Time Local Well-posedness for the

Time Local Well-posedness for the

Benjamin-Ono Equation with Large Initial Data

By

NaoyasuKita^∗and Jun-ichi Segata^∗∗

Abstract

This paper studies the time local well-posedness of the solution to the Benjamin- Ono equation. Our aim is to remove smallness condition on the initial data which was imposed in Kenig-Ponce-Vega’s work [13].

§1. Introduction

We consider the initial value problem for the Benjamin-Ono equation:

∂tu+Hx∂_x²u+u∂xu= 0, x, t∈R, u(x,0) =u0(x), x∈R, (1.1)

whereHxdenotes the Hilbert transform, i.e.,Hx=F⁻¹(−iξ/|ξ|)F. The equa- tion (1.1) arises in the study of long internal gravity waves in deep stratified fluid. For the physical background, see Benjamin [3] and Ono [18].

We present the time local well-posedness of (1.1). Namely, we prove the existence, uniqueness of the solution and the continuous dependence on the initial data. There are several known results about this problem. One of their concern is to overcome the regularity loss arising from the nonlinearity. Because of this difficulty, the contraction mapping principle via the associated integral

Communicated by H. Okamoto. Received March 22, 2004. Revised October 19, 2004.

2000 Mathematics Subject Classification(s): 35Q53.

Key words: Benjamin-Ono equation, time local well-posedness, smoothing effect.

∗Faculty of Education and Culture, Miyazaki University, Nishi 1-1, Gakuen kiharudai, Miyazaki 889-2192, Japan.

e-mail: [email protected]

∗∗Graduate School of Mathematics, Kyushu University and Mathematical Institute, Tohoku University, Aoba, Sendai, 980-8578, Japan.

e-mail: [email protected]

equation does not work as long as we consider the estimates only in the Sobolev spaceH_x^s,0, whereH_x^s,αis defined by

H_x^s,α={f ∈ S(R);fH_x^s,α <∞}

with fH_x^s,α =x^αDx^sfL²_x, x^α = (1 +x²)^α/2 and Dx^s=F⁻¹ξ^sF. Indeed, Molinet-Saut-Tzvetkov [17] negatively proved the solvability of the in- tegral equation inH_x^s,0for any s∈R.

Saut [21] proved the global well-posedness for (1.1) in H_x³. Abdelouhab- Bona-Felland-Saut [1] and Iorio [8] proved the time global existence and unique- ness of the solution inH_x^s,0withs >3/2. Their proofs are based on the energy method in which the estimate of ∂xuL^∞_T(L^∞_x) gives the regularity constraint of the initial data. Ponce [19] obtained the global unique solution in Hx^3/2,0

by the combination of energy method and dispersive structure of linear part in (1.1). More recently, Koch-Tzvetkov [14] have studied the local well-posedness withs >5/4 due to the cut off technique ofFu(ξ). Furthermore, Kenig-Koenig [10] proved the local well-posedness with s >9/8. We remark here that it is possible to minimize the regularity ofu₀by inducing another kind of function space. In fact, Kenig-Ponce-Vega [13] construct a time local solution via the integral equation by applying the smoothing property like

Dx

t 0

V(t−t)F(t)dtL^∞_x(L²_T)≤CFL¹_x(L²_T),

where uL^px(L^r_T) = (uL^r[0,T])L^px(R), D_x = F⁻¹|ξ|F and V(t) = exp(−tHx∂_x²). They obtained the time local well-posedness in H_x^s,0 (s > 1) for the cubic nonlinearity (Their argument is also applicable to the quadratic case if u0 satisfies u0 ∈ H_x^s,0 (s > 1) and the additional weight condition).

In their result, however, the smallness of the initial data is required. This is because the inclusionL¹_x(L^∞_T )·L^∞_x (L²_T)⊂L¹_x(L²_T) yieldsuL¹_x(L^∞_T) in the nonlinearity and we can not expect thatuL¹_x(L^∞_T)→0 even whenT →0.

Our concern in this paper is to remove this smallness condition of u0. Before presenting the rough sketch of our idea, we introduce the function space YT in which the solution is constructed:

YT ={u: [0, T]×R→R;|||u|||Y_T <∞}, where |||u|||YT = u_L∞

T(H_x^s,0∩H_x^s1,α1) + x^−ρD_x^s+1/2u_L^1/ε

x (L²_T) + D_x^µx^α1uL²_x(L^∞_T) with ρ, µ > 0 sufficiently small and 0 < ε < ρ.

We first consider the modified equation such that

∂tuν+Hx∂_x²uν+uν∂xην∗uν = 0, uν(0, x) =u0(x),

(1.2)

where η_ν(x) = ν⁻¹η(x/ν) withη ∈ C₀^∞,

η(x)dx = 1 and ν ∈(0,1]. Then, the existence of u_ν in Y_T easily follows and it is continuated as long as uν(t)_Hx^s,0_∩Hx^s1,α1 < ∞. Note that |||uν|||Y_T is continuous with respect to T. To seek for the a priori estimate of |||uν|||Y_T, we deform (1.2). Letϕ∈C₀^∞(R) and writeuν∂xην∗uν =ϕ∂xην∗uν+ (uν−ϕ)∂xην∗uν. Note here that, ifϕ is close tou₀, one can makeu_ν−ϕsufficiently small whent→0. To control the heavy term ϕ∂_xη_ν∗u_ν, we employ the gauge transform (see section 2) so that this quantity is, roughly speaking, absorbed in the linear operator. Then, our desired a priori estimate follows via the integral equation. As for the con- vergence of nonlinearityuν∂xην∗uν →u∂xu, we also consider the estimate of uν−uν in Y_T which is slightly weaker thanYT (see Proposition 6.1). Let us now state our main theorem.

Theorem 1.1. (i)Letu0∈H_x^s,0∩H_x^s1,α1 ≡X^swiths1+α1< s, 1/2<

s₁ and 1/2 < α₁ <1. Then, for some T = T(u₀) >0, there exists a unique solution to (1.1)such thatu∈C([0, T];X^s)∩Y_T.

(ii) Let u(t) be the solution to (1.1) with the initial data u₀ satisfying u₀− u0X^s < δ. Ifδ >0 is sufficiently small,then there exist someT ∈(0, T) and C >0 such that

u−uL^∞_T(X^s)≤Cu₀−u0X^s, x^−ρD_x^s+1/2(u−u)_L^1/ε

x (L²

T)≤Cu₀−u₀X^s.

In Theorem 1.1, the conditions on the initial data are determined by the estimate of maximal function, where, we call f(·, x)L^∞_T the maximal func- tion of f(t, x). Concretely speaking, the quantity uL¹_x(L^∞_T) is bounded by C(u0_H^s,0_x +u0H_x^s1,α1) (see Lemma 4.2).

Remark1. Only for the existence, one can further minimize the regular- ity of the initial data. Abdelouhab-Bona-Fell-Saut [1], Ginibre-Velo [7], Saut [21] and Tom [23] proved the global existence of weak solutions in L²_x,Hx^1/2,0

andH_x^1,0, respectively. Recently, Tao [22] has studied the global well-posedness inH_x^1,0 but theL²-stability of the data-to-solution map holds while the initial data belongs toH_x^1,0, i.e.,u(t)−u(t)L² ≤Cu₀−u0L². More recently, Kato [9] has proved the well-posedness by supposing thatu0∈H_x^swiths >1/2 and roughly speaking,u0satisfies the zero average condition

u0(x)dx= 0.

We also remark that Koch-Tzvetkov [17] and Biagioni-Linares [5] nega- tively proved the strong stability like

u(t)−u(t)H^s,0_x ≤Cu₀−u0H_x^s,0 fors >0,

if there is no weight condition onu₀andu₀. Though our result requires slightly large regularity in comparison with Tao’s work, it suggests that the additional weight condition yields the strong stability of the data-to-solution map in the sense that its target space coincides with that of initial data.

Remark2. The upper bound ofα₁ is required in the proof of weighted norm estimates (see section 4) and especially, in the estimate of [x^α1,Hx∂_x²].

It is possible to relax this weight condition. However, for the simplicity of our argument, we do not handle this kind of generalization in this paper. Let us also remark that the persistence of the solution fails ifα1≥3 (see Iorio [8]).

We close this section by introducing several notations and reviewing typical facts on the pseudo-differential operators. The Fourier transform (2π)^−1/2 e^−ixξf(x)dxis denoted by Ff or ˆf. B(X;Y) stands for the class of bounded operators fromX to Y. For simplicity, we often writeB(X;X) =B(X). The norm of the summation spaceX+Y is given byfX+Y = inf{gX+hY;g+ h = f}. We call a(x, ξ) ∈ C^∞(R×R) belongs to the symbol class S^m if sup_x,ξξ^−m+j|∂_ξ^j∂_x^ka(x, ξ)|<∞. For this symbol, the pseudo-differential op- eratora(x, i⁻¹∂_x) is defined by

a(x, i⁻¹∂_x)f= (2π)^−1/2

e^ixξa(x, ξ) ˆf(ξ)dξ.

Letσ(a(x, i⁻¹∂x)) be the symbol ofa(x, i⁻¹∂x). It is well-known (cf. Kumano- go [16], Stein [20]) that, ifa(x, ξ)∈S andb(x, ξ)∈S^m, then we have

σ(a(x, i⁻¹∂_x)b(x, i⁻¹∂_x))∈S^+mandσ([a(x, i⁻¹∂_x), b(x, i⁻¹∂_x)])∈S^+m−1, where [A, B] =AB−BA. These properties follow from the symbolic expansion formula like

σ(a(x, i⁻¹∂_x)b(x, i⁻¹∂_x)) (1.3)

=

N−1 j=0

1

2πj!i^j∂^j_ξa(x, ξ)∂^j_xb(x, ξ)

+ 1

2π(N−1)!i^NOs- e^{−i(x−y)(ξ−ζ)}∂ζa(x, ζ)

× 1

0

(1−θ)^N−1∂_x^Nb(θy+ (1−θ)x, ξdθ

dydζ,

where Os- stands for the oscillatory integral with respect to y andζ. The expansion formula (1.3) is also applicable even in the case a(x, ξ) = ξ^σ

and b(x, ξ) = x^α, which gives the equivalence of x^αD_x^σfL^px (resp.

x^αD_x^σfL^px(L^r_T)) and D_x^σx^αfL^px (resp. x^αD_x^σfL^px(L^r_T)). For the symbola(x, ξ)∈S^m,|a|^(m)_N denotes the semi-norm defined by

|a|^(m)_N = max

j+k≤Nsup

x,ξ

ξ^−m+j|∂^j_ξ∂_x^ka(x, ξ)|.

We note that, fora(x, ξ)∈S^m,b(x, ξ)∈S^m and arbitraryN >0, there exist some N >0 andC >0 such that

|σ(a(x, i⁻¹∂x)b(x, i⁻¹∂x))|^(m+mN ⁾≤C|a|^(m)N |b|^(mN⁾.

Furthermore, fora(x, ξ)∈S^m,a(x, i⁻¹∂x)∈ B(L²_x) if m≤0 anda(x, i⁻¹∂x)∈ B(L^p_x) (p∈[1,∞]) ifm <0. In these estimates, we see that the operator norms a(x, i⁻¹∂x)_B(L^p_x)anda(x, i⁻¹∂x)_B(L^p_x(L^r_T))are estimated by|a|^(m)N for some N >0. We also denotet

0V(t−τ)F(τ)dτ byG(t)F.

§2. Gauge Transform

In this section, we transform (1.2) appropriately for the a priori estimate.

We write

∂tuν+Hx∂_x²uν+ϕην∗∂xuν

(2.1)

+ (uν−ϕ)ην∗∂xuν= 0,

with ϕ∈C₀^∞(R) to be chosen closely to u0 in H_x^s,0∩H_x^s1,α1. We next define the gauge transformation of pseudo-differential operator with the symbol:

Kν(x, ξ) = exp π

2 iξ

|ξ|(1−ψ(ξ))ˆη(νξ) x

−∞

ϕ(y)dy

,

whereψ∈C₀^∞(R) satisfies ψ(ξ) =

1, if|ξ|<1, 0, if|ξ|>2.

(2.2)

Note thatK_ν(x, ξ)∈S⁰ uniformly inν ∈(0,1]. Applying K_ν ≡K_ν(x, i⁻¹∂_x) to (2.1) and lettingvν=Kνuν, we have

∂_tv_ν+Hx∂_x²v_ν+K_ν(u_ν−ϕ)η_ν∗∂_xu_ν+R_ν(ϕ, u_ν) = 0, (2.3)

whereR_ν(ϕ, u) = (K_νϕη_ν∗∂_x+[K_ν,Hx∂_x²])u. Note that the symbol ofK_νϕη_ν∗

∂x+ [Kν,(1−ψ(i⁻¹∂x))Hx∂_x²] belongs to S⁰ uniformly inν ∈(0,1] since the

top symbol of the commutator is −∂_xK_ν(x, ξ)∂_ξ((1−ψ)ξ|ξ|). The desired a priori estimate ofu_ν will be obtained in terms ofv_ν by transforming (2.3) into the integral equation. Henceforth, we are led to the preliminaries about several estimates ofV(t) etc. (These are given in the next section.)

§3. Preliminary

We introduce several linear estimates. The first lemma gives the smoothing effects due to Kenig-Ponce-Vega [13] which overcome a loss of regularity in the nonlinearity.

Lemma 3.1. Let p∈[2,∞]. Then we have D^1/2_x ^−1/pV(t)φL^p_x(L²_T)≤CT^1/pφL²_x, (3.1)

D_x^1−1/pG(t)FL^p_x(L²_T)≤CT^1/pFL¹_x(L²_T), (3.2)

D_x^1/2G(t)FL^∞_T(L²_x)≤CFL¹_x(L²_T). (3.3)

Proof of Lemma 3.1. The case p = ∞ and (3.3) are given in [13]. By (3.1) and (3.3), it is easy to see that, forλ∈R,

D_x^iλV(t)φL²_x(L²_T)≤T^1/2φL²_x, (3.4)

D^1/2+iλ_x V(t)φL^∞_x(L²_T)≤CφL²_x, (3.5)

D^1/2+iλG(t)FL²_x(L²_T)≤CT^1/2FL¹_x(L²_T). (3.6)

whereC >0 is independent ofλ. Also, in [13], the following estimate appears:

D_x^1+iλG(t)FL^∞_x(L²_T)≤Cλ^NFL¹_x(L²_T), (3.7)

where N is a large positive integer. Then, applying Stein’s interpolation for analytic families of operators to the pairs (3.4)–(3.5) and (3.6)–(3.7), we obtain the desired estimates forp=∞.

We next state the Strichartz estimates (for the proof, see [6, p. 377] and refer to [25]). These inequalities will be used for the weighted norm estimates.

Lemma 3.2. Let p_j andr_j (j= 1,2) satisfy 0≤2/r_j = 1/2−1/p_j ≤ 1/2. Then, we have

V(t)φL^r_T¹(L^px¹)≤CφL²_x, (3.8)

G(t)FL^r_T¹(L^p_x¹)≤CF

L^r_T²(L^p_x²), (3.9)

where1/p2+ 1/p₂= 1/r2+ 1/r₂ = 1.

We call u(·, x)L^∞_T the maximal function of u. Concerning the estimates of maximal functions, we have the following.

Lemma 3.3. Let σ >1/2 andT ∈[0,1]. Then, we have V(t)φL²_x(L^∞_T)≤Cφ_Hx^σ,0,

(3.10)

G(t)FL²_x(L^∞_T)≤CFL¹_T(H_x^σ,0), (3.11)

G(t)FL²_x(L^∞_T)≤CD_x^σFL¹_T(L¹_x). (3.12)

Proof of Lemma 3.3. The estimate (3.10) is due to Vega [24]. From Minkowski’s inequality, (3.11) follows. To prove (3.12), we first show that the integral kernel ofDx^−σexp(tHx∂_x²) (denoted byK(t, x−y)) is estimated as

|K(t, x−y)| ≤C

|x−y|^−σ if|x−y|>1,

|x−y|^−1+σif|x−y| ≤1, (3.13)

whereC >0 does not depend ont∈(0, T]. Letz=x−y and write K(t, z) = (2π)⁻¹

_∞

−∞

exp(−itξ|ξ|+izξ)ξ^−σdξ

= (2π)⁻¹ _∞

0

exp(−itξ²+izξ)ξ^−σdξ + (2π)⁻¹

0

−∞

exp(itξ²+izξ)ξ^−σdξ

≡K+(t, z) +K₋(t, z).

We only consider the estimate ofK+(t, z) sinceK₋(t, z) is similarly estimated.

Changing the integral variable, we can write K+(t, z) =

e^iz2/4tz_∞

−1e^−itz2ξ2z(ξ+ 1)^−σdξ ifz >0,

−e^iz2/4tz₋1

−∞e^−itz2ξ2z(ξ+ 1)^−σdξifz <0, wherez=z/2t. Let us mainly consider the casez >0 step by step.

(The case z > 1) The identity ∂_ξξe^−itz2ξ2 = (1−2itz²ξ²)e^−itz2ξ2 and integration by parts give

|K+(t, z)| ≤z|1−2itz²|⁻¹+z _∞

−1

|ξ∂ξ((1−2itz²ξ²)⁻¹z(ξ+ 1)^−σ)|dξ

≤Cz^−σ.

(The case 0 < z ≤1 and tz² >1) Let χ₋₁ ∈ C₀^∞(R) with χ₋₁(ξ) = 1 nearξ=−1, and let ˜χ₋₁= 1−χ₋₁. Then, we see that

|K+(t, z)| ≤ z

_∞

−1

e^−itz2ξ2χ₋1(ξ)z(ξ+ 1)^−σdξ +

z _∞

−1

e^−itz2ξ2χ˜₋1(ξ)z(ξ+ 1)^−σdξ

≡ |K+,1(t, z)|+|K+,2(t, z)|.

To estimate |K+,1(t, z)|, we use the identity ∂ξ(e^−itz2ξ2 − e^−itz2) =

−2itz²ξe^−itz2ξ2 and integration by parts. This yields (3.14)

|K+,1(t, z)| ≤Ct⁻¹z⁻¹ _∞

−1

(∂ξξ⁻¹χ₋1)z(ξ+ 1)^−σdξ +

_∞

−1

ξ⁻¹χ₋1|e^−itz2ξ2−e^−itz2|zz(ξ+ 1)^−σ−1dξ

≤Ct⁻¹z^−1−σ+Ct⁻¹z^−1−σ 0

−1

|e^−itz2ξ2−e^−itz2|(ξ+ 1)^−σ−1dξ.

The integral in (3.14) is bounded byC(tz)^σ since, for 0< R <1 andσ > σ, we have

0

−1

|e^−itz2ξ2−e^−itz2|(ξ+ 1)^−σ−1dξ

≤C 0

−1+R

(ξ+ 1)^−σ−1dξ+C

₋1+R

−1

(tz²)^σ(ξ+ 1)^σ−σ−1dξ

≤C(R^−σ+ (tz²)^σR^σ−σ)

withR= (σ/(σ−σ))^1/σ(tz²)⁻¹. Thus, noting thatz >2t^1/2in this case, we have |K+,1(t, z)| ≤Cz^−1+σ. The estimate |K+,2(t, z)| ≤Cz^−1+σ follows from the identity∂ξξe^−itz2ξ2= (1−2itz²ξ²)e^−itz2ξ2 and integration by parts.

(The case0< z≤1andtz²≤1) Changing the integral variable, we have another expression ofK₊(t, z) such that

K₊(t, z) =e^iz2/4tt^−1/2 _∞

−t^1/2z

e^−iξ2t^−1/2(ξ+t^1/2z)^−σdξ

=e^iz2/4tt^−1/2 _∞

−t^1/2z

e^−iξ2χ₀(ξ)t^−1/2(ξ+t^1/2z)^−σdξ +e^iz2/4tt^−1/2

_∞

−t^1/2z

e^−iξ2(1−χ₀(ξ))t^−1/2(ξ+t^1/2z)^−σdξ

≡K+,3(t, z) +K+,4(t, z),

where χ₀ ∈ C₀^∞(R) with χ₀(ξ) = 1 in the neighborhood of [−1,0]. It is easy to see that |K_+,3(t, z)| ≤ Ct^{−(1−σ)/2} ≤ Cz^−1+σ. Making use of the identity ∂ξe^−iξ2 = −2iξe^−2iξ2 and integration by parts, we can show that

|K+,4(t, z)| ≤Ct^{−(1−σ)/2} ≤Cz^−1+σ. Thus, (3.13) follows. Hence, (3.13) and Young’s inequality yield (3.12).

In the nonlinear estimates, the fractional order differentiation will be ap- plied to the quadratic term in (2.3). To handle this, we require the Leibnitz’

type rule for the fractional order derivatives due to Kenig-Ponce-Vega [12, Ap- pendix].

Lemma 3.4. Let σ∈(0,1) andσ₀, σ₁∈[0, σ]with σ=σ₀+σ₁. Also, let q ∈ [1,∞) and q₀, q₁, r₀, r₁ ∈ (1,∞) with 1/q = 1/q₀+ 1/q₁ and 1/2 = 1/r₀+ 1/r₁. Then, we have

(3.15)

D_x^σ(f g)−(D^σ_xf)g−f(D^σ_xg)L^qx(L²_T)≤CD_x^σ1fL^q_x⁰(L^r_T⁰)D^σ_x²gL^q_x¹(L^r_T¹). When we apply the Leibnitz’ rule for the fractional order derivative to the nonlinearity, we encounter the estimate of lower order derivatives likeDx^s−1/2u and∂xu. The following lemma and its corollary help us control these quantities.

In particular, we require the case q₀ = 1, q₁ =∞, r₀=∞and r₁= 2 (the end point case of the interpolation).

Lemma 3.5. Let σ0, σ1 > 0, α0, α1 ∈ R and q0, q1, r0, r1 ∈ [1,∞].

Also, letσ= (1−θ)σ0+θσ1,α= (1−θ)α0+θα1, 1/q= (1−θ)/q0+θ/q1 and 1/r= (1−θ)/r0+θ/r1 withθ∈[0,1]. Then,forf ∈ S(R;C^∞[0, T]),we have

D_x^σx^αfL^q_x(L^r_T)

(3.16)

≤

sup

λ∈R

e^−λ2D^σ_x^{0+iλ(σ1−σ0)}x^{α0+iλ(α1−α0)}fL^q_x⁰(L^r_T⁰)

1−θ

×

sup

λ∈R

e^1−λ2D^σ_x^{1+iλ(σ1−σ0)}x^{α1+iλ(α1−α0)}fL^q_x¹(L^r_T¹)

θ

.

Proof of Lemma3.5. We first define the complex valued functionF(z) by F(z) =e^z2

R×[0,T]

g_z(t, x)D^σ(z)_x x^α(z)f(t, x)dtdx, whereσ(z) = (1−z)σ₀+zσ₁,α(z) = (1−z)α₀+zα₁ and

gz(t, x) =g(·, x)^q_Lr^{/q(z)−r/r(z)} T

|g(t, x)|^r/r(z) sgng(t, x)

withg∈ S(R;C^∞[0, T]), 1/q(z) = (1−z)/q₀+z/q₁ and 1/r(z) = (1−z)/r₀+ z/r₁ (the prime denotes the H¨older conjugate). Then, F(z) is holomorphic in the strip S = {z ∈ C; 0 < Rez < 1} and continuous in S. In addition, lim_|Imz|→∞|F(z)| = 0 in virtue of the multiplication e^z2. According to the three line theorem, we see that

|F(z)| ≤M₀^1−RezM₁^Rez, (3.17)

whereMj= sup_λ∈R|F(j+iλ)|(j= 0,1). By applying H¨older’s inequality, (3.18)

M_j≤ g^q/qj

L^q_x(L^r_T)sup

λ

e^j−λ2D_x^{σj+iλ(σ1−σ0)}x^{αj+iλ(α1−α0)}f_L^qj

x(L^rj_T)

.

Combining (3.17)–(3.18) and (L^q_x(L^r_T)) ∼ L^q_x(L^r_T) with z = θ, we obtain Lemma 3.5.

Corollary 3.6. In addition to the assumptions in Lemma3.5,letµ >0.

Then,we have

Dx^σx^αfL^q_x(L^r_T)

(3.19)

≤CDx^σ0+µx^α0f¹_L^−q0^θ

x(L^r_T⁰)Dx^σ1+µx^α1f^θ_L^q1 x (L^r_T¹).

Proof of Corollary 3.6. By estimating the integral kernels of operators, we see, for instance, that

D_x^{σ0+iλ(σ1−σ0)}x^{iλ(α1−α0)}D_x^−(σ0+µ)_B(L^q_x⁰(L^r_T⁰))≤Cλ^N

withN sufficiently large. Then, Lemma 3.5 yields the desired result.

In our argument, the pseudo-differential operatorKν often appears. We note thatKν ∈ B(L¹_x) uniformly inν ∈(0,1] since the symbol ofKν contains the gap for ξ =±∞if ν = 0. The following lemma states thatK_ν ∈ B(L^p_x) (1< p <∞) and its operator norm is estimated in terms ofϕX^s.

Lemma 3.7. Let p∈(1,∞)andν ∈(0,1). Then,we have Kν_B(L^p_x)≤C,

(3.20)

where the positive constant C is independent ofν ∈(0,1]and does not diverge as ϕ→u₀ in X^s. Furthermore, in the above inequalities, we may replace L^p_x by L^p_x(L^r_T)withr∈(1,∞).

Proof of Lemma3.7. Note thatK_ν(x, ξ) =L_ν,+(x, ξ)χ₊(ξ) +L_ν,−χ₋(ξ), where

Lν,±(x, ξ) = exp

±i π

2(1−ψ(ξ))ˆη(νξ) x

−∞

ϕdy

and χ₊(ξ) (resp. χ₋(ξ)) is the characteristic function on (0,∞) (resp.

(−∞,0)). It is well-known that χ_±(i⁻¹∂x) ∈ B(L^p_x), and thus it suffices to show that Lν,±(x, i⁻¹∂x)∈ B(L^p_x). We write

Lν,±(x, ξ) =ψ(ξ/2)Lν,±(x, ξ) + (1−ψ(ξ/2))Lν,±(x, ξ) (3.21)

≡Lν,±,1(x, ξ) +Lν,±,2(x, ξ).

By the integration by parts, the integral kernels of Lν,±,1(x, i⁻¹∂x) (denoted byLν,±,1[x, y]) are estimated as

|L_ν,±,1[x, y]|= (2π)⁻¹

e^i(x−y)ξL_ν,±,1(x, ξ)dξ

≤Cexp(CϕX^s)x−y^−N, whereN >0 is sufficiently large. Also, note that

Lν,±,2(x, ξ) (3.22)

= (1−ψ(ξ/2)) exp

±i(π/2)^1/2(1−ψ(ξ))ˆη(νξ) x

−∞

ϕ(y)dy

= (1−ψ(ξ/2)) +(1−ψ(ξ/2))

exp

±i(π/2)^1/2η(νξ)ˆ x

−∞

ϕ(y)dy

−1

,

where we remarked that, if 1−ψ(ξ/2) = 0, then 1−ψ(ξ) = 1. Furthermore, the symbol exp(i⁻¹(π/2)^1/2η(νξ)ˆ x

−∞ϕ(y)dy)−1 yields the integral operator with the kernel bounded byCexp(CϕX^s)ν⁻¹(x−y)/ν^−N, and 1−ψ(ξ/2) yieldsL^p_x-bounded operator. Hence, we see thatL_ν,±,2(x, i⁻¹∂_x)∈ B(L^p_x) and (3.20) follows.

§4. Weighted Norm Estimates

In this section, we derive several linear estimates in the weighted norms, which bring us the persistence of the solution.