Short-range scattering of Hartree type fractional NLS

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Short-range scattering of Hartree type fractional NLS Yonggeun Cho Department of Mathematics, and Institute of Pure and Applied Mathematics, Chonbuk National University, Jeonju 561-756, Republic of Korea Received 1 February 2016; revised 18 July 2016

Abstract In this paper we consider scattering problem for Hartree type fractional NLS with |∇|α (1 < α < 2) and potential V ∼ |x|−γ . We show small data scattering in a weighted space for the short range 6−2α 4−α < γ < 2. The difficulty arises from the non-locality and non-smoothness of |∇|α . To overcome it we utilize the method of commutator estimate based on Balakrishnan’s formula. © 2016 Elsevier Inc. All rights reserved. MSC: 35Q55; 35Q40 Keywords: Fractional NLS; Hartree type potential; Short-range interaction; Small data scattering

1. Introduction In this paper we consider a small data scattering problem of the fractional nonlinear Schrödinger equation (fNLS) with Hartree type potential:  i∂t u = |∇|α u + (V ∗ |u|2 )u in R1+d , (1.1) u(0) = ϕ ∈ H s (Rd ), where d ≥ 1, 1 < α < 2, s ≥ 0 and V is a complex-valued measurable function on Rd . Here α |∇|α = (−) 2 = F −1 |ξ |α F is the fractional derivative of order α and ∗ denotes the space convolution. By Duhamel’s formula, (1.1) is written as an integral equation E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jde.2016.09.025 0022-0396/© 2016 Elsevier Inc. All rights reserved.

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u(t) = e

−it|∇|α

t (ϕ) − i



e−i(t−t )|∇| (V ∗ |u(t  )|2 )u(t  ) dt  , α

(1.2)

0

where the linear propagator e−it|∇| f is the solution to the linear problem i∂t z = |∇|α z with initial datum f . Then it is formally given by α

e

−it|∇|α

f =F

−1 −it|ξ |α

e

Ff = (2π)

−d



α ei(x·ξ −t|ξ | ) f(ξ ) dξ.

(1.3)

Rd

 Here f= Ff is the Fourier transform of f such that f(ξ ) = Rd e−ix·ξ f (x) dx and we denote  its inverse Fourier transform by F −1 g(x) = (2π)−d Rd eix·ξ g(ξ ) dξ . During the decade fractional NLS including semi-relativistic equations have been extensively studied to describe natural phenomena in the context of fractional quantum mechanics [18,19], system of bosons [11], system of long-range lattice interaction [16], water waves [22], turbulence [3] and so on. Heuristically, Hartree nonlinearity can be interpreted as an interaction between particles with potential V [11]. The potential V taken into account is of short-range interaction as follows: (S-R1) V is differentiable on Rd \ {0} and for 1 < γ < d it satisfies that |V (x)| + |x||∇V (x)| ≤ C|x|−γ , ∀x = 0.

(1.4)

If 0 < γ ≤ 1, then V is referred to be of long-range interaction. We adopt the terminologies short-/long-range interaction from the scattering theory of Hartree equations (α = 2). In this paper we are concerned with small data scattering of (1.1) for a class of potentials V . Definition 1.1. We say a solution u to (1.1) scatters (to u± ) in a Hilbert space H if there exist α ϕ± ∈ H (with u± (t) = e−it|∇| ϕ± ) such that α

lim eit|∇| u(t) − ϕ± H = 0.

t→±∞

If V has a long range, it was shown in [6] that scattering may not occur even in L2 . The short-range scattering in H s can be shown simply by Strichartz estimates on the real line when 2 < γ < d and s > s∗ := γ −α 2 since the dispersion of solution is fast. This is also the case for Hartree and semi-relativistic equation. See [12,8,13,5]. But in case when 1 < γ ≤ 2, the dispersion of solution to (1.1) is not good for Strichartz estimate on the whole time. One may usually think of two ways (radial symmetry, weighted normed space) to observe scattering. Under the radial assumption the global Strichartz estimate can cover the range 1 < γ ≤ 2 in part. To be 2d more precise, small data scattering in H s∗ is possible when α, γ are restricted to 2d−1 ≤α<2 and α ≤ γ < d. For this see [5]. In [7] even a large data scattering in energy space is treated 2d under radial symmetry when γ = 2α (energy-critical) and 2d−1 < α < 2. The other way is to use a weighted normed space as in Hartree [14] and semi-relativistic [13] equations. If the initial data is in a weighted space, then the solution could be dispersive enough to scatter.

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In this paper we establish a way of weighted space for the equation (1.1) with V in a short range. For this purpose we introduce a weighted space  H (a, b ; s) = {f ∈ H s  P≤1 (xf ) ∈ H˙ a , P>1 (xf ) ∈ H˙ b }, a, b, s ≥ 0 with norm defined by

f H (a,b ;s) := P≤1 (xf ) H˙ a + P>1 (xf ) H˙ b + f H s . P≤1 and P>1 are low and high frequency projection operators. To give a precise definition ∞ (1/2 ≤ |ξ | ≤ 2) with 0 ≤ π ≤ 1, π = 1 let us introduce Littlewood–Paley function π ∈ C0,rad 3 3 on { 4 ≤ |ξ | ≤ 2 } and π π = π , where  π (ξ ) = π(ξ/2) + π(ξ ) + π(2ξ ). Then the Littlewood– N = Paley operator PN is defined by F(PN f )(ξ ) = π(ξ/N )f for anydyadic number N . Let P  PN/2 + PN + P2N . Then PN PN = PN . We denote P≤ρ := 1 − N >ρ PN , P>ρ := N >ρ PN and Pρ1 <·≤ρ2 := ρ1
provided f is sufficiently smooth. By density Jf = xf + iαt|∇|α−2 ∇f for any f ∈ H (a, b; s) if a, b ≥ 0 and s ≥ α − 1. Then we define the function space XJ (a, b ; s) associated with J by 

 ∞ ˙b ˙a XJ (a, b ; s) := v ∈ Cb (R; H s )P≤1 (Jv) ∈ L∞ t H , P>1 (Jv) ∈ Lt H s. and its norm by v XJ (a,b ;s) := P≤1 (Jv) L∞ ˙ a + P>1 (Jv) L∞ ˙ b + v L∞ t H t H t H Contrary to NLS or semi-relativistic equation the low frequency part of Ju is defined by the homogeneous Sobolev norm not by L2 norm. (If L2 norm were available, then the whole argument would be much simpler.) This is due to the bad behavior of J arising from the non-locality of |∇|α−2 ∇. More precisely, the equation (1.1) has no Galilean invariance. Thus the evolution equation of Ju has an inhomogeneous with nonlocal operator |∇|α−2 ∇. By  term E associated taking J on both sides of (1.1), since J, i∂t − |∇|α = 0, we have

 (i∂t − |∇|α )Ju = (V ∗ |u|2 )Ju + E, E = iαt |∇|α−2 ∇, V ∗ |u|2 u. To get L2 estimate of Ju it is necessary to show E ∈ L1t L2x . E can bedecomposed in several ways by an acceptable commutator term E∈ L1t L2x and an error, e.g. iαt Rx , V ∗ |u|2 |∇|α−1 u, where Rx = F −1 (ξ/|ξ |)F . Unfortunately we cannot show for the present that such error is in L1t L2x within the short range 1 < γ < 2 since the commutator with the singular integral operator Rx does not give any smoothing effect in L2 . In Lp we can expect additional time decay as in 2d Corollary 2.8 and can show it for some 2 < p < d−2(2−α) when γ0 < γ < 2 for some 1 < γ0 < 2. These issues will be treated in the appendix. Similar situation arises in semi-relativistic equation

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 1 1 with E = it ∇(1 − )− 2 , V ∗ |u|2 u. But thanks to the smoothness of (1 − )− 2 one can obtain L1t L2x estimate for some b, s. See [13] and also [20]. We need to overcome the non-smoothness of |∇|α−2 ∇ and want to get an intuition from Lp variant. The idea to meet both demands is to add |∇|c to E and to investigate suitable exponents a, b, c and s guaranteeing |∇|c E L1 L2  t

x

u 3XJ (a,b;s) . Eventually, we will show that a = c = γ2 (2 − α), b = 1 + γ2 (2 − α) and s = 1 + γ2 . This is the reason for homogeneous Sobolev norm. From now let us set γα =

γ (2 − α) 2

and H (γα ) := H (γα , 1 + γα ; 1 +

γ γ ) and XJ (γα ) := XJ (γα , 1 + γα ; 1 + ). 2 2

Then the following is our main result. Theorem 1.2. Let 1 < α < 2, 6−2α 4−α < γ < 2 and d ≥ 3. Suppose that V satisfies (S-R1) and ϕ ∈ H (γα ) has sufficiently small H (γα ) norm. Then there exists a unique solution u ∈ XJ (γα ) scattering in H (γα ). By changing the role of α and γ we get an equivalent statement as follows. Theorem 1.3. Let 1 < γ < 2, max( 6−4γ 2−γ , 1) < α < 2 and d ≥ 3. Suppose that V satisfies (S-R1) and ϕ ∈ H (γα ) has sufficiently small H (γα ) norm. Then there exists a unique solution u ∈ XJ (γα ) scattering in H (γα ). To show Theorem 1.2 we extend the idea mentioned above to the trilinear operator Hγ defined by t Hγ (u1 , u2 , u3 ) := −i



e−i(t−t )|∇| (Vγ (u1 u2 ) u3 )(t  ) dt  , α

0

where Vγ (f ) := V ∗ f for V satisfying (S-R1) with trilinear estimate: for any u1 , u2 , u3 ∈ XJ (γα )

6−2α 4−α

Hγ (u1 , u2 , u3 ) XJ (γα ) ≤ C

3

< γ < 2. We will show the following

uj XJ (γα ) ,

(1.5)

j =1

which gives us an immediate proof of Theorem 1.2. Adding |∇|γα to J causes some cumbersome commutator estimates. To raise the efficiency of proof we use Balakrishnan’s formula such that

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∞

|∇| , f = cs s

 s ρ 2 (ρ − )−1 − , f (ρ − )−1 dρ,

5

(1.6)

0

where cs = sin(πs/2) > 0. For the proof see [1,17,2]. This formula was utilized in [17,2], which π enables us to transform low/high frequency analysis into simple integral estimates as in the spectral theory. One of the most serious commutator terms is ∗

−iαt|∇|−γα

 2−α  |∇| , Vγ (u1 u2 ) |∇|α−2 ∇u3

for which a possible bound is  C

γ

(1 + |t|)−γ +(1− 2 )(2−α) dt

3

uj XJ (γα ) .

j =1

R

This is finite only when γ > 6−2α 4−α . (See the arguments above (4.7).) While carrying out estimates, we encounter the norm uj for small ε > 0. To apply time decay estimate 2d L d−(γ +ε)

(Corollary 2.8) γ should be less than 2. These are presumably caused by technical issues. In the future work we hope to resolve those issues and hence the remaining scattering problem of 1 < γ ≤ 6−2α 4−α or γ = 2. Finally we make a remark for the case γ = 2. We modify V behaving like |x|−2 slightly into a potential close to |x|−2 but behaving better in the following sense: (S-R2) There exist positive real numbers 0 < γ1 , γ2 < d, 0 < ρ1 < ρ2 , complex numbers c1 , c2 and complex-valued functions ψ1 , ψ2 ∈ L1x , ψ0 ∈ L1x ∩ L∞ x such that V0 := Pρ1 <·≤ρ2 V = ψ0 , V1 := P≤ρ1 V = c1 |x|−γ1 + |x|−γ1 ∗ ψ1 ,

(1.7)

V2 := P>ρ2 V = c2 |x|−γ2 + |x|−γ2 ∗ ψ2 . Then as a corollary we have the following. Corollary 1.4. Suppose that V satisfies (S-R2) with γ1 = 2 and 0 < γ2 < 2. Let max(γ2 , 6−2α 4−α ) < γ < 2 and γα be as in Theorem 1.2. If ϕ ∈ H (γα ) and ϕ H (γα ) is sufficiently small, then there exists a unique solution u ∈ XJ (γα ) scattering in H (γα ). [6] considered another type of V such that 2 < γ1 < d and γ2 = 2 and showed the scattering p p in H s , s > 2−α 2 by using U − V method. To show Corollary 1.4 one can observe from (1.7) that for a smooth cut-off function η   0 , V0 = |ξ |−(d−γ ) (η(ξ/ρ2 ) − η(ξ/ρ1 ))ψ   1 (ξ )) , V1 = |ξ |−(d−γ ) |ξ |2−γ η(ξ/ρ1 )|ξ |−(d−2) (c1 + ψ

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  2 (ξ )) . V2 = |ξ |−(d−γ ) |ξ |γ0 −γ (1 − η(ξ/ρ2 ))|ξ |−(d−γ0 ) (c2 + ψ Since γ2 < γ < 2, the Fourier inversion leads us to V = |x|−γ ∗ ψ for some ψ ∈ L1x . The remaining thing is only to observe that the convolution by integrable function ψ is harmless in the whole analysis. We leave this part to the readers. This paper is organized as follows: In section 2 we introduce Sobolev lemmas and time decay estimates. In section 3 we prove Theorem 1.2 via (1.5). Section 4 is devoted to the proof of trilinear estimate (1.5). In the final section we make further remark on L2 estimate for E . Further notations. s

s

s

• Fractional derivatives: |∇|s = (−) 2 = F −1 |ξ |s F , (1 − ) 2 = F −1 (1 + |ξ |2 ) 2 F for s > 0. • Function spaces: H˙ rs = |∇|−s Lr , H˙ s = H˙ 2s , Hrs = (1 − )−s/2 Lr , H s = H2s , Lr = Lrx (Rd ) for some s ∈ R and 1 ≤ r ≤ ∞. For vector-valued function f = (f1 , · · · , fn ) we use the norm

f X := nj=1 fj X . q q q q q q q • Mixed-normed spaces: LI Lrx = Lt (I ; Lrx (Rd )), LI,x = LI Lx , Lt Lrx = LR Lrx and ∞ Cb (R; X) = (Lt ∩ C)(R; X).     • , is the complex inner product in L2 . For vector-valued functions f ; g =    1≤j ≤n fj , gj for f = (f1 , · · · , fn ), g = (g1 , · · · , gn ). • As usual different positive constants depending only on d, α are denoted by the same letter C, if not specified. A  B and A  B means that A ≤ CB and A ≥ C −1 B, respectively for some C > 0. A ∼ B means that A  B and A  B. 2. Preliminaries 2.1. Sobolev type lemmas Here we introduce well-known Sobolev lemmas. Lemma 2.1 (Hardy–Littlewood–Sobolev inequality (HLS)). Let 0 < β < d and 1 < p < ∞. Then we have

|∇|−β Lp →Lq  1 if

1 1 β = − . q p d

For this see [21]. As a corollary we have Lemma 2.2 (Sobolev inequality). Let 0 < δ < d2 . Then for any f ∈ H˙ δ

f

2d

L d−2δ

L∞ version of HLS is the following.

 f H˙ δ .

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Lemma 2.3 (HLS-∞). Let 0 < γ < d and 0 < ε < d − γ . Then 1

1

|x|−γ ∗ (f g) L∞  f 2

2d L d−(γ −ε)

1

f 2

1

g 2

2d L d−γ

2d L d−γ

g 2

2d

L d−(γ +ε)

.

Proof. The lemma follows from the well-known inequality

|x|−γ ∗ |h|2 L∞  h

2d

L d−(γ −ε

)

h

for 0 < ε  < d − γ . 1 In fact, by letting h = |f g| 2 and applying (2.1) with ε  = 1

ε 2

(2.1)

)

we have

1

|x|−γ ∗ (f g) L∞ ≤ |x|−γ ∗ (|f g| 2 )2 L∞  |f g| 2 1

2d

L d−(γ +ε

1

2d

L d−(γ −ε/2)

|f g| 2

2d

L d−(γ +ε/2)

1

 f g 2

d L d−(γ −ε/2)

By Hölder’s inequality with pairs such that d−(γ +ε) we get the desired inequality. 2 2d

f g 2

d

d−(γ −ε/2) d

=

L d−(γ +ε/2)

.

d−(γ −ε) 2d

+ d−γ 2d and

d−(γ +ε/2) d

=

d−γ 2d

+

The following is on the fractional Hardy–Sobolev inequality. For this see Theorem 2.5 of [15]. Lemma 2.4 (Hardy–Sobolev inequality (HS)). Let 0 < δ < d2 . Then for any f ∈ H˙ δ

|x|−δ f L2  f H˙ δ . The last one is on the comparison between two fractional derivatives: Lemma 2.5. Let 0 < θ < 2 and 1 < p < ∞. Then there holds for any ρ > 0 θ

|∇|θ (ρ − )−1 Lp →Lp  ρ −1+ 2 . This follows from dilation such that θ |∇|θ (ρ − )−1 f (x) = ρ −1+ 2 |∇|θ (1 − )−1 f

 1 ρ2

ρ

− 12

(x),

fδ (x) = δ −d f (x/δ)

and the estimate |∇|θ (1 − )−1 Lp →Lp  1. For the latter see [21]. 2.2. Time decay estimates 2d

Lemma 2.6. Let 0 < β < d/2 and φ ∈ H β ∩ |∇|−β(2−α) L d+2β . Then

e−it|∇| φ α

2d

L d−2β

 (1 + |t|)−β ( φ H β + |∇|β(2−α) φ

2d

L d+2β

).

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Proof of Lemma 2.6. If |t| ≤ 1, then by Sobolev inequality we have

e−it|∇| φ α

2d

L d−2β

 φ H β .

If |t| > 1, then we use the time decay estimate of [9] to get

e−it|∇| φ α

2d

L d−2β

This completes the proof of Lemma 2.6.

 |t|−β |∇|β(2−α) φ

2d

L d+2β

.

2

Lemma 2.7. If 0 < β < 1 and φ ∈ H (β(2 − α), β(2 − α) ; β), then

e−it|∇| φ α

2d

L d−2β

 (1 + |t|)−β φ H (β(2−α),β(2−α) ; β) .

Proof of Lemma 2.7. The lemma follows from Lemma 2.6 and weighted estimate

|∇|β(2−α) φ

2d

L d+2β

 |φ H β(2−α) + xφ H˙ β(2−α) .

The latter estimate can be shown as follows:

|∇|β(2−α) φ

2d

L d+2β

≤ (1 + |x|)−1 Ld/β (1 + |x|)|∇|β(2−α) φ L2  |∇|β(2−α) φ L2 + x|∇|β(2−α) φ L2  L2 + |∇|β(2−α) xφ L2  |∇|β(2−α) φ L2 + |ξ |β(2−α)−1 φ  |∇|β(2−α) φ L2 + |x|1−β(2−α) φ L2 + |∇|β(2−α) xφ L2 (by HS)  |∇|β(2−α) φ L2 + |x|−β(2−α) xφ L2 + |∇|β(2−α) xφ L2  |∇|β(2−α) φ L2 + |∇|β(2−α) xφ L2 (by HS).

2

Corollary 2.8. If 0 < β < 1 and for each t ∈ R u(t) ∈ XJ (β(2 − α), β(2 − α) ; β), then

u

2d

L d−2β

 (1 + |t|)−β u XJ (β(2−α),β(2−α) ; β) .

Proof of Corollary 2.8. From Lemma 2.7 it follows that

u

= e−it|∇| [eit|∇| u] α

2d L d−2β

α

 (1 + |t|)−β eit|∇| u H (β(2−α),β(2−α) ; β) α

2d L d−2β

 (1 + |t|)−β ( u H β + Ju H˙ β(2−α) ).

2

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3. Proof of Theorem 1.2 Once we showed (1.5), the proof is straightforward. Let YJλ be such that     YJλ = v ∈ XJ (γα ) v XJ (γα ) ≤ λ . Then YJλ is a complete metric space with the metric d(u, v) := u − v XJ (γα ) . Let us define the nonlinear functional α by α (u) := e−it|∇| (ϕ) + Nα (u), α

t  α where Nα (u) = −i 0 e−i(t−t )|∇| (V ∗ |u(t  )|2 )u(t  ) dt  . Then it can be readily shown that it is a contraction on YJλ , provided ϕ H (γα ) and ρ are sufficiently small. In fact, for any u ∈ YJλ let u1 = u, u2 = u, u3 = u. Then by (1.5) we have

α (u) XJ (γα ) ≤ C( ϕ H (γα ) + ρ 3 ). If C ϕ H (γα ) ≤

ρ 2

and Cρ 2 ≤

1 10 ,

then α maps YJλ to itself. Since for any u, v ∈ YJλ

Nα (u) − Nα (v) = Hγ (u, u, u − v) + Hγ (u − v, u, v) + Hγ (v, u − v, v), using (1.5) again, we get

α (u) − α (v) XJ (γα ) = Nα (u) − Nα (v) XJ (γα )  3Cρ 2 u − v XJ (γα ) 1 ≤ u − v XJ (γα ) . 2 The scattering follows from (4.10) below by defining ϕ ± by ±∞  α ϕ =ϕ−i eit |∇| (V ∗ |u|2 )u dt  . ±

0

4. Trilinear estimates This section is devoted to the proof of trilinear estimates (1.5). Let us set γα∗ = (1 −

γ )(2 − α) 2

so that γα + γα∗ = 2 − α and further set γα+ :=

γ +ε (2 − α). 2

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4.1. Sobolev norm bound We first observe the following Sobolev norm bound for Hγ : Lemma 4.1. Let s > we have

γ 2

and 0 < ε < min(2 − γ , 2s − γ ). Then for any u1 , u2 , u3 ∈ XJ (γα , γα+ ; s)

Hγ (u1 , u2 , u3 ) H s 

3

j =1

uj XJ (γα ,γα+ ;s) .

Proof of Lemma 4.1. Using Leibniz rule for fractional derivatives and Lemma 2.3, the Sobolev norm can be estimated as follows:

Hγ (u1 , u2 , u3 ) H s   Vγ (u1 u2 ) u3 H s dt R

 



Vγ (u1 u2 ) L∞ u3 H s + Vγ (u1 u2 )

 R

 

1

1

u1 2



2d L d−(γ −ε)

R

1

u1 2

2d L d−γ

1

u2 2

2d L d−γ

u2 2

2d

L d−(γ +ε)

2d

L d−γ

dt

u3 H s

⎞ + u1 u2 H s

u3

d γ d− 2

 

1

1

u1 2



u3

H s2d γ

2d L d−(γ −ε)

R

1

u1 2

2d L d−γ

+ u1 + u1

2d L d−γ

2d

Hs

u2 2

2d

L d−γ

2d

L d−(γ +ε)

u2 H s u3

u2

⎠ dt

1

u2 2

L d−γ

2d

L d−γ

u3

u3 H s

2d

L d−γ

 dt.

2d

L d−γ

Let us observe from Corollary 2.8 that

v

2d

L d−γ

 (1 + |t|)−γ /2 v XJ (γα ,γα ; γ ) 2

and ε

v

2d L d−(γ −ε)

1− γε

≤ v Lγ 2 v

2d

L d−γ

 (1 + |t|)−(γ −ε)/2 v XJ (γα ,γα ; γ ) . 2

(4.1)

Since ε < 2 − γ , we can also observe that

v

2d

L d−(γ +ε)

 (1 + |t|)−(γ +ε)/2 v XJ (γα+ ,γα+ ;(γ +ε)/2) .

(4.2)

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Therefore using the fact that v XJ (γα+ ,γα+ ;(γ +ε)/2)  v XJ (γα ,γα+ ;s) and γ > 1 we have 

Hγ (u1 , u2 , u3 ) H s 

(1 + |t|)−γ

3

j =1

R

uj XJ (γα ,γα+ ;s) dt 

3

j =1

uj XJ (γα ,γα+ ;s) .

2

4.2. Low and high frequency contribution of J We now consider the low and high Sobolev norm estimates for JHγ . Proposition 1.5 follows immediately from these and Lemma 4.1. Proposition 4.2. Let 6−2α 4−α < γ < 2 and 0 < ε  min(γ − any u1 , u2 , u3 ∈ XJ (γα ) we have

6−2α 4−α , 2 − γ , α

− 1, 2 − α). Then for

|∇|γα JHγ (u1 , u2 , u3 ) L∞ 2 t Lx ⎛ ⎞

u1 XJ (γα ,γα+ ;(γ +ε)/2) u2 XJ (γα ,γα ; γ ) 2 ⎠ u3 XJ (γα ,1+γα ,1) , or ⎝

u1 XJ (γα ,γα ; γ ) u2 XJ (γα ,γα+ ;(γ +ε)/2)

(4.3)

2

|∇|1+γα JHγ (u1 , u2 , u3 ) L∞ 2  t Lx

3

uj XJ (γα ) .

(4.4)

j =1

Through the next two subsections we prove Proposition 4.2. 4.3. Proof of (4.3) By definition of J we have t |∇| JHγ (u1 , u2 , u3 ) = γα



∗

e−i(t−t )|∇| |∇|−γα |∇|2−α J(Vγ (u1 u2 ) u3 ) dt  . α

0

From the formula such that for any smooth functions f, g we have   |∇|2−α J(f g) = |∇|2−α , f Jg − iαt |∇|2−α , f |∇|α−2 ∇g + iαt (∇f )g + f |∇|2−α Jg it follows that ∗

|∇|−γα |∇|2−α JHγ (u1 , u2 , u3 ) =

4   j =1 0

where

t

e−it|∇| (t − t  )Hj (t  ) dt  , α

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12

  2−α |∇| , Vγ (u1 u2 ) Ju3 ,   ∗ H2 = −iαt|∇|−γα |∇|2−α , Vγ (u1 u2 ) |∇|α−2 ∇u3 , ∗

H1 = |∇|−γα

 ∗ H3 = iαt|∇|−γα (∇Vγ (u1 u2 ) )u3 ,  ∗ H4 = |∇|−γα Vγ (u1 u2 ) |∇|2−α Ju3 . To treat the commutators in H1 and H2 we use Balakrishnan’s formula (1.6). We first deal with H1 as follows: ∞

H1 L2 

 (ρ − )−1 − , Vγ (u1 u2 ) (ρ − )−1 Ju3

2−α 2

ρ 0

∞ 

2−α 2

ρ

(ρ − )−1 ∇Vγ (u1 u2 ) · ∇(ρ − )−1 Ju3

0

∞ +

ρ

2−α 2

2d

∗

dρ

L d+2γα

2d

∗

dρ

L d+2γα

(ρ − )−1 (−Vγ (u1 u2 ) )(ρ − )−1 Ju3

0

2d

∗

dρ

L d+2γα

=: I1 + II 1 . The first inequality follows from HLS with 1 d + 2γα∗ γα∗ = − . 2 2d d We treat I1 by dividing integral region into two parts as follows: 1 I1 =

∞ +

0

=: L1 + H1 . 1

For L1 we apply Lemma 2.5 with θ = β to be chosen later and have that 1 L1 

ρ

2−α 2

  β ρ −1+ 2 |∇|−β ∇Vγ (u1 u2 ) · ∇(ρ − )−1 Ju3

0

By HLS of |∇|−β with γα∗ +β d

we have

d+2γα∗ 2d

=

d+2(γα∗ +β) 2d

−

β d

2d

∗

dρ.

L d+2γα

and Hölder’s inequality with

d+2(γα∗ +β) 2d

=

1 2

+

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1 L1 

ρ

β−α 2

ρ

β−α 2

∇Vγ (u1 u2 ) · ∇(ρ − )−1 Ju3

2d ∗

L d+2(γα +β)

0

1 

∇Vγ (u1 u2 )

0

13

dρ

∇|∇|−γα (ρ − )−1 |∇|γα Ju3

d ∗

L γα +β

L2

dρ.

Now by using Lemma 2.5 with θ = 1 − γα and HLS with γα∗ + β d − (γ + 1 − γα∗ − β) d − (γ + 1) = − , d d d we get 1 L1 

ρ

β−α 2

ρ

β−α−1−γα 2

ρ −1+

1−γα 2

∇Vγ (u1 u2 )

0

1 

u1 u2

0

d

∗

d ∗

L γα +β

L d−(γ +1−γα −β)

|∇|γα Ju3 L2 dρ

|∇|γα Ju3 L2 dρ.

By taking β = α − 1 + γα + ε we get from (4.1) and (2.8) 1 L1 

ρ

−2+ε 2

dρ u1

2d

L d−(γ −ε)

u2

2d

L d−(γ −ε)

|∇|γα Ju3 L2

0

(4.5)

 (1 + |t|)−(γ −ε)

3

j =1

uj XJ (γα ,γα ; γ ) . 2

Similarly we can treat H1 with β = α − 1 + ε < α as follows: ∞ H1 

ρ

β−α 2 −1

∇Vγ (u1 u2 )

1

 u1 u2

d ∗ L d−(γ +1−γα −γα −β)

 (1 + |t|)−(γ −ε)

2

j =1

d

∗

L γα +γα +β

|∇|Ju3

2d

L d−2γα

|∇|1+γα Ju3 L2 = u1 u2

dρ

d

L d−(γ −ε)

|∇|1+γα Ju3 L2

uj XJ (γα ,γα ; γ ) |∇|1+γα Ju3 L2 . 2

Thus I1  (1 + |t|)−(γ −ε)

2

j =1

uj XJ (γα ,γα ; γ ) u3 XJ (γα ,1+γα ; γ ) . 2

2

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14

Now we deal with II 1 . For this we use the duality and integration by parts to avoid the second derivatives acted on Vγ (u1 u2 ) which may cause the higher regularity than requested for uj . Let 2d us set p = d−2γ ∗ . Then we have α

∞ II 1 =

ρ

2−α 2

ρ

2−α 2

∞

     (ρ − )−1 (−Vγ (u1 u2 ) )(ρ − )−1 Ju3 , f  dρ

sup

     (∇Vγ (u1 u2 ) )(ρ − )−1 Ju3 ; ∇(ρ − )−1 f  dρ

f Lp ≤1

0

≤

sup

f Lp ≤1

0

∞ +

ρ

2−α 2

sup

f Lp ≤1

0

     (ρ − )−1 (∇Vγ (u1 u2 ) ) · ∇(ρ − )−1 Ju3 , f  dρ

=: D1 + E1 . For D1 we use Lemma 2.5 with θ = 1 to get ∞ D1 ≤

ρ

1−α 2

∇Vγ (u1 u2 ) (ρ − )−1 Ju3 Lp dρ.

0

With β = α − 1 + γα + ε, we deduce by Hölder’s inequality and Sobolev inequality that ∞ D1 ≤

ρ

1−α 2

∇Vγ (u1 u2 )

0

∞ 

ρ

1−α 2

∇Vγ (u1 u2 )

0

1 

ρ

1−α 2

ρ −1+

β−γα 2

+

ρ

1−α 2

1−α 2

−1+

β−γα 2

2d

L d−2β

d ∗

L γα

|∇|γα Ju3 L2

d ∗

L γα

|∇|β Ju3 L2 .

= −1 + 2ε . Thus from HLS with

γα∗ + β d − (γ + 1 − γα∗ − β) d − (γ + 1) = − d d d and (4.1) we get that

dρ

|∇|β (ρ − )−1 Ju3 L2 dρ

ρ −1 dρ ∇Vγ (u1 u2 )

1

Our choice of β shows that

d ∗

L γα

(ρ − )−1 Ju3

dρ ∇Vγ (u1 u2 )

0

∞

d ∗

L γα +β

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Y. Cho / J. Differential Equations ••• (••••) •••–•••

D1  ∇Vγ (u1 u2 )  u1 u2

d ∗

L γα +β

15

u3 XJ (γα ,α−1+ε+γα ;1)

u3 XJ (γα ,α−1+ε+γα ;1)

d

L d−(γ −ε)

2

 (1 + |t|)−(γ −ε)

j =1

uj XJ (γα ,γα ; γ ) u3 XJ (γα ,1+γα ;1) . 2

Since E1  I1 we obtain 2

II 1  (1 + |t|)−(γ −ε)

j =1

uj XJ (γα ,γα ; γ ) u3 XJ (γα ,1+γα ;1) , 2

and since γ − ε > 1, 

H1 L2 dt 

2

j =1

R

uj XJ (γα ,γα ; γ ) u3 XJ (γα ,1+γα ;1) .

Let us consider H2 . As above with p = ∞

H2 L2  |t|

ρ

2−α 2

(4.6)

2

2d d−2γα∗

(ρ − )−1 ∇Vγ (u1 u2 ) · ∇|∇|α−2 ∇(ρ − )−1 u3

Lp



dρ

0

∞ + |t|

ρ

2−α 2

(ρ − )−1 (−Vγ (u1 u2 ) )|∇|α−2 ∇(ρ − )−1 u3

Lp



dρ

0

=: I2 + II 2 . We divide the integral region of I2 and write down ⎛ I2 = |t| ⎝

1 0

For some 0 < ε  1 and 0 < β < 1 − inequality that 1 L2 

ρ

2−α 2

ρ

ε−α 2

γ 2

⎞ ∞ + ⎠ =: |t|(L2 + H2 ). 1

we get from Lemma 2.5 with θ = ε, HLS and Hölder’s

ε

ρ −1+ 2 ∇Vγ (u1 u2 ) · ∇|∇|α−2 ∇(ρ − )−1 u3

Lp



dρ

0

1  0

∇Vγ (u1 u2 ) · ∇|∇|α−2 ∇(ρ − )−1 u3

2d ∗

L d+2(γα +ε)

dρ

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16

1 

ρ

ε−α 2

∇Vγ (u1 u2 ) L

0

|∇|α (ρ − )−1 u3

d γ ∗ 2 +β+γα +ε

L

dρ.

2d γ d−2( 2 +β)

By Lemma 2.5 with θ = α and HLS 1 L2 

ρ

ε−α 2

α

ρ −1+ 2 dρ ∇Vγ (u1 u2 ) L

0

 u1 u2 L

d γ d−(γ +1− 2 −γα∗ −ε−β)

L

d γ d−( 2 +1−γα∗ −ε−β)

 u1 u2

u3 L

u3 L

d γ ∗ 2 +β+γα +ε

u3 L

2d γ d−2( 2 +β)

2d γ d−2( 2 +β)

2d γ d−2( 2 +β)

.

By taking β = 1 − γ2 − γα∗ − ε (β > 0 if ε < 3 − α − γ2 (α − 1) and clearly β + by Corollary 2.8 that 2

L2 

uj

j =1

2d d−γ

 (1 + |t|)−

3γ 2

u3 L −β

2

j =1

 (1 + |t|)−

3γ 2

−β

2

j =1

H2 

uj XJ (γα ,γα ; γ ) u3 XJ (γα +β(2−α),γα +β(2−α); γ +β) 2

2

uj XJ (γα ,γα ; γ ) u3 XJ (γα ,1+γα ,1) . 2

γ 2

− γα∗ − ε as follows:

α

ρ − 2 ∇Vγ (u1 u2 ) · ∇|∇|α−2 ∇(ρ − )−1 u3

1

∞ 

2d

∗

dρ

L d+2γα

α

ρ − 2 ∇Vγ (u1 u2 ) · |∇|α−(1−β(2−α)) (ρ − )−1 |∇|1−β(2−α) u3

1

∞ 

α

ρ − 2 ∇Vγ (u1 u2 ) L

1

∞ 

ρ −1−

1−β(2−α) 2

2

j =1

d γ ∗ 2 +γα +β

dρ u1 u2 L

1



< 1), we have

2d γ d−2( 2 +β)

Now H2 can be treated with β = 1 − ∞

γ 2

uj

2d

L d−γ

ρ −1+

α−(1−β(2−α)) 2

2d

|∇|1−β(2−α) u3 L

d γ d−(γ +1− 2 −γα∗ −β)

|∇|1−β(2−α) u3 L

2d γ d−2( 2 +β)

.

|∇|1−β(2−α) u3 L

∗

dρ

L d+2γα

2d γ d−2( 2 +β)

2d γ d−2( 2 +β)

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Y. Cho / J. Differential Equations ••• (••••) •••–••• γ 2

Since β > 0 and

17

+ β < 1 as above, we have

H2  (1 + |t|)−

3γ 2

2

−β

j =1

uj XJ (γα ,γα ; γ ) 2

× ( |∇|γα +β(2−α) J|∇|1−β(2−α) u3 L2 + u3  (1 + |t|)−

3γ 2

2

−β

j =1

 (1 + |t|)−

3γ 2

2

−β

j =1

H

γ 2 +β

uj XJ (γα ,γα ; γ ) ( |∇|1+γα Ju3 L2 + u3 2

H

)

γ 2 +β

)

uj XJ (γα ,γα ; γ ) u3 XJ (γα ,1+γα ;1) . 2

We used Lemma 2.5 with θ = α − (1 − β(2 − α)) for the third inequality. Thus we have from the estimates of L2 and H2 that I2  (1 + |t|)−

3γ 2

−β+1

2

j =1

uj XJ (γα ,γα ; γ ) u3 XJ (γα ,1+γα ;1) . 2

For II 2 let us invoke the duality argument used to II 1 and write down ∞ II 2 = |t|

ρ

2−α 2

f Lp ≤1

0

where p =

2d d−2γα∗ .

∞ II 2 = |t|

ρ

sup

Using integration by parts we split the inner product as follows:

2−α 2

sup

f Lp ≤1

0

∞ + |t|

     (ρ − )−1 (−Vγ (u1 u2 ) )|∇|α−2 ∇(ρ − )−1 u3 , f  dρ,

ρ

2−α 2

     (∇Vγ (u1 u2 ) )|∇|α−2 ∇(ρ − )−1 u3 ; ∇(ρ − )−1 f  dρ

sup

f Lp ≤1

0

     (ρ − )−1 (∇Vγ (u1 u2 ) ) · ∇|∇|α−2 ∇(ρ − )−1 u3 , f  dρ

=: D2 + E2 . Now by Hölder’s inequality and Lemma 2.5 with θ = 1 we get ∞ D2 ≤ |t|

ρ 0

1−α 2

⎛ 1 ∞⎞   α−2 −1 ⎝

∇Vγ (u1 u2 ) |∇| ∇(ρ − ) u3 Lp dρ = |t| + ⎠. 0

1

By consecutive application of Hölder’s inequality for β > 0, 0 < ε  1 and Lemma 2.5 with θ = α − 1 + ε to the low integration part, we have

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18

1

1 

0

ρ

1−α 2

∇Vγ (u1 u2 )

0

1 

ρ

1−α 2



ρ

1−α 2

L

2d γ ∗ 2 +β+ε+γα

∇Vγ (u1 u2 )

0

1

L

2d γ ∗ 2 +β+ε+γα

ρ −1+

α−1+ε 2

γ 2

2d γ d−2( 2 +β+ε)

|∇|α−1+ε (ρ − )−1 u3 L

2d γ ∗ 2 +β+ε+γα

L

u3 L

2d γ d−2( 2 +β)

2d γ d−2( 2 +β)

.

− γα∗ − ε and apply HLS to ∇Vγ (u1 u2 ) to get

1  u1 u2

L

∇Vγ (u1 u2 )

0

Choose β = 1 −

|∇|α−2 ∇(ρ − )−1 u3

u3

d L d−γ

L

0

 (1 + |t|)−(

2d γ d−2( 2 +β)

3γ 2

+β)

2

j =1

uj XJ (γα ,γα ; γ ) u3 XJ (γα ,1+γα ;1) . 2

The high integration part can be treated as follows: for some 0 < ε  1 and 0 < β < 1 − ∞

∞ 

1

ρ

1−α 2

ε

∇Vγ (u1 u2 ) L

1

d γ ∗ 2 +β+γα

|∇|α−1−ε (ρ − )−1 (1 − ) 2 u3 L

Using Lemma 2.5 with θ = α − 1 − ε and by taking β = 1 − ∞

∞ 

1

ρ

1−α 2

ρ −1+

α−1−ε 2

L

L



j =1

γ 2

+β <1

d γ ∗ 2 +β+γα

(1 − ) 2 u3 L

2d γ d−2( 2 +β)

ε

 u1 u2 2

− γα∗ so that

.

ε

∇Vγ (u1 u2 )

1

γ 2

2d γ d−2( 2 +β)

uj

d γ d−(γ +1−( 2 +β+γα∗ ))

(1 − ) 2 u3 L

2d γ d−2( 2 +β)

ε

2d L d−γ

 (1 + |t|)−(

3γ 2

(1 − ) 2 u3 L

+β)

2

j =1

 (1 + |t|)−(

3γ 2

+β)

2

j =1

2d γ d−2( 2 +β)

ε

uj XJ (γα ,γα ; γ ) |∇|γα +β(2−α) J(1 − ) 2 u3 L2 2

uj XJ (γα ,γα ; γ ) u3 XJ (γα ,1+γα ;1) . 2

For the last inequality we used the inequality

|∇|γα +β(2−α) J(1 − ) 2 u3 L2  |∇|γα +β(2−α) (1 − ) 2 Ju3 L2 + u3 ε

ε

H

γ 2 +β+ε

.

γ 2

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19

All estimates together with the fact E2  I2 give us that 



H2 L2 dt 

R

Since γ >

(1 + |t|)−(

3γ 2

+β)+1

uj XJ (γα ,γα ; γ ) u3 XJ (γα ,1+γα ;1) . 2

j =1

R 6−2α 3γ 4−α , 2

2

dt

+ β − 1 = γ − γα∗ = γ − (1 − γ2 )(2 − α) > 1. Therefore 

H2 L2 dt 

2

j =1

R

As for H3 we have for 0 < β < 1 −

H3 L2  |t| ∇Vγ (u1 u2 ) u3  |t| u1 u2

γ 2

that 2d

∗

L d+2γα

d γ d−(γ +1− 2 −γα∗ −β)

L

d γ d−( 2 +1−γα∗ −β)

− γα∗ . Then β +

H3 L2  (1 + |t|)−

(4.7)

2

L

 |t| u1 u2 Let us choose β = 1 −

γ 2

uj XJ (γα ,γα ; γ ) u3 XJ (γα ,1+γα ;1) .

3γ 2

 ∇Vγ (u1 u2 ) L

u3 L

u3 L γ 2

d γ ∗ 2 +γα +θ

u3 L

2d γ d−2( 2 +β)

2d γ d−2( 2 +β)

2d γ d−2( 2 +β)

.

= 1 − γα∗ < 1, Corollary 2.8 yields

−β

2

( |∇|γα Juj L2 + uj

j =1

× ( |∇|γα +β(2−α) u3 L2 + u3

H

γ 2

)

γ 2 +β

).

H

Therefore 

H3 L2 dt 

2

j =1

R

uj XJ (γα ,γα ; γ ) u3 XJ (γα ,1+γα ;1) .

(4.8)

2

We now consider the final term H4 . We use duality argument as follows: ∗

H4 L2 = |∇|−γα (Vγ (u1 u2 ) |∇|2−α Ju3 ) L2   ∗ ∗    = sup  |∇|γα Ju3 , |∇|γα Vγ (u1 u2 ) |∇|−γα f 

f L2 ≤1

 ∗  |∇|γα Ju3 L2 Vγ (u1 u2 ) L∞ + |∇|γα Vγ (u1 u2 ) × 

sup

f L2 ≤1



 d ∗ L γα

∗

f L2 + |∇|−γα f



2d ∗ L d−2γα



∗

 |∇|γα Ju3 L2 Vγ (u1 u2 ) L∞ + |∇|γα Vγ (u1 u2 )

d ∗ L γα

.

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20

By HLS-∞, (4.1) and (4.2) we get

Vγ (u1 u2 ) L∞

⎧ 1 1 1 1 ⎪

u1 2 2d u1 2 2d u2 2 2d u2 2 2d , ⎪ ⎪ ⎨ L d−γ L d−γ L d−(γ −ε) L d−(γ +ε)  or ⎪ 1 1 1 1 ⎪ ⎪ ⎩ u1 2 2d u1 2 2d u2 2 2d u2 2 2d L d−(γ +ε)

 (1 + |t|)−γ

L d−γ

L d−γ

L d−(γ −ε)

⎧ , ⎪ ⎨ u1 XJ (γα ,γα ; γ2 ) u2 XJ (γα ,γα+ ; γ +ε 2 ) or ⎪ ⎩ u1 γ XJ (γα ,γα+ ; γ +ε ) u2 XJ (γα ,γα ; ) 2

2

and by HLS ∗

|∇|γα Vγ (u1 u2 )

d ∗

L γα

 u1 u2

d

L d−γ

 u1

2d

L d−γ

u2

2d

L d−γ

 (1 + |t|)−γ u1 XJ (γα ,γα ; γ ) u2 XJ (γα ,γα ; γ ) . 2

2

Thus we obtain

H4 L2  (1 + |t|)−γ

⎧ ⎫ ,⎪ ⎪ ⎨ u1 XJ (γα ,γα ; γ2 ) u2 XJ (γα ,γα+ ; γ +ε ⎬ ) 2 or

|∇|γα Ju3 L2 . ⎪ ⎪ ⎩ u1 ⎭ γ XJ (γα ,γα+ ; γ +ε ) u2 XJ (γα ,γα ; ) 2

2

Therefore  R

⎧ ⎫ ,⎪ ⎪ ⎨ u1 XJ (γα ,γα ; γ2 ) u2 XJ (γα ,γα+ ; γ +ε ⎬ ) 2

H4 L2 dt  or

u3 XJ (γα ,1+γα ;1) . ⎪ ⎪ ⎩ u1 + γ +ε u2 X (γ ,γ ; γ ) ⎭ XJ (γα ,γα ; ) J α α 2

2

This proves (4.3). 4.4. Proof of (4.4) We move onto the proof of (4.4). Let us first observe that

|∇|1+γα JHγ (u1 , u2 , u3 ) L2  |∇|γα JHγ (u1 , u2 , u3 ) L2 + |∇|γα JHγ (∇u1 , u2 , u3 ) L2 + |∇|γα JHγ (u1 , ∇u2 , u3 ) L2 + |∇|γα JHγ (u1 , u2 , ∇u3 ) L2 =: |∇|γα JHγ (u1 , u2 , u3 ) L2 + A1 + A2 + A3 . From (4.3) we readily have that

|∇|γα JHγ (u1 , u2 , u3 ) L2 + A1 + A2 

3

j =1

uj XJ (γα ,1+γα ;1) .

(4.9)

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21

For A3 we need a more subtlety. A3 

4  

j L2 dt,

H

j =1 R

where   1 = |∇|−γα∗ |∇|2−α , Vγ (u1 u2 ) J∇u3 , H   2 = −iαt|∇|−γα∗ |∇|2−α , Vγ (u1 u2 ) |∇|α−2 ∇ 2 u3 , H   3 = iαt|∇|−γα∗ (∇Vγ (u1 u2 ) )∇u3 , H   4 = |∇|−γα∗ Vγ (u1 u2 ) |∇|2−α J∇u3 . H By (4.9) we get 

4 L2 dt 

H

2

uj XJ (γα ,1+γα ;1) |∇|γα J∇u3 L2 ≤

j =1

R

3

uj XJ (γα ,1+γα ;1) .

j =1

3 we have As for H 3 L2  |t| ∇Vγ (u1 u2 ) ∇u3 p  ∇Vγ (u1 u2 )

H L L

 |t| u1 u2 L

d γ d−(γ +1− 2 −γα∗ )

L

d γ d−( 2 +1−γα∗ )

 |t| u1 u2

∇u3

∇u3

d γ ∗ 2 +γα

∇u3

2d

L d−γ

2d

L d−γ 2d

L d−γ

.

Set β := (1 − γ2 − γα∗ )/2. Then our choice of γ gives that γ < 2( γ2 + β) < 2. Then Corollary 2.8 yields that 

3 L2 dt

H

R



(1 + |t|)

 R

Since

−( 3γ 2 +2β)

3γ 2

2



γ

|∇|( 2 +β)(2−α) Juj L2 + uj

j =1

+ 2β = 1 + γ − γα∗ > 2 for γ >  R

6−2α 4−α ,

3 L2 dt 

H

3

j =1

H

γ 2 +β

we get

uj XJ (γα ,1+γα ;1) .



u3 XJ (γα ,1+γα ;1) dt.

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22

1 we follow the argument for H1 and have that For H 1 L2 

H

∞ ρ

2−α 2

(ρ − )−1 ∇Vγ (u1 u2 ) · ∇(ρ − )−1 J∇u3

Lp



dρ

0

∞ +

ρ

2−α 2

(ρ − )−1 (−Vγ (u1 u2 ) )(ρ − )−1 J∇u3

Lp



dρ

0

II 1 . =: I1 +  We divide integral region of I1 and have I1 =

1

∞ +

0

1 + H 1 . =: L

1

From (4.5) we have 3

1  (1 + |t|)−γ L

uj XJ (γα ,1+γα ;1) .

j =1

1 like H1 with β = α − 1 − ε as follows: We treat H 1  H

∞ ρ

β−α 1 2 −2

∇Vγ (u1 u2 )

1

 u1 u2

d ∗ L d−(γ +1−γα −γα −β)

 (1 + |t|)−(γ −ε)

3

d

∗

L γα +γα +β

J∇u3

2d

L d−2γα

|∇|γα J∇u3 L2 = u1 u2

dρ

d

L d−(γ −ε)

|∇|1+γα Ju3 L2

uj XJ (γα ,1+γα ;1) .

j =1

Thus we get  R

I1 dt 

3

uj XJ (γα ,1+γα ;1) .

j =1

By duality and integration by parts we have for  II 1 that  II 1 =

∞ ρ 0

2−α 2

sup

f Lp ≤1

     ∇Vγ (u1 u2 ) (ρ − )−1 J∇u3 , ∇(ρ − )−1 f  dρ

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∞ +

ρ

2−α 2

sup

f Lp ≤1

0

23

     (ρ − )−1 ∇Vγ (u1 u2 ) · ∇(ρ − )−1 J∇u3 , f  dρ

1 . 1 + E =: D 1 we have from Lemma 2.5 with θ = 1 that 1 can be treated similarly to I1 . As for D E 1  D

∞ ρ

1−α 2

∇Vγ (u1 u2 ) (ρ − )−1 J∇u3

Lp

dρ



0

1 =

∇Vγ (u1 u2 ) 0

(ρ − )−1 J∇u3

d ∗ L β1 +γα

∞ +

∇Vγ (u1 u2 ) 1

2d

L d−2β1

(ρ − )−1 J∇u3

d ∗ L β2 +γα

dρ

2d

L d−2β2

dρ,

where β1 = α − 1 + γα + ε which is 1 − γα∗ + ε and β2 = 1 − γα∗ − ε. Then Sobolev inequality gives 1  D

1

1−α 2

ρ

∇Vγ (u1 u2 )

0

∞ +

ρ

1−α 2

d

∗

L β1 +γα

∇Vγ (u1 u2 )

1

|∇|β1 (ρ − )−1 J∇u3

d

∗

L β2 +γα

L2

|∇|β2 (ρ − )−1 J∇u3

L2

.

Applying Lemma 2.5 with θ = β1 − γα∗ or β2 − γα∗ , we have 1  D

1

ρ −1

1−α+β1 −γα∗ 2

dρ u1 u2

0

∞ +

ρ −1

1−α+β2 −γα∗ 2

d ∗ L d−(γ +1−β1 −γα )

dρ u1 u2

1

Since −1 +

1−α+β1 −γα∗ 2

1  D

2  j =1

uj

2d L d−(γ −ε)

 (1 + |t|)−(γ −ε)

+

3

j =1

2

j =1

d

∗

L d−(γ +1−β2 −γα )

1−α+β2 −γα∗ 2

= −1 + 2ε , −1 +

|∇|γα J∇u3 L2

uj

= −1 − 

ε 2

2d L d−(γ +ε)

uj XJ (γα ,1+γα ;1) .

|∇|γα J∇u3 L2 .

and ε < α − 1, we get

u3 XJ (γα ,1+γα ;1)

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24

Thus 

 II 1 dt 

3

uj XJ (γα ,1+γα ;1) .

j =1

R

2 . Integration by parts gives Finally it remains to handle H 2 L2  |t|

H

∞ ρ

2−α 2

(ρ − )−1 ∇Vγ (u1 u2 ) · ∇|∇|α−2 ∇(ρ − )−1 ∇u3

Lp



dρ

0

∞ + |t|

ρ

2−α 2

(ρ − )−1 (Vγ (u1 u2 ) )|∇|α−2 ∇(ρ − )−1 ∇u3

Lp



dρ

0

=: I2 +  II 2 . We start with I2 . ⎛ 1 ∞⎞   2 ). 2 + H I2 = |t| ⎝ + ⎠ =: |t|(L 0

1

For some 0 < ε  1 we obtain 2  L

1 ρ

ε−α 2

ρ

ε−α 2

∇Vγ (u1 u2 ) ·

0

1 

∇|∇|α−2 ∇ ∇u3 ρ −

2d ∗

L d+2(γα +ε)

α

ρ −1+ 2 dρ ∇Vγ (u1 u2 ) L

0

 u1 u2 L

d γ d−(γ +1− 2 −γα∗ −ε)

L

d γ d−( 2 +1−γα∗ −ε)

 u1 u2

∇u3

∇u3

d γ ∗ 2 +γα +ε

3γ 2

+2β)

2d

2d

L d−γ

3

j =1

2 we use duality and integration by parts. For H

2d

L d−γ

L d−γ

.

3 , 0 < Set β := (1 − γ2 − γα∗ − ε)/2. Then as previously for H sufficiently small ε. Thus 2  (1 + |t|)−( L

∇u3

dρ

γ 2

+ β < 1 and

uj XJ (γα ,1+γα ;1) .

3γ 2

+ 2β > 2 for

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2 ≤ H

∞ ρ

2−α 2

ρ

2−α 2

∞ ≤

sup

     ∇|∇|α−2 ∇(ρ − )−1 ∇u3 , ∇Vγ (u1 u2 ) (ρ − )−1 f  dρ

sup

     ∇|∇|α−2 ∇(ρ − )−1 u3 , ∇ 2 Vγ (u1 u2 ) (ρ − )−1 f  dρ

f Lp ≤1

1

f Lp ≤1

1

∞ +

ρ

25

2−α 2

sup

f Lp ≤1

1

     ∇|∇|α−2 ∇(ρ − )−1 u3 , ∇Vγ (u1 u2 ) ∇(ρ − )−1 f  dρ

=: B1 + B2 . To handle B1 we carry out estimate similar to H2 by replacing u1 u2 with pairs ∇(u1 u2 ) and get ∞ B1 

(ρ − )−1 ∇ 2 Vγ (u1 u2 ) ∇|∇|α−2 ∇(ρ − )−1 u3

Lp



dρ

1

 (1 + |t|)−(

3γ 2

+β)

3

uj XJ (γα ,1+γα ;1) .

j +1

Note that

3γ 2

+ β > 2. For B2 let  ε = ε(2 − α) with ε  1. Then

∞ B2 

ρ

2−α 2

ρ −1+

α−1− ε 2

1

ρ − 2 dρ |∇|1+ε u3 L

1

2d γ d−2( 2 −ε)

∇Vγ (u1 u2 ) L

d γ ∗ 2 −ε+γα

γ

 (1 + |t|)−( 2 −ε) u1 u2 L

d γ d−(γ +1− 2 −γα∗ +ε)

γ

× ( |∇|( 2 −ε)(2−α) J|∇|1+ε u3 L2 + |∇|1+ε u3 γ

γ

∗

 (1 + |t|)−( 2 −ε) (1 + |t|)−( 2 +1−γα +ε)

2

j =1

∗

 (1 + |t|)−(γ +1−γα )

2

j =1

Since γ >

6−2α 4−α ,

H

γ 2 −ε

)

uj XJ (γα ,1+γα ;1) u3 XJ (γα ,1+γα ;1+ γ ) 2

uj XJ (γα ,1+γα ;1) u3 XJ (γα ,1+γα ;1+ γ ) . 2

γ + 1 − γα∗ > 2. This implies that  R

I2 L2 dt 

3

j =1

uj XJ (γα ,1+γα ;1+ γ ) . 2

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26

Now let us consider  II 2 . ⎛ 1 ∞⎞   ⎝  + ⎠. II 2 = |t| 0

1

For the low integration we have from the estimate for L2 that 1

1 =

0

ρ

2−α 2

(ρ − )−1 (∇ · Vγ (∇(u1 u2 ))|∇|α−2 ∇(ρ − )−1 ∇u3

Lp



dρ

0

 ∇(u1 u2 )  (1 + |t|)−(

d

L d−γ

3γ 2

+β)

u3 3

j =1

2d γ d−2( 2 +β) L

(β = 1 −

γ − γα∗ − ε, 0 < ε  1) 2

uj XJ (γα ,1+γα ;1+ γ ) . 2

For the high integration we follow the argument on H2 with the same β, ε as above. ∞  ∇(u1 u2 )

d

L d−γ

|∇|1−β(2−α) u3 L

1

 (1 + |t|)−(

3γ 2

+β)

3

j =1

We observed that

3γ 2

2d γ d−2( 2 +β)

uj XJ (γα ,1+γα ;1+ γ ) . 2

+ β > 2 previously. Therefore 

 II 2 dt 

3

uj XJ (γα ) .

j =1

R

This completes the proof of Proposition 4.2. 4.5. Time decay for scattering The scattering of nonlinear solution can be shown by the following error estimate: for some δ0 > 0 ⎛

e

it|∇|α

⎞ ±∞  α u(t) − ⎝ϕ − i eit |∇| (V ∗ |u|2 )u dt  ⎠ HJ (γα ) 0

= O(|t|−δ0 ) as t → ±∞.

(4.10)

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27

For this we have only to consider the case t > 0 and then we can readily get this from the proof of Proposition 4.2. In fact, let us define a trilinear operator by (γ (u1 , u2 , u3 ) := −i H

∞



e−i(t−t )|∇| (Vγ (u1 u2 ) u3 )(t  ) dt  . α

t

Then we have for t > 1 that (γ (u1 , u2 , u2 )

H

H

γ 1+ 2

(γ (u1 , u2 , u3 ) L2 + |∇|γα (1 + |∇|)JH

∞ 3

 (t  )−1−δ0 dt 

uj XJ (γα ) , j =1

t

where δ0 = min(γ − 1 − ε, 3γ 2 + β − 2) as appearing in the proof of Proposition 4.2. Acknowledgments I would like to express my gratitude to LG Yonam Foundation and Department of Mathematics at UCSD for their kind support during my visit. Especially I thank Ioan Bejenaru for enlightening discussion and Enno Lenzmann for informing of his work [2]. I learned Balakrishnan’s formula from [2]. In addition, I am very grateful to the anonymous referee for his/her valuable comments. This work was supported in part by NRF (NRF-2015R1D1A1A09057795). Appendix A p

A.1. L1t Lx bound of E We first show the Lp bound of E . It can be rewritten as  E = iαt|∇|α−2 ((∇V ∗ |u|2 )u) + iαt |∇|α−2 , V ∗ |u|2 ∇u. 2d Let β be such that 0 < β < min(1, d−2 2 ) and p = d−2(β+2−α) . Then we have from HLS and commutator estimate for fractional integral [4] that for 0 < ε  1  

E Lp  |t| ∇(V ∗ |u|2 )u 2d + V ∗ |u|2 BMO ∇u 2d L d−2β L d−2β    |t| ∇(V ∗ |u|2 ) Ld u + V ∗ |u|2 L∞ ∇u 2d 2d L d−2β L d−2(1+β)    |t| u 2 2d ∇u 2d + u

u

∇u 2d 2d 2d L d−γ

L d−2β

L d−(γ −ε)

L d−(γ +ε)

L d−2β

 (1 + |t|)−(γ +β−1) u 2XJ (γα ) u XJ (1+β(2−α),1+β(2−α);1+β) , where BMO is the space of functions of bounded mean oscillation and f BMO  f L∞ . If we choose β = γ2 for d ≥ 4 and β = 12 − ε for d = 3, then we obtain

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28

E L1 Lpx  u 3XJ (γα ) for γ0 < γ < 2, t

where γ0 =

3 2

if d = 3 and γ0 = 1 if d ≥ 4.

A.2. L1t L2x bound of modified commutator  Now let us consider the modified commutator E= iαtRx |∇|α−1 , V ∗ |u|2 u. We show  1 2  u 3

E XJ (0,1+γα ;1) if 3 − α < γ < 2. L L t

x

By Balakrishnan’s formula (1.6)   L2  |t| |∇|α−1 , V ∗ |u|2 u L2

E  ∞ α−1  |t| ρ 2 (ρ − )−1 ∇V ∗ |u|2 · ∇(ρ − )−1 u L2 dρ 0

∞ +

ρ

α−1 2

(ρ − )−1 (−)V ∗ |u|2 (ρ − )−1 u L2 dρ



0

=: |t|(E1 + E2 ). Splitting integral E1 into E1±

1

+

ρ

α−1 2

1 

∞

0

1

=: E1− + E1+ , we have for some 0 < ε  1

ρ −1+

2−α∓ε 2

|∇|−(2−α∓ε) ((∇V ∗ |u|2 u)) L2 dρ

0

1 

ε

ρ −1+ 2 ∇V ∗ |u|2 Ld u

2d

L d−2(α−1±ε)

0

 u 2

2d

L d−γ

u

2d

L d−2(α−1∓ε)

 (1 + |t|)−(γ +α−1∓ε) u 3XJ (0,1+γα ;1) . By duality and integration by parts we have the same conclusion for E2 . Since γ > 3 − α, for sufficiently small ε we get the desired estimate. A.3. Difference between E and E By direct calculation we observe that  E − E = iαt Rx , V ∗ |u|2 |∇|α−1 u.

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29

We want the difference to be in L1t L2x . However, in view of Theorem 1 of [10] we may have at best for γ < 2 that  L2  |t| V ∗ |u|2 BMO |∇|α−1 u L2  (1 + |t|)−γ +1 u 3

E − Rx E XJ (0,1+γα ;1) . x The last term is not integrable on the real line if γ < 2. For the present we cannot show that the difference is trivial. We hope that this will be just a technical difficulty to be overcome. References [1] A.V. Balakrishnan, Fractional powers of closed operators and the semigroups generated by them, Pacific J. Math. 10 (1960) 419–437. [2] T. Boulenger, D. Himmelsbach, E. Lenzmann, Blowup for fractional NLS, preprint, arXiv:1509.08845. [3] D. Cai, A.J. Majda, D.W. McLaughlin, E.G. Tabak, Dispersive wave turbulence in one dimension, Phys. D 152/153 (2001) 551–572, Advances in nonlinear mathematics and science. [4] S. Chanillo, A note on commutators, Indiana Univ. Math. J. 31 (1982) 7–16. [5] Y. Cho, H. Hajaiej, G. Hwang, T. Ozawa, On the Cauchy problem of fractional Schrödinger equation with Hartree type nonlinearity, Funkcial. Ekvac. 56 (2013) 193–224. [6] Y. Cho, G. Hwang, T. Ozawa, On small data scattering of Hartree equations with short-range interaction, Commun. Pure Appl. Anal. 15 (5) (2016) 1809–1823. [7] Y. Cho, G. Hwang, T. Ozawa, On the focusing energy-critical fractional nonlinear Schrödinger equations, preprint. [8] Y. Cho, T. Ozawa, On the semi-relativistic Hartree type equation, SIAM J. Math. Anal. 38 (4) (2006) 1060–1074. [9] Y. Cho, T. Ozawa, S. Xia, Remarks on some dispersive estimates, Commun. Pure Appl. Anal. 10 (4) (2011) 1121–1128. [10] R.R. Coifman, R. Rochberg, G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. (2) 103 (3) (1976) 611–635. [11] J. Fröhlich, E. Lenzmann, Mean-field limit of quantum Bose gases and nonlinear Hartree equation, in: Sémin. Équ. Dériv. Partielles, exp. no. XIX, Ecole Polytechnique, 2004, 26 pp. [12] J. Ginibre, T. Ozawa, Long range scattering for nonlinear Schrödinger and Hartree equations in space dimension n ≥ 2, Comm. Math. Phys. 151 (1993) 619–645. [13] N. Hayashi, P.I. Naumkin, T. Ogawa, Scattering operator for semirelativistic Hartree type equation with a short range potential, Differential Integral Equations 28 (2015) 1085–1104. [14] N. Hayashi, Y. Tsutsumi, Scattering theory for Hartree type equations, Ann. Inst. Henri Poincaré Phys. Théor. 46 (1987) 187–213. [15] I.W. Herbst, Spectral theory of the operator (p 2 + m2 )1/2 − Ze2 /r, Comm. Math. Phys. 53 (3) (1977) 285–294. [16] K. Kirkpatrick, E. Lenzmann, G. Staffilani, On the continuum limit for discrete NLS with long-range lattice interactions, Comm. Math. Phys. 317 (3) (2013) 563–591. [17] J. Krieger, E. Lenzmann, P. Raphaël, Nondispersive solutions to the L2-critical half-wave equation, Arch. Ration. Mech. Anal. 209 (1) (2013) 61–129. [18] N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A 268 (2000) 298–305. [19] N. Laskin, Fractals and quantum mechanics, Chaos 10 (2000) 780–790. [20] F. Pusateri, Modified scattering for the Boson star equation, Comm. Math. Phys. 332 (2014) 1203–1234. [21] E.M. Stein, Singular Integral and Differentiability Properties of Functions, Princeton Math. Ser., vol. 30, Princeton Univ. Press, 1970. [22] C. Sulem, P.L. Sulem, Self-focusing and Wave Collapse, Springer, 1993.