Note that for the classical solution of (1.1) we have the conservation lawsR Ru(t, x)dx= R Ru0(x)dx andku(t)kL2 =ku0kL2

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Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 77, pp. 1–34.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

KDV TYPE ASYMPTOTICS FOR SOLUTIONS TO HIGHER-ORDER NONLINEAR SCHR ¨ODINGER EQUATIONS

PAVEL I. NAUMKIN, ISAHI S ´ANCHEZ-SU ´AREZ

Abstract. We consider the Cauchy problem for the higher-order nonlinear Schr¨odinger equation

i∂tu−a

3|∂_x|³u−b

4∂_x⁴u=λi∂x(|u|²u), (t, x)∈R⁺×R, u(0, x) =u0(x), x∈R,

wherea, b >0,|∂x|^α =F⁻¹|ξ|^αF andF is the Fourier transformation. Our purpose is to study the large time behavior of the solutions under the non-zero mass conditionR

u0(x)dx6= 0.

1. Introduction

We consider the Cauchy problem for the higher-order nonlinear Schr¨odinger equation

i∂_tu−a

3|∂x|³u− b

4∂_x⁴u=i∂_x(|u|²u), (t, x)∈R⁺×R, u(0, x) =u0(x), x∈R,

(1.1) where a, b >0,|∂x|^α=F⁻¹|ξ|^αF andF is the Fourier transformation defined by Fφ = ^√¹

2π

R

Re^−ixξφdx, and its inverse by F⁻¹φ = ^√¹

2π

R

Re^ixξφ(ξ)dξ. Note that we have the relationu(−t, x) =u(t,−x), so we only consider the caset >0. Note that for the classical solution of (1.1) we have the conservation lawsR

Ru(t, x)dx= R

Ru0(x)dx andku(t)k_L2 =ku0k_L2.

Equation (1.1) arises in the context of high-speed soliton transmission in long- haul optical communication system [13]. Also it can be considered as a particular form of the higher order nonlinear Schr¨odinger equation introduced by [42] to de- scribe the nonlinear propagation of pulses through optical fibers. This equation also represents the propagation of pulses by taking higher dispersion effects into account than those given by the Schr¨odinger equation (see [10, 11, 25, 28, 38, 32, 43, 49]).

Higher order nonlinear Schr¨odinger equations have been widely studied recently.

For the local and global well-posedness of the Cauchy problem we refer to [5, 6, 40]

and references cited therein. The dispersive blow-up was obtained in [2]. The existence and uniqueness of solutions to (1.1) were proved in [1, 19, 30, 33, 34, 39, 47, 50], and the smoothing properties of solutions were studied in [3, 8, 12, 30, 33,

2010Mathematics Subject Classification. 35B40, 35Q35.

Key words and phrases. Nonlinear Schr¨odinger equation; large time asymptotic behavior;

critical nonlinearity; self-similar solutions.

c

2020 Texas State University.

Submitted June 2, 2018. Published July 22, 2020.

1

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34, 36, 37, 48]. The blow-up effect for a special class of slowly decaying solutions of the Cauchy problem (1.1) was studied in [1].

In this article we are interested in the case of non zero total mass of the initial data

Z

R

u₀(x)dx6= 0.

Then by (1.1) we obtain the non-zero total mass for the solution R

Ru(t, x)dx = R

Ru0(x)dx 6= 0 for all t > 0. We develop the factorization technique originated in our previous papers [22, 21, 26, 20, 44, 45, 46]. The case of zero total mass R

Ru(t, x)dx = R

Ru0(x)dx = 0 is easier and the corresponding results can be obtained following the approach in [24].

We denote the Lebesgue space byL^p={φ∈S⁰:kφkL^p<∞}, with norms kφkL^p=Z

|φ(x)|^pdx^1/p

for 1≤p <∞andkφkL^∞ = sup_x∈R|φ(x)|. The weighted Sobolev space isH_p^m,s= {ϕ∈ S⁰ :kφk_H^m,s_p =khxi^shi∂xi^mφkL^p <∞}, where m, s∈R, 1 ≤p≤ ∞, hxi=

√

1 +x², hi∂xi=p

1−∂_x². We also use the notations H^m,s=H₂^m,s, H^m=H^m,0 shortly, if it does not cause any confusion. LetC(I;B) be the space of continuous functions from an interval I to a Banach space B. Different positive constants might be denoted by the same letterC.

In [29], it was proved the existence of the self-similar solutions of the formu= t^−1/3fm(xt^−1/3) to the reduced equation

i∂_tu−1

3|∂_x|³u=i∂_x(|u|²u), (1.2) which is defined by the mean valuem=R

Rf_m(x)dx6= 0.

Now we state the main result of this article. We show that the asymptotic behavior of the solutions to (1.1) resembles that of the KdV equation, which was studied intensively (see [15, 18, 23]). We denote by µ(x) the root of the equation Λ⁰(ξ) =ξ|ξ|(a+b|ξ|) =xfor allx∈R.

Theorem 1.1. Suppose that R

Ru0(x)dx 6= 0, and that the initial data u0 ∈H^1,1 have a sufficiently small norm ku0k_H1,1 ≤ ε. Then there exists a unique global solutionu∈C([0,∞);H^1,1)to the Cauchy problem (1.1). Furthermore there exists W+∈L^∞ such that the large time behavior satisfies

u(t, x) = M

pitΛ⁰⁰(µ(^x_t))W₊ µ(x t)

exp iµ(^x_t)

|Λ⁰⁰(µ(^x_t))|

W₊ µ(x t)

2logt +O(t^−1/3ht^1/3µ(x

t)i^−3/4) +O(t⁻¹³^−δ)

on the domain |x| ≥ t¹³^+2γ and u(t, x) =t^−1/3f_m(xt^−1/3) +O(t⁻¹³^−δ) on the domain|x| ≤t¹³^+2γ, whereγ, δ >0 are small andt^−1/3fm(xt^−1/3)is the self-similar solution to equation (1.2)with m=R

Rfm(x)dx=R

Ru0(x)dx6= 0.

We organize the rest of the paper as follows. Section 2 is devoted to the factorization formulas and L²-boundedness of pseudodifferential operators. Then in Sections 3 and 4 we obtain estimates for the operator V and V^∗, respectively, in

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the uniform norm and theL²-estimates of commutators. In Section 5, we prove a priori estimates of the solutionu(t) in the norm

kukXT = sup

t∈[1,T]

kϕkb L^∞+t^−γk∂xJu(t)kL²+t^−γk∂⁻¹_x Ibu(t)kL²

+t^−γk∂_x⁻¹Pbu(t)k_L2+t^−γku(t)k_H1

,

where ϕb = F U(−t)u(t), U(t) = F⁻¹e^−itΛ(ξ)F, Λ(ξ) = ^a₃|ξ|³+ ^b₄ξ⁴, J = x− tΛ⁰(−i∂x),Ib=∂b+it∂bΛ(−i∂x),Pb = 3t∂t+∂xx+b∂b. Section 6 is devoted to the proof of Theorem 1.1.

2. Preliminaries

2.1. Factorization techniques. We define the free evolution group U(t) =F⁻¹e^−itΛ(ξ)F, Λ(ξ) =^a₃|ξ|³+^b₄ξ⁴. We write

U(t)F⁻¹φ=Dt

r|t|

2π Z

R

eit(xξ−Λ(ξ))φ(ξ)dξ,

where Dtφ = |t|−1/2φ(^x_t) is the dilation operator. There is a unique station- ary point ξ = µ(x) in the integral R

Reit(xξ−Λ(ξ))φ(ξ)dξ, which is defined as the root of the equation Λ⁰(ξ) = ξ|ξ|(a+b|ξ|) = x for all x ∈ R. Thus we obtain Λ⁰(µ(x)) = x for all x ∈ R. Also Λ⁰⁰(ξ) = 2|ξ|(a+ ³₂b|ξ|) > 0 for all ξ ∈ R. We have Λ⁰(ξ) = O(ξ²hξi), therefore µ(x) = O({x}^1/2hxi^1/3). Define the scaling operator (Bφ)(x) = φ(µ(x)). Hence U(t)F⁻¹φ = D_tBMVφ, where M =e−it(Λ(η)−ηΛ⁰(η)), the operatorV(t)φ=

q|t|

2π

R

Re^{−itS(ξ,η)}φ(ξ)dξ and the phase function S(ξ, η) = Λ(ξ)−Λ(η)−Λ⁰(η)(ξ−η). Denote A_k = M^k_tΛ00¹(η)∂_ηM^k, k = 0,1. We have A1 =A0+iη, also A1V = Viξ, [iη,V] = −A0V, therefore we obtain the commutator∂ηV =−tΛ⁰⁰(η)[iη,V]. Since∂ξS(ξ, η) = Λ⁰(ξ)−Λ⁰(η), we obtain the commutatorit[Λ⁰(η),V]φ=−V∂ξφ. Also we use the representation for the inverse evolution group F U(−t)φ=V^∗MB⁻¹D_t⁻¹, where the inverse dilation operatorD_t⁻¹φ=|t|^1/2φ(xt) and the inverse scaling operator (B⁻¹φ)(η) =φ(Λ⁰(η)), and the conjugate operator

V^∗(t)φ= r|t|

2π Z

R

e^itS(ξ,η)φ(η)Λ⁰⁰(η)dη.

We haveiξV^∗φ=V^∗A1φ. Hence [iξ,V^∗] =V^∗A0.

Define a new dependent variableϕb=F U(−t)u(t). SinceF U(−t)L=∂tF U(−t) withL=∂t+iΛ, whereΛ= Λ(−i∂x) =F⁻¹Λ(ξ)F, applying the operatorF U(−t) to equation (1.1)Lu=∂x(|u|²u), we obtain

∂_tϕb=F U(−t)Lu=F U(−t)∂x(|U(t)F⁻¹ϕ|b²U(t)F⁻¹ϕ)b

=iξt⁻¹V^∗(|Vϕ|b²Vϕ),b (2.1)

since the nonlinearity is gauge invariant. Also we mention that the operator J = U(t)xU(−t) =x−tΛ⁰, withΛ⁰= Λ⁰(−i∂x) =F⁻¹Λ⁰(ξ)F, plays a crucial role in the large time asymptotic estimates. Note that J commutes with L, i.e. [J,L] = 0.

Also we see that the symbol Λ(ξ) = ^a₃|ξ|³+ ₄^bξ⁴ satisfies the identities ξ∂_ξΛ− b∂bΛ = 3Λ and ξ∂ξΛ +a∂aΛ = 4Λ. Hence we have the commutator relations [Pba, e^−itΛ(ξ)] = [Pbb, e^−itΛ(ξ)] = 0, withPba= 4t∂_t−ξ∂ξ−a∂a,Pbb= 3t∂_t−ξ∂ξ+b∂_b.

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We also use the operators Pa = 4t∂t+∂xx−a∂a, Pb = 3t∂t+∂xx+b∂b and Ia = ∂a +it∂aΛ(−i∂x), Ib = ∂b +it∂bΛ(−i∂x), and the commutator relations [L,Ia] = [L,Ib] = 0 and [L,Pa] = 4L, [L,Pb] = 3L hold. Using the relation u(t) =U(t)F⁻¹ϕb=F⁻¹e^−itΛ(ξ)ϕ, we obtainb

Pau=F⁻¹Pbae^−itΛ(ξ)ϕb=F⁻¹[Pba, e^−itΛ(ξ)]ϕb+F⁻¹e^−itΛ(ξ)Pcaϕb=U(t)F⁻¹Pbaϕ,b andPbu=F⁻¹Pbbe^−itΛ(ξ)ϕb=F⁻¹[Pbb, e^−itΛ(ξ)]ϕb+F⁻¹e^−itΛ(ξ)Pcbϕb=U(t)F⁻¹Pbbϕ.b Note thatIau=IaU(t)F⁻¹ϕb=F⁻¹(∂_a+it∂_aΛ(ξ))Eϕb=F⁻¹E∂_aϕb=U(t)F⁻¹∂_aϕ,b and Ibu = IbU(t)F⁻¹ϕb = F⁻¹(∂_b +it∂_bΛ(ξ))Eϕb = F⁻¹E∂_bϕb = U(t)F⁻¹∂_bϕ.b Hence kIaukL² =k∂bϕkb L² and kIbukL² = k∂bϕkb L². Also we have the identities P_a = 4tL+∂_xJ −aI_a andP_b= 3tL+∂_xJ +bI_b.

2.2. Boundedness of pseudodifferential operators. Define the pseudodifferential operator

a(x, D)φ≡ Z

R

e^ixξa(x, ξ)bφ(ξ)dξ.

There are many papers devoted to theL²-estimates of pseudodifferential operator a(x, D) (see [4, 7, 9, 27]). Below we will use the following results on the L²- boundedness of pseudodifferential operatora(x, D) (see [27]).

Lemma 2.1. Let the symbol a(x, ξ) be such that sup_x,ξ∈_R|∂_x^k∂_ξ^la(x, ξ)| ≤ C for k, l= 0,1. Then ka(x, D)φk_L2

x ≤Ckφk_L2.

Lemma 2.2. Let the symbol a(x, ξ) be such that sup_x∈Rk∂^l_ξa(x, ξ)k_L²

ξ ≤ C for l= 0,1. Then ka(x, D)φkL²_x ≤CkφkL².

Next we consider the time dependent pseudodifferential operator a(t, x, D)φ≡

Z

R

e^ixξa(t, x, ξ)bφ(ξ)dξ.

Lemma 2.3. Let the symbola(t, x, ξ) be such that sup

x,ξ∈R,t≥1

|{ξ}^−νhξi^ν(ξ∂ξ)^ka(t, x, ξ)| ≤C fork= 0,1,2, whereν∈(0,1). Thenka(t, x, D)φk_L2

x≤Ckφk_L2 for allt≥1.

Proof. We define the kernelK(t, x, y) =R

Re^iyξa(t, x, ξ)dξ, then we write a(t, x, D)φ=

Z

R

e^ixξa(t, x, ξ)bφ(ξ)dξ= Z

R

K(t, x, y)φ(x−y)dy.

Integrating two times by parts via the identitye^iyξ =H∂ξ(ξe^iyξ) withH = (1 + iξy)⁻¹ we obtain K(t, x, y) = R

Re^iyξξ∂ξ(Hξ∂ξ(Ha(x, ξ)))dξ. By the condition in this lemma, we find|ξ∂ξ(Hξ∂ξ(Ha(t, x, ξ)))| ≤ ^C{ξ}_hξyi^ν^hξi2^−ν for allx, y, ξ∈R. Hence we obtain the estimate

|K(t, x, y)| ≤C Z 1

0

ξ^νdξ hξyi² +C

Z ∞

1

ξ^−νdξ hξyi²

≤C|y|^−1−ν Z |y|

0

η^νhηi⁻²dη+C|y|^ν−1 Z ∞

|y|

η^−νhηi⁻²dη

≤C|y|^ν−1hyi^−2ν.

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Then by Young’s inequality we have ka(t, x, D)φkL²=k

Z

R

K(t, x, y)φ(x−y)dykL²

≤Ck Z

R

|y|^ν−1hyi^−2ν|φ(x−y)|dykL²≤C

|x|^ν−1hxi^−2ν

_L₁kφkL² ≤CkφkL².

The proof is complete.

Similarly, by considering the conjugate operator.

Lemma 2.4. Let the symbola(t, x, ξ) be such that sup

x,ξ∈R,t≥1

|{x}^−νhxi^ν(x∂_x)^ka(t, x, ξ)| ≤C

fork= 0,1,2, whereν∈(0,1). Thenka(t, x, D)φkL²_x≤CkφkL² for allt≥1.

3. Estimates for the operator V

Let χ ∈ C⁴(R) be such that χ(x) = 1 for |x| ≤ 1 and χ(x) = 0 for |x| ≥ 2.

Define the cut off functionsχj(z)∈C⁴(R),j= 1,2,3, such thatχ1+χ2+χ3≡1, χ2(z) = 0 for z ≤ ¹₃ or z ≥3 and χ2(z) = 1 for ²₃ ≤ z ≤ ³₂, χ1(z) = 1−χ2(z) for −³₂ < z < ²₃ and χ1(z) = 0 forz ≥ ²₃ or z ≤ −3 and χ3(z) = 1−χ2(z) for z > ³₂, χ3(z) = 1−χ1(z) for z < −³₂ and χ3(z) = 0 for −³₂ ≤ z ≤ ³₂. Denote Ψ1(t, ξ, η) = (1−χ(ηt^1/3))χ1(_η^ξ), Ψ2(t, ξ, η) = (1−χ(ηt^1/3))χ2(^ξ_η), Ψ3(t, ξ, η) = (1−χ(ηt^1/3))χ₃(_η^ξ), Ψ₄(t, ξ, η) =χ(ηt^1/3)χ(¹₃ξt^1/3) and Ψ₅(t, ξ, η) =χ(ηt^1/3)(1− χ(¹₃ξt^1/3)) and consider the operators

Vj(t)φ= r|t|

2π Z

R

e^{−itS(ξ,η)}Ψj(t, ξ, η)φ(ξ)dξ forj= 1,2,3,4,5.

3.1. Estimates for commutators. We first obtain theL²-estimate for the oper- atorV1.

Lemma 3.1. Let the weightP ∈C²(R\0) be such that ∂_η^kP(η) =O(|η|^α¹^−k)for k= 0,1,2, withα1≤2. Assume that the weight Q≡1 ifα2= 0, orQ∈C²(R\0) satisfies the estimate ∂_ξ^kQ(ξ) = O(|ξ|^α²^−k), k = 0,1,2, if α2 ≥ 1. Suppose that 2≤α1+α2<5/2. Then

|Λ⁰⁰|^1/2P tV₁Qφ

_L₂ ≤Ck∂_ξφk_L2+C|t|¹³⁽⁵²^−α¹^−α²⁾|φ(0)|

for allt≥1.

Proof. Integrating by parts we obtain P tV1Qφ=C 1−χ(ηt^1/3)

P(η)|t|^1/2 Z

R

e^{−itS(ξ,η)}∂ξ

Q(ξ)χ1(^ξ_η)

∂_ξS(ξ, η) φ(ξ) dξ

=C|t|^1/2 Z

R

e^{−itS(ξ,η)}q1(t, η, ξ)φξ(ξ)dξ +C|t|^1/2

Z

R

e^{−itS(ξ,η)}q2(t, η, ξ)φ(ξ)−φ(0)

ξ dξ

+C|t|^1/2φ(0) Z

R

e^{−itS(ξ,η)}q3(t, η, ξ)dξ,

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whereqk(t, η, ξ) = (1−χ(t^1/3η))P(η)(ξ∂ξ)^k−1(Q(ξ)χ1(_η^ξ)_∂ ¹

ξS(ξ,η)),k= 1,2, q3(t, η, ξ) = (1−χ(t^1/3η))P(η)∂ξ(Q(ξ)χ1(ξ

η) 1

∂ξS(ξ, η)).

Next we change η=µ(x) (recalling that the inverse scaling operator (B⁻¹φ)(η) = φ(Λ⁰(η)) andµ(x) is the root of the equation Λ⁰(η) =x), we obtain

P tV1Qφ=MB⁻¹|t|^1/2 Z

R

e^itxξq1(t, µ(x), ξ)e^−itΛ(ξ)φξ(ξ)dξ +MB⁻¹|t|^1/2

Z

R

e^itxξq2(t, µ(x), ξ)e^−itΛ(ξ)φ(ξ)−φ(0)

ξ dξ

+φ(0)MB⁻¹|t|^1/2 Z

R

e^itxξq₃(t, µ(x), ξ)e^−itΛ(ξ)dξ Also we change the variable of integrationξ=t^−1/3ξ⁰; then

P tV1Qφ=MB⁻¹D⁻¹_t_2/3Z

R

e^ixξq1(t, µ(xt^−2/3), ξt^−1/3)D_t1/3e^−itΛ(ξ)φξ(ξ)dξ +

Z

R

e^ixξq2(t, µ(xt^−2/3), ξt^−1/3)D_t1/3e^−itΛ(ξ)φ(ξ)−φ(0)

ξ dξ

+φ(0) Z

R

e^ixξq3(t, µ(xt^−2/3), ξt^−1/3)D_t1/3e^−itΛ(ξ)dξ ,

here D⁻¹_t_2/3φ(x) = |t|^1/3φ(xt^2/3) and D_t1/3φ(ξ) = |t|^−1/6φ(ξt^−1/3). Define the pseudodifferential operators ak(t, x, D)φ ≡ R

Re^ixξak(t, x, ξ)φ(ξ)dξb with symbols ak(t, x, ξ) =qk(t, µ(xt^−2/3), ξt^−1/3). Then we obtain

P tV₁Qφ=MB⁻¹D⁻¹_t_2/3

a₁(t, x, D)F⁻¹D_t1/3e^−itΛ(ξ)φ_ξ(ξ) +a₂(t, x, D)F⁻¹D_t1/3e^−itΛ(ξ)φ(ξ)−φ(0)

ξ +φ(0)a3(t, x, D)F⁻¹D_t1/3e^−itΛ(ξ)

.

Let us prove theL²-boundedness of the operatorsak(t, x, D),k= 1,2. We obtain ak(t, x, ξ) = 1−χ t^1/3µ(xt^−2/3)

P(µ(xt^−2/3))

×(ξ∂ξ)^k−1Q(ξt^−1/3)χ1(ξt^−1/3/µ(xt^−2/3)) Λ⁰(ξt⁻¹³)−xt^−2/3

.

Note that µ(x) =O(|x|^1/2) for small |x|, therefore (1−χ(t^1/3µ(xt^−2/3)))6= 0 for

|x| ≥C >0,t≥1. Also we have χ1(z)6= 0 for−3< z <2/3, hence

|Λ⁰(ξt^−1/3)−xt^−2/3|=

Z ξt^−1/3

µ(xt⁻²3)

Λ⁰⁰(η)dη ≥

Z ²₃µ(xt^−2/3)

µ(xt^−2/3)

Λ⁰⁰(η)dη

≥ 1

3|µ(xt^−2/3)|Λ⁰⁰(2

3µ(xt^−2/3))

≥C|Λ⁰(µ(xt^−2/3))|=C|x|t^−2/3.

Note that |∂_x^k∂_ξ^la_k(t, x, ξ)| ≤ C for all x, ξ ∈ R, t ≥ 1,k, l = 0,1, if 2 ≤ α₁+ α₂ ≤3. Therefore by Lemma 2.1 we havekak(t, x, D)φkL²_x ≤CkφkL². Thus the

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pseudodifferential operators ak(t, x, D) areL²-bounded fork= 1,2. Then we find that

k|Λ⁰⁰|¹²MB⁻¹D⁻¹_t_2/3a₁(t, x, D)F⁻¹D_t1/3e^−itΛ(ξ)φ_ξ(ξ)k_L2 η

≤ ka₁(t, x, D)F⁻¹D_t1/3e^−itΛ(ξ)φ_ξ(ξ)k_L2 x

≤CkF⁻¹D_t1/3e^−itΛ(ξ)φ_ξ(ξ)kL² =Ck∂ξφkL²

and by the Hardy inequality

|Λ⁰⁰|¹²MB⁻¹D_t⁻¹_2/3a2(t, x, D)F⁻¹D_t1/3e^−itΛ(ξ)φ(ξ)−φ(0) ξ kL²_η

≤ ka2(t, x, D)F⁻¹D_t1/3e^−itΛ(ξ)φ(ξ)−φ(0) ξ

_L₂

x

≤CkF⁻¹D_t1/3e^−itΛ(ξ)φ(ξ)−φ(0) ξ k_L²

ξ =Ckφ(ξ)−φ(0) ξ k_L²

ξ ≤Ck∂ξφkL². Now let us prove the L² - L^∞ boundedness of the pseudodifferential operator a3(t, x, D). We have

a3(t, x, ξ) =q3 t, µ(xt^−2/3), ξt^−1/3

=

1−χ t^1/3µ(xt^−2/3)

P µ(xt^−2/3)

×t^1/3∂_ξQ(ξt^−1/3)χ1(ξt^−1/3/µ(xt^−2/3)) Λ⁰(ξt^−1/3)−xt^−2/3

.

We consider two symbolsa3,1(t, x, ξ) =a3(t, x, ξ)χ(ξ) anda3,2(t, x, ξ) =a3(t, x, ξ)(1−

χ(ξ)). Note that |hxi^ν(x∂x)^jhξi^δa3,1(t, x, ξ)| ≤C|t|¹³^(3−α¹^−α²⁾for allx, ξ∈R,t≥ 1,j= 0,1,2 with someν ∈(0,1), ifα2= 0 orα2≥1,α1≤2, 2≤α1+α2<5/2.

Therefore by Lemma 2.4 we find that ka3,1(t, x, D)φk_L2

x≤C|t|¹³^(3−α¹^−α²⁾khξi^−δφkb _L2 ≤C|t|¹³^(3−α¹^−α²⁾kφkb L^∞. Also we have |∂_x^k∂_ξ^lhξi^δa3,2(t, x, ξ)| ≤C|t|¹³^(3−α¹^−α²⁾ for allx, ξ ∈R, t ≥1, k, l= 0,1, if 2≤α1+α2<5/2,δ >1/2. Therefore by Lemma 2.1 we find that

ka3,2(t, x, D)φk_L2

x≤C|t|¹³^(3−α¹^−α²⁾khξi^−δφkb _L2 ≤C|t|¹³^(3−α¹^−α²⁾kφkb L^∞. Then

|Λ⁰⁰|^1/2φ(0)MB⁻¹D⁻¹

t²3

a3(t, x, D)F⁻¹D

t¹3e^−itΛ(ξ) _L₂

η

≤ |φ(0)|ka3(t, x, D)F⁻¹D_t1/3e^−itΛ(ξ)kL²_x

≤C|t|¹³^(3−α¹^−α²⁾|φ(0)|kD_t1/3e^−itΛ(ξ)kL^∞_ξ

≤C|t|¹³⁽⁵²^−α¹^−α²⁾|φ(0)|,

which yields the result of the lemma.

In the next lemma we estimate the commutator [h,V2].

Lemma 3.2. Let the weights P ∈ C¹(R\0) and Q ∈ C²(R\0) be such that

∂_η^kP(η) =O(|η|^α¹^−k),k= 0,1, and∂_ξ^kQ(ξ) =O(|ξ|^α²^−k),k= 0,1,2. Suppose that

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h(ξ)∈C⁴(R\0)is such that|∂_ξ^kh(ξ)| ≤C|ξ|^α³^−k forξ∈R\0,0≤k≤4. Assume that α1, α2∈R,α3≥1,2≤α1+α2+α3<5/2. Then

k|Λ⁰⁰|^1/2P t[h,V2]Qφk_L2 ≤Ck∂ξφk_L2+C|t|¹³⁽⁵²^−α¹^−α²^−α³⁾|φ(0)|

for allt≥1.

Proof. Integrating by parts we obtain P t[h,V2]Qφ

=C

1−χ ηt^1/3

P(η)|t|^1/2 Z

R

h(η)−h(ξ)

∂ξS(ξ, η)

Q(ξ)φ(ξ)χ2(ξ η)dξ

=C|t|^1/2 Z

R

e^{−itS(ξ,η)}q1(η, ξ)φξ(ξ)dξ+C|t|^1/2 Z

R

e^{−itS(ξ,η)}q2(η, ξ)φ(ξ)−φ(0)

ξ dξ

+Cφ(0)|t|^1/2 Z

R

e^{−itS(ξ,η)}q3(η, ξ)dξ,

where qk(η, ξ) = (1−χ(ηt^1/3))P(η)(ξ∂ξ)^k−1(Q(ξ)χ2(_η^ξ)^{h(η)−h(ξ)}_∂

ξS(ξ,η) ), q3(η, ξ) = (1− χ(ηt^1/3))P(η)∂ξ(Q(ξ)χ2(^ξ_η)^{h(η)−h(ξ)}_∂

ξS(ξ,η) ). Next we change η =µ(x) and the variable of integrationξ=t^−1/3ξ⁰, then we obtain

P t[h,V2]Qφ=MB⁻¹D⁻¹_t_2/3Z

R

Z

R

e^ixξq2 t, µ(xt^−2/3), ξt^−1/3

D_t1/3e^−itΛ(ξ)φ(ξ)−φ(0)

ξ dξ

+φ(0) Z

R

e^ixξq3 t, µ(xt^−2/3), ξt^−1/3

D_t1/3e^−itΛ(ξ)dξ . We define the pseudodifferential operators ak(t, x, D)φ ≡ R

Re^ixξak(t, x, ξ)φ(ξ)dξb with symbolsak(t, x, ξ) =qk(t, µ(xt^−2/3), ξt^−1/3). Then we obtain

P t[h,V2]Qφ=MB⁻¹D⁻¹_t_2/3

a1(t, x, D)F⁻¹D_t1/3e^−itΛ(ξ)φξ(ξ) +a2(t, x, D)F⁻¹D_t1/3e^−itΛ(ξ)φ(ξ)−φ(0)

ξ +φ(0)a3(t, x, D)F⁻¹D_t1/3e^−itΛ(ξ)

.

We will prove theL²-boundedness of ak(t, x, D). Note thatχ2 ξt^−1/3 µ(xt^−2/3)

6= 0 for 1/3<_t_1/3_µ(xt^ξ_−2/3₎<3. Using the identities Λ⁰(ξ)−x= (ξ−µ(x))R1

0 Λ⁰⁰(ξ+(µ(x)−

ξ)τ)dτ, andh(µ(x))−h(ξ) = (µ(x)−ξ)R1

0 h⁰(ξ+ (µ(x)−ξ)τ)dτ, we obtain a_k(t, x, ξ) =−

1−χ t^1/3µ(xt^−2/3)

P µ(xt^−2/3)

(ξ∂_ξ)^k−1

χ₂ ξt^−1/3/µ(xt^−2/3)

×Q(ξt^−1/3) R1

0 h⁰(ξt^−1/3+ (µ(xt^−2/3)−ξt^−1/3)τ)dτ R1

0 Λ⁰⁰(ξt^−1/3+ (µ(xt^−2/3)−ξt^−1/3)τ)dτ

for k = 1,2. We have |∂_x^k∂_ξ^la_k(t, x, ξ)| ≤ C for allx, ξ ∈ R, t ≥ 1, k, l = 0,1, if α₁, α₂∈R,α₃≥1, 2≤α₁+α₂+α₃<5/2. Therefore by Lemma 2.1 we find that

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kak(t, x, D)φkL²_x ≤CkφkL². Thus the pseudodifferential operators ak(t, x, D) are L²-bounded fork= 1,2. Then as above we have

|Λ⁰⁰|¹²MB⁻¹D_t⁻¹2/3a₁(t, x, D)F⁻¹D_t1/3e^−itΛ(ξ)φ_ξ(ξ)

L²_η≤Ck∂ξφkL²

and

|Λ⁰⁰|¹²MB⁻¹D⁻¹_t_2/3a2(t, x, D)F⁻¹D_t1/3e^−itΛ(ξ)φ(ξ)−φ(0) ξ

_L₂

η

≤Ck∂ξφk_L2. Now let us prove the L² - L^∞ boundedness of the pseudodifferential operator a3(t, x, D). We have

a3(t, x, ξ) =−

1−χ t^1/3µ(xt^−2/3)

P µ(xt^−2/3) t^1/3∂ξ

Q(ξt^−1/3)

×χ2

ξt^−1/3 µ(xt^−2/3)

R1

0 h⁰(ξt^−1/3+ (µ(xt^−2/3)−ξt^−1/3)τ)dτ R1

0 Λ⁰⁰(ξt^−1/3+ (µ(xt^−2/3)−ξt^−1/3)τ)dτ

. Note that|∂_x^k∂_ξ^lhξi^δa3(t, x, ξ)| ≤C|t|¹³^(3−α¹^−α²⁾for allx, ξ∈R,t≥1,k, l= 0,1, if α1, α2∈R,α3≥1, 2≤α1+α2+α3<5/2,δ >1/2. Therefore by Lemma 2.1, we have

ka3(t, x, D)φkL²_x≤C|t|¹³^(3−α¹^−α²^−α³⁾khξi^−δφkb L² ≤C|t|¹³^(3−α¹^−α²^−α³⁾kφkb L^∞. Then

|Λ⁰⁰|^1/2φ(0)MB⁻¹D⁻¹

t²3

a3(t, x, D)F⁻¹D

t¹3e^−itΛ(ξ) _L₂

η

≤ |φ(0)|ka3(t, x, D)F⁻¹D_t1/3e^−itΛ(ξ)k_L2 x

≤C|t|¹³^(3−α¹^−α²^−α³⁾|φ(0)|kD_t1/3e^−itΛ(ξ)kL^∞_ξ

≤C|t|¹³⁽⁵²^−α¹^−α²^−α³⁾|φ(0)|,

which yields the result of the lemma.

In the next lemma we estimate the operatorV₃.

Lemma 3.3. Let the weights P ∈ C¹(R\0) and Q ∈ C²(R\0) be such that

∂_η^kP(η) =O(|η|^α¹^−k),k= 0,1, and∂_ξ^kQ(ξ) =O(|ξ|^α²^−k),k= 0,1,2. Assume that α1 ≥ 0, α2 ∈ R, 2 ≤α1+α2 ≤5/2. Then

|Λ⁰⁰|^1/2P tV3Qφ

_L₂ ≤Ck∂ξφk_L2+ C|t|¹³⁽⁵²^−α¹^−α²⁾|φ(0)|for all t≥1.

Proof. Integrating by parts we obtain P tV3Qφ=CP(η)|t|^1/2

Z

R

1

∂_ξS(ξ, η)Q(ξ)φ(ξ)Ψ3(t, ξ, η)dξ

=C|t|^1/2 Z

R

Z

R

e^{−itS(ξ,η)}q2(t, η, ξ)φ(ξ)−φ(0)

ξ dξ

+Cφ(0)|t|^1/2 Z

R

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whereqk(t, η, ξ) =P(η) ξ∂ξ^k−1

(Q(ξ)Ψ3(t, ξ, η)_∂ ¹

ξS(ξ,η)), q₃(t, η, ξ) = P(η)∂_ξ(Q(ξ)Ψ₃(t, ξ, η)_∂ ¹

ξS(ξ,η)). Next we change η = µ(x), and the variable of integrationξ=t^−1/3ξ⁰, then we obtain

P tV3Qφ=MB⁻¹D⁻¹_t_2/3Z

R

Z

R

e^ixξq2(t, µ(xt⁻²³), ξt^−1/3)D_t1/3e^−itΛ(ξ)φ(ξ)−φ(0)

ξ dξ

+φ(0) Z

R

e^ixξq3 t, µ(xt^−2/3), ξt^−1/3

D_t1/3e^−itΛ(ξ)dξ . Define the pseudodifferential operatorsak(t, x, D)φ≡R

Re^ixξak(t, x, ξ)bφ(ξ)dξ with symbolsak(t, x, ξ) =qk t, µ(xt^−2/3), ξt⁻¹³

. Then we obtain P tV3Qφ=MB⁻¹D⁻¹_t_2/3

a1(t, x, D)F⁻¹D_t1/3e^−itΛ(ξ)φξ(ξ) +a2(t, x, D)F⁻¹D_t1/3e^−itΛ(ξ)φ(ξ)−φ(0)

ξ +φ(0)a3(t, x, D)F⁻¹D_t1/3e^−itΛ(ξ)

.

Let us prove theL²-boundedness of the operatorsa_k(t, x, D),k= 1,2. We have ak(t, x, ξ) =

1−χ t^1/3µ(xt^−2/3)

P(µ(xt^−2/3))

×(ξ∂ξ)^k−1Q(ξt^−1/3)χ3(ξ/(t^1/3µ(xt^−2/3))) Λ⁰(ξt^−1/3)−xt^−2/3

.

Note thatµ(x) =O(|x|^1/2) for small|x|, thereforeχ(t^1/3µ(xt^−2/3))6= 0 for|x| ≤C, t≥1. Alsoχ3(ξt^−1/3/µ(xt^−2/3))6= 0 for _t_1/3_µ(xt^ξ_−2/3₎>3/2, hence

|Λ⁰(ξt^−1/3)−xt^−2/3|=|

Z ξt^−1/3

µ(xt^−2/3)

Λ⁰⁰(η)dη| ≥ |

Z ξt^−1/3

2 3ξt^−1/3

Λ⁰⁰(η)dη|

≥ 1

3|ξt^−1/3|Λ⁰⁰ 2 3ξt^−1/3

≥C|Λ⁰(ξt^−1/3)|.

Note that|∂_x^k∂^l_ξak(t, x, ξ)| ≤C for allx, ξ∈R,t≥1,k, l= 0,1, if 2≤α1+α2≤3.

Therefore by Lemma 2.1 we find that kak(t, x, D)φk_L2

x ≤ C|t|kφk_L2. Thus the pseudodifferential operatorsa_k(t, x, D) areL²-bounded fork= 1,2. Then as above we have

|Λ⁰⁰|¹²MB⁻¹D⁻¹_t_2/3a₁(t, x, D)F⁻¹D_t1/3e^−itΛ(ξ)φ_ξ(ξ) _L₂

η

≤Ck∂_ξφk_L2, |Λ⁰⁰|¹²MB⁻¹D⁻¹_t_2/3a2(t, x, D)F⁻¹D_t1/3e^−itΛ(ξ)φ(ξ)−φ(0)

ξ _L₂

η

≤Ck∂ξφk_L2. Now let us prove the L² - L^∞ boundedness of the pseudodifferential operator a3(t, x, D). We have

a3(t, x, ξ) =

1−χ t^1/3µ(xt^−2/3)

P µ(xt^−2/3) t^1/3

×∂_ξQ(ξt^−1/3)χ3(ξ/(t^1/3µ(xt^−2/3))) Λ⁰(ξt^−1/3)−xt^−2/3

.

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Note that|∂_x^k∂_ξ^lhξi^δa3(t, x, ξ)| ≤C|t|¹³^(3−α¹^−α²⁾for allx, ξ∈R,t≥1,k, l= 0,1, if 2≤α1+α2+δ≤3,δ >1/2. Therefore by Lemma 2.1 we have

ka3(t, x, D)φk_L2

x ≤C|t|¹³^(3−α¹^−α²⁾khξi^−δφkb _L2 ≤C|t|¹³^(3−α¹^−α²⁾kφkb _L∞. Thus as above we obtain

k|Λ⁰⁰|^1/2φ(0)MB⁻¹D⁻¹

t²3

a3(t, x, D)F⁻¹D

t¹3e^−itΛ(ξ)kL²_η≤C|t|¹³⁽⁵²^−α¹^−α²⁾|φ(0)|.

Next we obtain theL²-estimate for the operatorV4.

Lemma 3.4. Let the weight P be such that P(η) = O(|η|^α¹), with α1 > −1.

Assume that the weightQsatisfies the estimateQ(ξ) =O(|ξ|^α²), ifα2>−1. Then |Λ⁰⁰|^1/2PV4Qφ

L^p≤C|t|⁻¹³⁽¹²^+α¹^+α²⁾⁻^3p¹(k∂ξφkL²+|t|¹⁶|φ(0)|) for allt≥1, where1≤p≤ ∞.

Proof. Using the estimate|φ(ξ)−φ(0)| ≤C|ξ|^1/2k∂ξφk_L2, we have k|Λ⁰⁰|^1/2PV4QφkL^p

≤C|t|^1/2k|Λ⁰⁰|^1/2P(η)χ(ηt^1/3) Z

R

e^{−itS(ξ,η)}χ(1

3ξt^1/3)Q(ξ)φ(ξ)dξkL^p

≤C|t|^1/2k|Λ⁰⁰|^1/2P(η)χ(ηt^1/3)kL^pkχ(1

3ξt^1/3)Q(ξ)φ(ξ)kL¹

≤C|t|^1/2k∂ξφkL²k|η|¹²^+α¹k_Lp(|η|≤2t^−1/3)k|ξ|¹²^+α²k_L1(|ξ|≤6t^−1/3)

+C|t|^1/2|φ(0)|k|η|¹²^+α¹k_Lp(|η|≤2t^−1/3)k|ξ|^α²k_L1(|ξ|≤6t^−1/3)

≤C|t|⁻¹³⁽¹²^+α¹^+α²⁾⁻^3p¹ (k∂_ξφk_L2+|t|^1/6|φ(0)|).

In the next lemma we estimate the operatorV5.

Lemma 3.5. Let the weights P andQ∈C³(R\0) be such that P(η) =O(|η|^α¹), and ∂^k_ξQ(ξ) = O(|ξ|^α²^−k), k = 0,1,2,3. Assume that α₁ ≥ 0, α₂ ∈ R, −1 ≤ α1+α2≤2. Then

|Λ⁰⁰|^1/2PV5Qφ

_L₂ ≤Ct⁻^1+α^{1 +}³ ^α²(k∂ξφk_L2+t¹⁶|φ(0)|) for allt≥1.

Proof. Integrating by parts we obtain PV5Qφ=Cχ(ηt^1/3)P(η)|t|^1/2

Z

R

1

∂_ξS(ξ, η)Q(ξ) 1−χ(1

3ξt^1/3) φ(ξ)dξ

=C|t|^1/2 Z

R

Z

R

e^{−itS(ξ,η)}q2(t, η, ξ)φ(ξ)−φ(0)

ξ dξ

+Cφ(0)|t|^1/2 Z

R

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whereqk(t, η, ξ) =t⁻¹χ(ηt^1/3)P(η)(ξ∂ξ)^k−1(Q(ξ)(1−χ(¹₃ξt^1/3))_∂ ¹

ξS(ξ,η)),q3(t, η, ξ) = t⁻¹χ(ηt^1/3)P(η)∂ξ(Q(ξ)(1−χ(¹₃ξt^1/3))_∂ ¹

ξS(ξ,η)). Next we changeη =µ(x), and the variable of integrationξ=t^−1/3ξ⁰, then we obtain

PV5Qφ=MB⁻¹D_t⁻¹_2/3Z

R

Z

R

e^ixξq2(t, µ(xt^−2/3), ξt^−1/3)D_t1/3e^−itΛ(ξ)φ(ξ)−φ(0)

ξ dξ

+φ(0) Z

R

e^ixξq3(t, µ(xt^−2/3), ξt^−1/3)D_t1/3e^−itΛ(ξ)dξ . Define the pseudodifferential operatorsak(t, x, D)φ≡R

Re^ixξak(t, x, ξ)bφ(ξ)dξ with symbolsa_k(t, x, ξ) =q_k(t, µ(xt^−2/3), ξt⁻¹³). Then we obtain

PV₅Qφ=MB⁻¹D_t⁻¹_2/3

a₁(t, x, D)F⁻¹D_t1/3e^−itΛ(ξ)φ_ξ(ξ) +a₂(t, x, D)F⁻¹D_t1/3e^−itΛ(ξ)φ(ξ)−φ(0)

ξ +φ(0)a3(t, x, D)F⁻¹D_t1/3e^−itΛ(ξ)

.

Let us prove theL²-boundedness of the operatorsa_k(t, x, D),k= 1,2. We have ak(t, x, ξ) =t⁻¹χ t^1/3µ(xt^−2/3)

P(µ(xt^−2/3))(ξ∂ξ)^k−1Q(ξt^−1/3)(1−χ(¹₃ξ)) Λ⁰(ξt^−1/3)−xt^−2/3

, Note thatµ(x) =O(|x|^1/2) for small|x|, thereforeχ(t^1/3µ(xt^−2/3))6= 0 for|x| ≤C, t≥1. Alsoχ₃(ξt^−1/3/µ(xt^−2/3))6= 0 for _t_1/3_µ(xt^ξ_−2/3₎>3/2, hence

|Λ⁰(ξt^−1/3)−xt^−2/3|=|

Z ξt^−1/3

µ(xt^−2/3)

Λ⁰⁰(η)dη| ≥ |

Z ξt^−1/3

2 3ξt^−1/3

Λ⁰⁰(η)dη|

≥ 1

3|ξt^−1/3|Λ⁰⁰(2

3ξt^−1/3)≥C|Λ⁰(ξt^−1/3)|.

Note that |hξi^ν(ξ∂ξ)^jak(t, x, ξ)| ≤ Ct⁻^1+α^{1 +}³ ^α² for all x, ξ ∈ R, t ≥ 1, j = 0,1,2 with someν ∈(0,1), if α1≥0,α2 ∈R,−1≤α1+α2 <2. Therefore by Lemma 2.3 we findkak(t, x, D)φk_L2

x≤Ct⁻^1+α^{1 +}³ ^α²kφk_L2. Then as above we have |Λ⁰⁰|^1/2MB⁻¹D⁻¹_t2/3a₁(t, x, D)F⁻¹D_t1/3e^−itΛ(ξ)φ_ξ(ξ)

L²_η≤Ct⁻^1+α^{1 +}³ ^α²k∂ξφkL²

and

|Λ⁰⁰|^1/2MB⁻¹D⁻¹_t_2/3a2(t, x, D)F⁻¹D_t1/3e^−itΛ(ξ)φ(ξ)−φ(0) ξ

_L₂

η

≤Ct⁻^1+α^{1 +}³ ^α²k∂ξφkL².

Now let us prove the L² - L^∞ boundedness of the pseudodifferential operator a3(t, x, D). We have

a3(t, x, ξ) =t⁻¹χ t^1/3µ(xt^−2/3)

P(µ(xt^−2/3))t^1/3∂ξ

Q(ξt^−1/3)(1−χ(¹₃ξ)) Λ⁰(ξt^−1/3)−xt^−2/3

, Note that |hξi^ν(ξ∂_ξ)^jhξi^δa₃(t, x, ξ)| ≤ C|t|⁻¹³^(α¹^+α²⁾ for all x, ξ ∈ R, t ≥ 1, j = 0,1,2 with some small ν ∈ (0,1), δ > 1/2. Therefore by Lemma 2.3 we find