma 2.7 to find |⟨𝜉⟩ℱ𝒰(−𝑡)𝒢(𝑢,⟨𝑖𝜕_𝑥⟩⁻¹𝑢²)−𝑡⁻¹⟨𝜉⟩^−1/2Ω(𝜉)𝑣(𝑡,𝜉)𝒱⁻¹(2𝑡)⟨𝜉⟩^3/2(𝒱(𝑡)𝑣)²|≤𝐶𝑡^−5/4‖⟨𝜉+2𝑦⟩^1/2⟨𝜉+𝑦⟩^1/2(⟨𝜉+2𝑦⟩+⟨𝜉+𝑦⟩)^1/4𝑣(𝑡,𝜉+2𝑦)Φ₁(𝑡,𝜉+𝑦)‖⟨𝜉+2𝑦⟩+⟨2(𝜉+𝑦)⟩_{𝐋∞𝜉𝐋2𝑦}+𝐶𝑡^−5/4‖⟨𝜉+2𝑦⟩^3/2⟨𝜉+𝑦⟩^3/2(⟨𝜉+2𝑦⟩+⟨𝜉+𝑦⟩)^3/4𝜕_𝑦𝑣(𝑡,𝜉+2𝑦)Φ₁(𝑡,𝜉+𝑦)‖⟨𝜉⟩+⟨𝜉+2𝑦⟩+⟨2(𝜉+𝑦)⟩_{𝐋∞𝜉𝐋2𝑦}+𝐶𝑡^−5/4‖𝑣‖³_𝐇1,3≤𝐶𝑡^−5/4(‖𝑣‖_𝐋^∞‖Φ₁‖_𝐇^0,1/4+‖Φ₁‖_𝐋^∞‖𝑣‖_𝐇^0,1/4+‖𝑣‖_𝐇∞^0,1‖Φ₁‖_𝐇^1,1/4+‖𝑣‖_𝐇^1,1‖Φ₁‖_𝐇∞^0,1/4+‖𝑣‖³_𝐇1,3)≤𝐶𝑡^−5/4(‖𝑣‖_𝐇^1,1‖Φ₁‖_𝐇^1,1/4+‖𝑣‖³_𝐇1,3) (2.123) since ⟨𝜉⟩𝜙₁(𝜉,𝑦)≡𝑣(𝑡,𝜉+2𝑦)Φ₁(𝑡,𝜉+𝑦)√.𝜋𝑖𝑡(⟨𝜉⟩+⟨𝜉+2𝑦⟩+⟨2(𝜉+𝑦)⟩) (2.124) By Lemma 2.2 we have ‖Φ₁(𝑡)‖_𝐇^1,1/4≤𝐶‖⟨𝜉⟩^−3/4𝜕_𝜉(𝒱⁻¹(2𝑡)⟨𝜉⟩^3/2(𝒱(𝑡)𝑣)²)‖_𝐋2≤𝐶‖⟨𝜉⟩^3/2+1/4+𝛾𝜕_𝜉(𝒱(𝑡)𝑣)²‖_𝐋2+𝐶‖⟨𝜉⟩^{3/2−3/4+𝛾}(𝒱(𝑡)𝑣)²‖_𝐋²≤𝐶‖𝑣‖²_𝐇1,2. (2.125) We now write the representation for Ψ(𝑡,𝜉) as Ψ(𝑡,𝜉)=⟨𝜉⟩^3/2⟨2𝜉⟩(2⟨𝜉⟩+⟨2𝜉⟩)𝑣(𝑡,𝜉)𝒱⁻¹(2𝑡)⟨𝜉⟩^3/2(𝒱(𝑡)𝑣)²=⟨𝜉⟩Ω(𝜉)|𝑣|²𝑣+𝑅, (2.126) where the remainder is 𝑅=⟨𝜉⟩^−1/2⟨2𝜉⟩Ω(𝜉)𝑣(𝑡,𝜉)⟨2𝜉⟩⁻¹×((𝒱⁻¹(2𝑡)−1)⟨𝜉⟩^3/2(𝒱(𝑡)𝑣)²+⟨𝜉⟩^3/2((𝒱(𝑡)𝑣)²−𝑣²)). (2.127) By the second estimate of Lemma 2.4 with 𝛽=5/4 and the first estimate of Lemma 2.4 with 𝛽=3/4, we obtain ‖⟨2𝜉⟩⁻¹(𝒱⁻¹(2𝑡)−1)⟨𝜉⟩^3/2(𝒱(𝑡)𝑣)²‖_𝐋^∞≤𝐶𝑡^−1/4(‖⟨𝜉⟩^3/4𝒱(𝑡)𝑣‖_𝐋^∞‖⟨𝜉⟩²𝜕_𝜉𝒱(𝑡)𝑣‖_𝐋2+‖⟨𝜉⟩𝒱(𝑡)𝑣‖²_𝐋4)≤𝐶𝑡^−1/4‖⟨𝜉⟩^3/4(𝒱(𝑡)−1)𝜙‖_𝐋∞‖𝑣‖_𝐇^1,2+𝐶𝑡^−1/4‖𝑣‖²_𝐇1,2≤𝐶𝑡^−1/4‖𝑣‖²_𝐇1,2 (2.128) for all 𝑡≥1. Also ‖⟨𝜉⟩^1/2((𝒱(𝑡)𝑣)²−𝑣²)‖_𝐋^∞≤𝐶(‖⟨𝜉⟩^1/4(𝒱(𝑡)−1)𝑣‖_𝐋^∞+‖⟨𝜉⟩^1/4𝑣‖_𝐋^∞)‖⟨𝜉⟩^1/4(𝒱(𝑡)−1)𝑣‖_𝐋^∞≤𝐶𝑡^−1/4‖𝑣‖²_𝐇1,2. (2.129) Thus we find the estimate for the remainder 𝑅(𝑡) as ‖𝑅(𝑡)‖_𝐋^∞≤𝐶‖⟨𝜉⟩^−1/2⟨2𝜉⟩Ω(𝜉)𝑣‖_𝐋^∞‖⟨2𝜉⟩⁻¹(𝒱⁻¹(2𝑡)−1)⟨𝜉⟩^3/2(𝒱(𝑡)𝑣)²‖_𝐋^∞+𝐶‖⟨𝜉⟩^−1/2⟨2𝜉⟩Ω(𝜉)𝑣‖_𝐋^∞‖⟨𝜉⟩^1/2((𝒱(𝑡)𝑣)²−𝑣²)‖_𝐋^∞≤𝐶𝑡^−1/4‖𝑣‖³_𝐇1,2. (2.130) Hence Ψ(𝑡,𝜉)=⟨𝜉⟩Ω(𝜉)|𝑣|²𝑣+𝑂(𝑡^−1/4‖𝑣‖³_𝐇1,2). (2.131) Therefore the asymptotics of the lemma is true. Lemma 2.8 is proved.

3. Proof of Theorem 1.1

We introduce the function space

𝐗_𝑇={𝜙∈𝐂([0,𝑇];𝐋²);‖𝜙‖_𝐗𝑇<∞}, (3.1) where

‖𝜙‖_𝐗𝑇=sup_{𝑡∈[0,𝑇]}(⟨𝑡⟩^−𝛾‖𝜙(𝑡)‖_𝐇3+⟨𝑡⟩^−𝛾‖𝒥𝜙(𝑡)‖_𝐇1+⟨𝑡⟩^−3𝛾‖𝒥𝜙(𝑡)‖_𝐇2+⟨𝑡⟩^1/2‖𝜙(𝑡)‖_𝐇1∞), (3.2) and 𝛾>0 is small.

The local existence in the function space 𝐗_𝑇 can be proved by a standard contraction mapping principle. We state it without a proof.

Theorem 3.1. Let 𝑢₀∈𝐇^3,1 and the norm ‖𝑢₀‖_𝐇^3,1=𝜀. Then there exist 𝜀₀>0 and 𝑇>1 such that for all 0<𝜀<𝜀₀ the initial value problem (1.1) has a unique local solution 𝑢∈𝐂([0,𝑇];𝐇^3,1) with the estimate ‖𝑢‖_𝐗𝑇<√𝜀.

Let us prove that the existence time 𝑇 can be extended to infinity which then yields the result of Theorem 1.1. By contradiction, we assume that there exists a minimal time 𝑇>0 such that the a priori estimate ‖𝑢‖_𝐗𝑇<√𝜀 does not hold; namely, we have ‖𝑢‖_𝐗𝑇≤√𝜀.

We apply the energy method to (1.14) ℒ(𝑢+𝑖𝜆𝒢(𝑢,𝑢))=−2𝑖|𝜆|²𝒢(𝑢,⟨𝑖𝜕_𝑥⟩⁻¹𝑢²), (i.e., multiplying both sides of the above equation by ⟨𝑖𝜕_𝑥⟩⁶(𝑢+𝑖𝜆𝒢(𝑢,𝑢)), taking the real part, and integrating over the space) to obtain

𝑑𝑑𝑡‖𝑢+𝑖𝜆𝒢(𝑢,𝑢)‖_𝐇³≤𝐶‖𝒢(𝑢,⟨𝑖𝜕_𝑥⟩⁻¹𝑢²)‖_𝐇³≤𝐶‖𝑢‖²_𝐇1∞‖𝑢‖_𝐇³≤𝐶𝜀^3/2⟨𝑡⟩^𝛾−1 (3.3) since by Lemma 2.1

‖𝒢(𝑢,⟨𝑖𝜕_𝑥⟩⁻¹𝑢²)‖_𝐇³≤𝐶‖𝒢(𝜕³_𝑥𝑢,𝑢²)‖_𝐋²+𝐶‖𝒢(𝑢,𝜕²_𝑥𝑢²)‖_𝐋²+𝐶‖𝒢(𝜕²_𝑥𝑢,𝑢²)‖_𝐋²+𝐶‖𝒢(𝜕_𝑥𝑢,𝑢²)‖_𝐋²≤𝐶‖𝑢‖²_𝐇1∞‖𝑢‖_𝐇³ (3.4) and ‖𝑢‖_𝐇¹_∞≤√𝜀⟨𝑡⟩^−1/2, ‖𝑢‖_𝐇³≤√𝜀⟨𝑡⟩^𝛾 by the estimate ‖𝑢‖_𝐗𝑇<√𝜀. Hence by Theorem 3.1 we have ‖𝑢+𝑖𝜆𝒢(𝑢,𝑢)‖_𝐇³≤𝐶𝜀⟨𝑡⟩^𝛾 and

‖𝑢‖_𝐇³≤‖𝑢+𝑖𝒢(𝑢,𝑢)‖_𝐇³+‖𝒢(𝑢,𝑢)‖_𝐇³≤𝐶𝜀⟨𝑡⟩^𝛾+𝐶‖𝑢‖_𝐇¹_∞‖𝑢‖_𝐇³≤𝐶𝜀⟨𝑡⟩^𝛾. (3.5)

Next we use the commutator relations [𝑥,⟨𝑖𝜕_𝑥⟩^𝛼]=𝛼⟨𝑖𝜕_𝑥⟩^𝛼−2𝜕_𝑥,[𝒫⟨𝑖𝜕_𝑥⟩^𝛼]=𝛼⟨𝑖𝜕_𝑥⟩^𝛼−2𝜕_𝑥𝜕_𝑡,𝜕_𝑡𝑢=−𝑖⟨𝑖𝜕_𝑥⟩𝑢+𝑖⟨𝑖𝜕_𝑥⟩⁻¹(𝑢²), and ℒ𝒫=(𝒫−𝑖⟨𝑖𝜕_𝑥⟩⁻¹𝜕_𝑥)ℒ, to getℒ(𝑢+𝑖𝒢(𝑢,𝑢))=−2𝑖|𝜆|²𝒢(𝑢,⟨𝑖𝜕_𝑥⟩⁻¹𝑢²). Hence ℒ𝒫(𝑢+𝑖𝜆𝒢(𝑢,𝑢))=−2𝑖|𝜆|²𝒫𝒢(𝑢,⟨𝑖𝜕_𝑥⟩⁻¹𝑢²)+2|𝜆|²⟨𝑖𝜕_𝑥⟩⁻¹𝜕_𝑥𝒢(𝑢,⟨𝑖𝜕_𝑥⟩⁻¹𝑢²). (3.6) Then by the energy method (i.e., multiplying both sides of the above equation by ⟨𝑖𝜕_𝑥⟩⁴𝒫(𝑢−𝒢(𝑢,𝑢)), taking the real part, and integrating over the space), by Lemma 2.1, using 𝜕_𝑡𝑢=−𝑖⟨𝑖𝜕_𝑥⟩𝑢+𝑖⟨𝑖𝜕_𝑥⟩⁻¹(𝑢²), we obtain

‖𝒫𝒢(𝑢,⟨𝑖𝜕_𝑥⟩⁻¹𝑢²)‖_𝐇²≤𝐶(‖𝒫𝑢‖_𝐇²+‖𝜕_𝑡𝑢‖_𝐇²)‖𝑢‖²_𝐇1∞≤𝐶(‖𝒫𝑢‖_𝐇²+‖𝑢‖_𝐇³)‖𝑢‖²_𝐇1∞. (3.7) Hence

𝑑𝑑𝑡‖𝒫(𝑢+𝑖𝜆𝒢(𝑢,𝑢))‖_𝐇²≤𝐶‖𝒫𝒢(𝑢,⟨𝑖𝜕_𝑥⟩⁻¹𝑢²)‖_𝐇²+𝐶‖𝒢(𝑢,⟨𝑖𝜕_𝑥⟩⁻¹𝑢²)‖_𝐇²≤𝐶(‖𝒫𝑢‖_𝐇²+‖𝑢‖_𝐇³)‖𝑢‖²_𝐇1∞≤𝐶𝜀⟨𝑡⟩⁻¹‖𝒫𝑢‖_𝐇²+𝐶𝜀²⟨𝑡⟩^𝛾−1. (3.8) Therefore by Theorem 3.1 it follows that

‖𝒫𝑢‖_𝐇²≤𝐶𝜀⟨𝑡⟩^𝛾. (3.9) The energy estimate and the identity ℒ𝑥=𝑥ℒ−𝑖⟨𝑖𝜕_𝑥⟩⁻¹𝜕_𝑥 imply that

𝑑𝑑𝑡‖𝑥(𝑢+𝑖𝜆𝒢(𝑢,𝑢))‖_𝐇²≤𝐶‖𝑥𝒢(𝑢,⟨𝑖𝜕_𝑥⟩⁻¹𝑢²)‖_𝐇²+‖𝑢+𝑖𝒢(𝑢,𝑢)‖_𝐇²≤𝐶‖𝑢‖²_𝐇1∞‖𝑥𝑢‖_𝐇²+𝐶‖𝑢‖²_𝐇1∞‖𝑢‖_𝐇²+𝐶𝜀⟨𝑡⟩^𝛾≤𝐶𝜀⟨𝑡⟩⁻¹‖𝑥𝑢‖_𝐇²+𝐶𝜀⟨𝑡⟩^𝛾, (3.10) which yields

‖𝑥𝑢‖_𝐇²≤𝐶𝜀⟨𝑡⟩^𝛾+1. (3.11) Then by the identity 𝒥=𝑖𝒫−𝑖𝑥ℒ−⟨𝑖𝜕_𝑥⟩⁻¹𝜕_𝑥 we obtain

‖𝒥(𝑢+𝑖𝜆𝒢(𝑢,𝑢))‖_𝐇²≤‖𝒫(𝑢+𝑖𝜆𝒢(𝑢,𝑢))‖_𝐇²+‖𝑥ℒ(𝑢+𝑖𝜆𝒢(𝑢,𝑢))‖_𝐇²+‖𝑢+𝑖𝜆𝒢(𝑢,𝑢)‖_𝐇²≤𝐶𝜀⟨𝑡⟩^𝛾+𝐶‖𝑥𝒢(𝑢,⟨𝑖𝜕_𝑥⟩⁻¹𝑢²)‖_𝐇²≤𝐶𝜀⟨𝑡⟩^𝛾+𝐶‖𝑢‖²_𝐇1∞‖𝑥𝑢‖_𝐇²+𝐶‖𝑢‖²_𝐇1∞‖𝑢‖_𝐇²≤𝐶𝜀⟨𝑡⟩^𝛾. (3.12) We use Lemma 2.8 to obtain for the new variable 𝑣(𝑡)=𝑒^{𝑖𝑡⟨𝜉⟩}𝑢(𝑡,𝜉)

⟨𝜉⟩ℱ𝒰(−𝑡)𝒢(𝑢,⟨𝑖𝜕_𝑥⟩⁻¹𝑢²1)=𝑡⟨𝜉⟩Ω(𝜉)|𝑣(𝑡,𝜉)|²𝑣(𝑡,𝜉)+𝑂(𝑡^−5/4‖𝑣‖³_𝐇1,3) (3.13) for 𝑡≥1 uniformly with respect to 𝜉∈𝐑, where Ω(𝜉)≡⟨𝜉⟩²/⟨2𝜉⟩(2⟨𝜉⟩+⟨2𝜉⟩). Hence for the new function 𝜙=𝑒^{𝑖𝑡⟨𝜉⟩}ℱ(𝑢+𝑖𝜆𝒢(𝑢,𝑢)) we get

𝜕_𝑡⟨𝜉⟩𝜙=−2𝑖|𝜆|²1𝑡⟨𝜉⟩Ω(𝜉)|𝜙(𝑡,𝜉)|²𝜙(𝑡,𝜉)+𝑂(𝑡^−5/4𝜀³⟨𝑡⟩^3𝛾) (3.14) in 𝐋^𝑟, 2≤𝑟≤∞. Then integrating, we obtain sup_𝑡≥1‖⟨𝜉⟩𝜙(𝑡)‖_𝐋∞≤𝐶𝜀. Hence

sup_𝑡≥1‖𝑣(𝑡)‖_𝐇∞^0,1≤𝐶𝜀. (3.15) By the decomposition of the free evolutions group we have the identity

⟨𝑖𝜕_𝑥⟩𝑢(𝑡)=(𝒟_𝑡𝑀(𝑡)ℬ𝒱(𝑡)+𝒟_𝑡𝒲(𝑡))⟨𝜉⟩ℱ𝑒^{𝑖⟨𝑖𝜕𝑥⟩𝑡}𝑢(𝑡)=𝒟_𝑡𝑀(𝑡)ℬ⟨𝜉⟩ℱ𝑒^{𝑖⟨𝑖𝜕𝑥⟩𝑡}𝑢(𝑡)+(𝒟_𝑡𝑀(𝑡)ℬ⟨𝜉⟩^−3/2⟨𝜉⟩^3/2(𝒱(𝑡)−1)+𝒟_𝑡𝒲(𝑡))⟨𝜉⟩ℱ𝑒^{𝑖⟨𝑖𝜕𝑥⟩𝑡}𝑢(𝑡). (3.16) By Lemmas 2.4 and 2.5 we find that the last term of the right-hand side of (3.16) is a remainder. Indeed we have the estimate

‖𝑢(𝑡)‖_𝐇¹_∞≤𝐶‖𝒟_𝑡𝑀(𝑡)ℬ⟨𝜉⟩ℱ𝑒^{𝑖⟨𝑖𝜕𝑥⟩𝑡}𝑢(𝑡)‖_𝐋^∞+𝐶𝜀⟨𝑡⟩^−5/4‖⟨𝜉⟩ℱ𝑒^{𝑖⟨𝑖𝜕𝑥⟩𝑡}𝑢(𝑡)‖_𝐇^1,2≤𝐶⟨𝑡⟩^−1/2sup_𝑡≥1‖𝑣(𝑡)‖_𝐇∞^0,1+𝐶⟨𝑡⟩^−5/4‖𝒥𝑢(𝑡)‖_𝐇², (3.17) which yields

sup_𝑡≥1⟨𝑡⟩^1/2‖𝑢(𝑡)‖_𝐇¹_∞≤𝐶𝜀. (3.18) From (3.15) and (3.18) it follows that

‖𝑢‖_𝐗𝑇≤𝐶𝜀<√𝜀, (3.19) which implies the desired contradiction. Thus there exists a unique global solution 𝑢∈𝐂([0,∞);𝐇^3,1) to (1.1) with the time decay estimate.

We now prove the asymptotics of solutions. By (3.14) as in the proof of Lemma 2.7 we have

‖𝜓(𝑡)−𝜓(𝑠)‖_𝐇∞0,1+‖𝜓(𝑡)−𝜓(𝑠)‖_𝐇0,1≤𝐶𝜀^3/2𝑡_𝑠𝜏^𝛾−5/4𝑑𝜏≤𝐶𝜀^3/2𝑠^𝛾−1/4 (3.20) with 𝛾∈(0,1/4), where

𝜓(𝑡)=𝑣𝑒^{−2𝑖|𝜆|2Ω|𝑣|2log𝑡}=ℱ𝑒^{𝑖⟨𝑖𝜕𝑥⟩𝑡}𝑢(𝑡)𝑒^{−2𝑖|𝜆|2Ω|ℱ𝑒𝑥𝑖⟨𝑖𝜕⟩𝑡𝑢(𝑡)|2log𝑡}. (3.21) Thus we see that there exists a unique final state 𝜓₊∈𝐇_∞^0,1∩𝐇^0,1 such that

‖𝜓(𝑡)−𝜓₊‖_𝐇∞0,1+‖𝜓(𝑡)−𝜓₊‖_𝐇0,1≤𝐶𝜀^3/2𝑡^𝛾−1/4. (3.22) We consider the asymptotics of the phase function

Φ(𝑡)=𝑖Ω^𝑡₁(|𝑣(𝜏)|²−|𝑣(𝑡)|²)𝑑𝜏𝜏=𝑖Ω^𝑡₁(|𝜓(𝜏)|²−|𝜓(𝑡)|²)𝑑𝜏𝜏. (3.23) By a direct calculation we have

Φ(𝑡)−Φ(𝑠)=𝑖Ω(^𝑡_𝑠(|𝜓(𝜏)|²−|𝜓(𝑡)|²)𝑑𝜏𝜏+(|𝜓(𝑡)|²−|𝜓(𝑠)|²)log𝑠), (3.24) where 1<𝑠<𝜏<𝑡. Hence

‖Φ(𝑡)−Φ(𝑠)‖_𝐇∞^0,1≤𝐶^𝑡_𝑠‖𝜓(𝜏)−𝜓(𝑡)‖_𝐇∞0,1(‖𝜓(𝜏)‖_𝐋∞+‖𝜓(𝑡)‖_𝐋∞)𝑑𝜏𝜏+𝐶‖𝜓(𝑠)−𝜓(𝑡)‖_𝐇∞0,1(‖𝜓(𝑠)‖_𝐋∞+‖𝜓(𝑡)‖_𝐋∞)log𝑠≤𝐶𝜀^5/2𝑡_𝑠𝜏^𝛾−5/4𝑑𝜏+𝐶𝜀^5/2𝑠^𝛾−1/4log𝑠, (3.25) from which it follows that there exists a unique real valued function Φ₊ such that

‖Φ(𝑡)−𝑖Φ₊‖_𝐇∞^0,1≤𝐶𝜀^3/2𝑡^𝛾−1/4. (3.26) Similarly, we find ‖Φ(𝑡)−𝑖Φ₊‖_𝐇^0,1≤𝐶𝜀^3/2𝑡^𝛾−1/4. Therefore we have the asymptotics of the phase function

𝑖Ω^𝑡₁|𝜓(𝜏)|²𝑑𝜏𝜏=𝑖Ω^𝑡₁|ℱ𝑒^{𝑖⟨𝑖𝜕𝑥⟩𝑡}𝑢(𝜏)|²𝑑𝜏𝜏=𝑖Φ₊+𝑖Ω|𝜓₊|²log𝑡+(Φ(𝑡)−𝑖Φ₊)+𝑖Ω(|ℱ𝑒^{𝑖⟨𝑖𝜕𝑥⟩𝑡}𝑢(𝑡)|²−|𝜓₊|²)log𝑡. (3.27) We also find

ℱ𝑒^{𝑖⟨𝑖𝜕𝑥⟩𝑡}𝑢(𝑡)−𝜓₊𝑒^{2|𝜆|2(𝑖Φ++𝑖Ω|𝜓+|2log𝑡)}=ℱ𝑒^{𝑖⟨𝑖𝜕𝑥⟩𝑡}𝑢(𝑡)−𝜓₊𝑒^{2|𝜆|2𝑖Ω∫𝑡1|𝜓(𝜏)|2(𝑑𝜏/𝜏)}+𝜓₊(𝑒^{2|𝜆|2𝑖Ω∫𝑡1|𝜓(𝜏)|2(𝑑𝜏/𝜏)}−𝑒^{2|𝜆|2𝑖Φ++2|𝜆|2𝑖Ω|𝜓+|2log𝑡})=𝜓(𝑡)𝑒^{2|𝜆|2𝑖Ω∫𝑡1|𝜓(𝑡)|2(𝑑𝜏/𝜏)}−𝜓₊𝑒^{2|𝜆|2𝑖Ω∫𝑡1|𝜓(𝜏)|2(𝑑𝜏/𝜏)}+𝜓₊(𝑒^{2|𝜆|2𝑖Ω∫𝑡1|𝜓(𝜏)|2(𝑑𝜏/𝜏)}−𝑒^{(2|𝜆|2𝑖Φ++2|𝜆|2𝑖Ω|𝜓+|2log𝑡)}). (3.28) Collecting these estimates, we obtain

‖ℱ𝑒^{𝑖⟨𝑖𝜕𝑥⟩𝑡}𝑢(𝑡)−𝜓₊𝑒^{2|𝜆|2𝑖Φ++2|𝜆|2𝑖Ω|𝜓+|2log𝑡}‖_𝐇0,1≤𝐶‖𝜓(𝑡)−𝜓₊‖_𝐇0,1+𝐶‖𝜓₊‖_𝐇0,1‖Φ(𝑡)−𝑖Φ₊‖_𝐋²+‖𝜓₊‖_𝐇0,1‖𝜓(𝑡)−𝜓₊‖_𝐋∞(‖𝜓₊‖_𝐋2+‖𝜓(𝑡)‖_𝐋2)log𝑡≤𝐶𝜀^3/2𝑡^𝛾−1/4, (3.29) and similarly

‖ℱ𝑒^{𝑖⟨𝑖𝜕𝑥⟩𝑡}𝑢(𝑡)−𝜓₊𝑒^{2|𝜆|2𝑖Φ++2|𝜆|2𝑖Ω|𝜓+|2log𝑡}‖_𝐇∞0,1≤𝐶𝜀^3/2𝑡^𝛾−1/4. (3.30) Therefore we have

‖ℱ𝑒^{𝑖⟨𝑖𝜕𝑥⟩𝑡}𝑢(𝑡)−𝑊₊𝑒^𝑖Ω|𝑊₊^|2log𝑡‖_𝐇^0,1≤𝐶𝜀^3/2𝑡^𝛾−1/4,‖ℱ𝑒^{𝑖⟨𝑖𝜕𝑥⟩𝑡}𝑢(𝑡)−𝑊₊𝑒^𝑖Ω|𝑊_+|²_log𝑡‖_𝐇∞^0,1≤𝐶𝜀^3/2𝑡^𝛾−1/4, (3.31) where 𝑊₊=𝜓₊exp(2|𝜆|²𝑖Φ₊). Estimate (3.31) means that

‖𝑢(𝑡)−𝑒^{−𝑖⟨𝑖𝜕𝑥⟩𝑡}ℱ⁻¹𝑊₊𝑒^{2|𝜆|2𝑖Ω|}𝑊₊^|2log𝑡‖_𝐇^1,0≤𝐶𝜀^3/2𝑡^𝛾−1/4. (3.32) Theorem 1.1 is proved.