https://doi.org/10.4134/JKMS.j170224 pISSN: 0304-9914 / eISSN: 2234-3008
SMALL DATA SCATTERING OF HARTREE TYPE FRACTIONAL SCHR ¨ ODINGER EQUATIONS IN DIMENSION
2 AND 3
Yonggeun Cho and Tohru Ozawa
Abstract. In this paper we study the small-data scattering of the d dimensional fractional Schr¨ odinger equations with d = 2, 3, L´ evy index 1 < α < 2 and Hartree type nonlinearity F (u) = µ(|x|
−γ∗ |u|
2)u with max(α,
2d−12d) < γ ≤ 2, γ < d. This equation is scaling-critical in ˙ H
sc, s
c=
γ−α2. We show that the solution scatters in H
s,1for any s > s
c, where H
s,1is a space of Sobolev type taking in angular regularity with norm defined by kϕk
Hs,1= kϕk
Hs+ k∇
Sϕk
Hs. For this purpose we use the recently developed Strichartz estimate which is L
2-averaged on the unit sphere S
d−1and utilize U
p-V
pspace argument.
1. Introduction
In this paper we consider the following Cauchy problem for Hartree type fractional Schr¨ odinger equations of the form:
i∂ t u = D α u + F (u) in R 1+d , u(x, 0) = ϕ(x) x ∈ R d , (1.1)
where D α = (−∆)
α2, 1 < α < 2, d = 2, 3 and F (u) = [V ∗ |u| 2 ]u with V = µ|x| −γ , 0 < γ < d, µ ∈ R \ {0}. The equation (1.1) has the scaling invariance structure in ˙ H sc, s c = γ−α 2 . Some basic notations are listed at the end of this section.
By Duhamel’s formula, (1.1) is written as an integral equation
(1.2) u = e −itDαϕ − i Z t
0
e −i(t−t0)D
α(F (u(t 0 ))) dt 0 .
Received April 2, 2017; Revised June 15, 2017; Accepted July 24, 2017.
2010 Mathematics Subject Classification. 35Q55, 35Q53.
Key words and phrases. Hartree type fractional Schr¨ odinger equation, small data scat- tering, angularly averaged Strichartz estimate, U
pand V
pspaces.
c
2018 Korean Mathematical Society
373
Here we define the linear propagator e −itDαgiven by the solution to the linear problem i∂ t v = D α v with initial datum v(0) = f . It is formally given by
e −itDαf = F −1 (e −it|ξ|αF (f )) = (2π) −d Z
F (f )) = (2π) −d Z
R
de i(x·ξ−t|ξ|α) f (ξ) dξ, b (1.3)
where b f = F (f ) denotes the Fourier transform of f and F −1 the inverse Fourier transform such that
F (f )(ξ) = Z
R
de −ix·ξ f (x) dx, F −1 (g)(x) = (2π) −d Z
R
de ix·ξ g(ξ) dξ.
The fractional Schr¨ odinger equations have been derived to describe natural phenomena for the system of long-range lattice interaction (1 < α < 2) [16], water waves (α = 1 2 , 3 2 ) [19], and fractional quantum mechanics (1 < α < 2) [18]. We are concerned with small-data scattering theory of (1.1). In this paper the notion of scattering is defined as follows.
Definition 1.1. We say that a solution u to (1.1) scatters (to u ± ) in a Hilbert space H if there exist ϕ ± ∈ H (with u ± (t) = e −itDαϕ ± ) such that lim t→±∞ ku(t) − u ± k H = 0.
There have been several scattering results for the Hartree type nonlinearity only for d ≥ 3. The small data scattering can be obtained by Strichartz esti- mates [2, 3] as γ > 2 and by weighted space estimates [1, 9] as 1 < γ ≤ 2. In the weighted space regime one can take advantage of the smoothing effect of Hartree potential in the high frequency analysis. However, we have to pay the cost d ≥ 3 and high regularity for using the smoothing of ∇V . To resolve the 2D or low regularity problem a different approach is required. The authors of this paper treated the cubic problem F (u) = µ|u| 2 u for d ≥ 3 in [3, 4] by using U p − V p space argument based on the endpoint Strichartz estimates which is angularly averaged. In this paper we apply the U p − V p argument to the 2, 3-d Hartree problem.
To state the main theorem let us introduce angularly regular Sobolev space H s,1 . It is the set of all H s functions whose angular derivative is also in H s . The norm is defined by kf k Hs,1 := kf k Hs+ k∇ S f k Hs. Here ∇ S is the gradient on the unit sphere and it can also be represented as x × ∇.
+ k∇ S f k Hs. Here ∇ S is the gradient on the unit sphere and it can also be represented as x × ∇.
Theorem 1.2. Let d = 2, 3, 1 < α < 2, max(α, 2d−1 2d ) < γ ≤ 2, γ < d and let s > s c . Then there exists δ > 0 such that for any ϕ ∈ H s,1 with kϕk Hs,1 ≤ δ, (1.1) has a unique solution u ∈ (C ∩ L ∞ )(R; H s,1 ) which scatters in H s,1 .
We show Theorem 1.2 essentially on the basis of the nonlinear estimates, namely Proposition 4.1 below. The crucial part of the nonlinear estimate occurs in the case where high-high frequency interactions make low frequency outputs.
Such case does not occur in the cubic problem. Thanks to the radial lemmas
(Lemma 2.8 and Lemma 2.9 below) and endpoint Strichartz estimates ((2.1),
(2.2), (2.3) below) we can circumvent the singularity in the low frequency part
in case that 2d−1 2d < α < 2, α < γ < 2 for d = 2 and α < γ ≤ 2 for d = 3.
The subcritical condition s > s c is necessary for the summability of square sum (3.2). If we consider the initial data in the Besov space B s 2,1c,1 with norm defined by kϕk B
sc,1
2,1
= kϕk Bsc
2,1
+ k∇ S ϕk Bsc
2,1
, one can easily get the small data GWP in B 2,1 sc,1 and scattering in H s
c,1 . See Remark 1 below.
The scattering in the sense of Definition 1.1 does not occur in the long-range case 0 < γ < 1 (see [8]). For the critical case γ = 1 it is highly expected that a modified scattering will occur. In fact, it turns out to be true for the 3-d case when α is close enough to 2 (see [5]). The short-range case 1 < γ ≤ α still remains open. These two issues deserves to be taken into account within a regime of weighted spaces or U p -V p spaces.
Due to the lack of analysis technique for high-high-low case, the mass critical case γ = α is not attained by the current argument. But one can think of a modification of potential near zero frequency as in [1, 3]. Here let us consider a potential V in the sense:
(H) V is radial and smooth functions on R d \{0} such that for any nonnegative integers k
|∇ k ξ ( b V (ξ))| . |ξ| −(d−α+)−k for |ξ| ≤ 1,
|∇ k ξ ( b V (ξ))| . |ξ| −(d−α)−k for |ξ| ≥ 1,
where α + = α + ε for an arbitrarily small ε > 0. The Yukawa potential V = µ e−m|x||x| (m > 0) is of type (H) since b V (ξ) = 4πµ(m 2 + |ξ| 2 ) −1 .
Now we introduce the second scattering result concerning the hybrid type potential.
Theorem 1.3. Let d = 2, 3 and 2d−1 2d < α < 2. Suppose that V satisfies (H).
If ϕ ∈ H s,1 for some s > α+2 −α and kϕk H
s,1 is sufficiently small, then there exists a unique solution u ∈ (C ∩ L ∞ )(R; H s,1 ) which scatters in H s,1 .
This paper is organized as follows: In Section 2 we give several preliminary lemmas on 2-d endpoint Strichartz estimates, U p − V p space, and radial func- tions. We prove Theorem 1.2 in Section 3 via nonlinear estimates. In Section 4 we provide a proof for the crucial nonlinear estimate. In the last section we prove Theorem 1.3.
Basic notations.
• Littlewood-Paley operators: (1) Homogeneous type. ˙ β ∈ C 0,rad ∞ with ˙ β ¨ β = ˙ β and ¨ β(ξ) = ˙ β(ξ/2) + ˙ β(ξ) + ˙ β(2ξ). F ( ˙ P N f )(ξ) = ˙ β(ξ/N ) b f for any dyadic number N . Let ¨ P N = ˙ P N/2 + ˙ P N + ˙ P 2N . Then ˙ P N P ¨ N = ˙ P N . (2) Inhomogeneous type. P 1 = 1 − P
N >1 P ˙ N and P N = ˙ P N for N > 1.
• Fractional derivatives: D s = (−∆)
s2= F −1 |ξ| s F , Λ s = (1 − ∆)
s2= F −1 (1 +
|ξ| 2 )
s2F for s > 0.
• Function spaces: H ˙ r s = D −s L r , H ˙ s = H ˙ 2 s , H r s = Λ −s/2 L r , H s = H 2 s , L r = L r x (R d ) for s ∈ R and 1 ≤ r ≤ ∞. The Besov space B s p,q is defined by the norm kf k Bp,qs = ( P
N ≥1 N qs kP N f k q L
px
)
1qfor s ∈ R, 1 ≤ p, q ≤ ∞.
• Mixed-normed spaces: For a Banach space X, u ∈ L q I X if and only if u(t) ∈ X for a.e. t ∈ I and kuk Lq
I
X := kku(t)k X k L
qI
< ∞. Especially, we denote L q I L r x = L q t (I; L r x (R d )), L q I,x = L q I L q x and L q t L r x = L q
R L r x .
• 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. Strichartz estimates
Let the pair (q, r) satisfy that 2 ≤ q, r ≤ ∞, 2 q + d r = d 2 and (q, r, d) 6=
(2, ∞, 2). Then we have for any 0 < α 6= 1 < 2 that N −2−αq ke −itDαP ˙ N ϕk Lq
P ˙ N ϕk Lq
t
L
rx. k ˙ P N ϕk L2
x. (2.1)
For this see [10].
The endpoint estimate can be extended to a wider range provided an angular average is taken into account. More precisely, let d = 2, 3, 1 < α < 2, and let 6 < r ≤ ∞, r ∗ = 2 if d = 2 and 10 3 < r < 6, 2 < r ∗ < 10−r 4r if d = 3. Then there holds
N −d−α2 +
drke −itDαP ˙ N ϕk L2
P ˙ N ϕk L2
t
L
rρL
r∗θ. k ˙ P N ϕk L2x. (2.2)
In particular, if d−α 2d < r < d−2 2d , then we have from (2.2) that
(2.3)
ke −itDαP 1 ϕk L2
t
L
rρL
r∗θ. X
0<N ≤1
ke −itDαP ˙ N ϕk L2
t
L
rρL
r∗θ. X
0<N ≤1
N
d−α2−
drk ˙ P N ϕk L2
x
. kP 1 ϕk L2
x. Here the norm L 2 t L r ρ L r θ∗ with r < ∞ is defined by
with r < ∞ is defined by
kuk L2
t
L
rρL
r∗θ= Z
R
Z
S
d−1|u(t, ρθ)| r∗dθ
r∗1
2
L
rρ(ρ
d−1dρ)
!
12.
If r = ∞, then we define L ∞ ρ = L ∞ ρ . For (2.2) see Theorem 1.1 and Corollary
2.9 of [12]. See also [6,13] for some general Strichartz estimates associated with
angular regularity.
2.2. U p and V p spaces
We list some useful properties of U p and V p spaces. For complete a descrip- tion see [14, 17].
Let Z be the set of all finite partitions −∞ < t 0 < · · · < t K ≤ ∞ of R. If t K = ∞, then we use the convention u(t K ) = 0 for all functions u : R → L 2 x . Definition 2.1. Let 1 ≤ p < ∞. A U p -atom is defined by a step function a : R → L 2 x of the form
a(t) =
K
X
k=1
χ [tk−1,t
k) (t)φ k−1 , where χ is the characteristic function,
{t k } ∈ Z, {φ k } K−1 k=0 ⊂ L 2 x with
K−1
X
k=0
kφ k k p L2
x = 1.
The atomic space U p (R; L 2 x ) is defined as the set of functions u : R → L 2 x of the form
U p (R; L 2 x ) =
u =
∞
X
j=1
λ j a j
a j are U p -atoms and {λ j } ∈ ` 1
,
with the norm
kuk Up := inf
∞
X
j=1
|λ j |
u = X λ j a j
.
Definition 2.2. Let 1 ≤ p < ∞. We define V p (R; L 2 x ) as the normed space of all functions v : R → L 2 x such that lim t→±∞ v(t) exist and for which the norm
kvk Vp := sup
{t
k}∈Z K
X
k=1
kv(t k ) − v(t k−1 )k p L2
x
!
1pis finite. V − p (R; L 2 x ) denotes the normed space of all function v ∈ V p (R; L 2 x ) with v(−∞) = 0. V −,rc p (R; L 2 x ) is the closed subspace of all right continuous V − p (R; L 2 x ) functions.
Lemma 2.3. (1) U p (R; L 2 x ) and V p (R; L 2 x ) are Banach spaces.
(2) For 1 ≤ p < q < ∞ the embedding U p (R; L 2 x ) ,→ U q (R; L 2 x ) ,→ L ∞ t L 2 x is continuous.
(3) Every u ∈ U p (R; L 2 x ) is right-continuous and lim t→−∞ u(t) = 0.
(4) For 1 ≤ p < ∞ the embedding U p (R; L 2 x ) ,→ V −,rc p (R; L 2 x ) is continuous.
(5) For 1 < p < ∞ (U p (R; L 2 x )) ∗ = V p0(R; L 2 x ), where 1 p + p 10 = 1.
= 1.
Let B(u, v) denote the duality form between U p (R; L 2 x ) and V p0(R; L 2 x ).
Lemma 2.4. Let 1 < p < ∞. Let u ∈ V − 1 (R; L 2 x ) be absolutely continuous on compact intervals and v ∈ V p0(R; L 2 x ). Then
B(u, v) = − Z ∞
−∞
u 0 (t), v(t) dt.
Now we introduce adapted space U α p and V α p to e −itDα.
Definition 2.5. We define U α p (R; L 2 x ) as the spaces of all functions u : R → L 2 x
such that e itDαu ∈ U p (R; L 2 x ) with norm kuk Uαp := ke itDαuk Up. Likewise, we define V α p (R; L 2 x ) and its norm kvk Vαp:= ke itDαvk Vp.
:= ke itDαuk Up. Likewise, we define V α p (R; L 2 x ) and its norm kvk Vαp:= ke itDαvk Vp.
. Likewise, we define V α p (R; L 2 x ) and its norm kvk Vαp:= ke itDαvk Vp.
vk Vp.
Lemma 2.3 is extended to the spaces U α p (R; L 2 x ) and V α p (R; L 2 x ).
Lemma 2.6. For any v ∈ L ∞ t L 2 x we have kvk L∞
t
L
2x. kvk Vα2.
Lemma 2.7 (Transfer principle). Let T : L 2 x → L 1 loc (R d ; C) be a linear operator satisfying that
kT (e −itDαφ)k Lq
t
X . kφk L
2xfor some 1 ≤ q ≤ ∞ and a Banach space X ⊂ L 1 loc . Then kT (u)k Lq
t
X . kuk U
αq.
Proof of Lemma 2.7. Without loss of generality we may assume that e itDαu(t) is a U q -atom with the partition t 0 , . . . , t K−1 and t K = ∞. That is to say,
e itDαu(t) =
K
X
k=1
χ [tk−1,t
k) φ k−1
for φ k ∈ L 2 with P K−1
k=0 kφ k k q L
2= 1. Then we have only to show that kT (u)k Lq
t
X . 1. In fact, by assumption we have kT (u)k q L
qt
X = kT (e −itD
αK
X
k=1
χ [tk−1,t
k) φ k−1 )k q L
q
tX
=
K
X
k=1
kT (e −itDαφ k−1 )k q Lq
[tk−1,tk)
X
.
K−1
X
k=0
kφ k k q L2
x . 1.
2.3. Some useful lemmata
Lemma 2.8. Let ψ, f be smooth and let ψ be radially symmetric. Then
∇ S (ψ ∗ f ) = ψ ∗ ∇ S f.
Lemma 2.9 (Lemma 7.1 of [7]). If ψ(x) is radially symmetric, then kψ ∗ f k LpρL
q
θ
≤ kf k Lp1
ρ
L
q1θkψk Lp2 x
for all p 1 , p 2 , p, q, q 1 ∈ [1, ∞] satisfying 1
p 1 + 1
p 2 − 1 = 1 p , 1
q 1 + 1
p 2 − 1 ≤ 1 q . The next is on the convolution inequality for sequence.
Lemma 2.10. For any f ∈ ` 1 (Z) and g ∈ ` 2 (Z) we have kf ∗ gk `2(Z) ≤ kf k `
1(Z) kgk `
2(Z) .
The final one is on the Sobolev inequality on the unit sphere.
Lemma 2.11. For any d − 1 < e r < ∞ kf k L∞θ . kf k Lrθe+ k∇ S f k Lre
+ k∇ S f k Lre
θ
, kf k Lre
θ
. kf k L2θ+ k∇ S f k L2
θ. For this see [11, 20].
. For this see [11, 20].
3. Proof of Theorem 1.2 Let us define the Banach space X s,1 for s ∈ R by
X s,1 := n
u : R → L 2
P N u, ∇ S2P N u ∈ U α 2 (R; L 2 x ) ∀N ≥ 1 o with the norm
kuk Xs,1 =
X
N ≥1
N 2s (kP N uk U2
α
+ kP N (∇ S2u)k U2
α) 2
) 2
1 2
.
Let X + s,1 be the restricted space defined by X + s,1 = n
u ∈ C([0, ∞); H s )
χ [0,∞) (t)u(t) ∈ X s,1 o with norm kuk Xs,1
+
:= kχ [0,∞) uk Xs,1.
Let D + (δ) be a complete metric space {u ∈ X + s,1
kuk Xs,1
+
≤ δ} equipped with the metric d(u, v) := ku − vk Xs,1
+
. Then we will show that the nonlinear functional Ψ(u) = e −itDαϕ + N α (u) is a contraction on D + (δ), where
N α (u) = −i Z t
0
e −i(t−t0)D
α[V ∗ |u| 2 ]u dt 0 . We will show that if s > s c = 2−α 2 ,
(3.1)
kN α (u)k Xs,1
+
. kuk 3 Xs,1
+
, kN α (u) − N α (u)k Xs,1
+
. (kuk X+s,1+ kvk Xs,1
+
) 2 ku − vk Xs,1
+
.
Clearly, ke −itDαϕk Xs,1
+
. kϕk Hs,1 and thus we can find δ small enough for Ψ to be a contraction mapping on D + (δ).
Since e itDαP N N α (u) and e itDαP N ∇ S N α (u) are in V −,rc 2 (R; L 2 ) from (4) of Lemma 2.3, and
P N ∇ S N α (u) are in V −,rc 2 (R; L 2 ) from (4) of Lemma 2.3, and
X
N ≥1
N 2s (ke itDαP N N α (u)k V2+ ke itDαP N ∇ S N α (u)k V2) 2 < ∞
+ ke itDαP N ∇ S N α (u)k V2) 2 < ∞
) 2 < ∞
from (3.1), lim t→+∞ e itDαN α (u) exists in H s,1 . Define a scattering state u + with
ϕ + := ϕ + lim
t→+∞ e itD
αN α (u).
By time symmetry we can argue in a similar way for the negative time. Thus we get the desired result.
Now it remains to show (3.1). Since the second part of (3.1) follows from the argument of the first part, we omit it. We may assume that u(t) = 0 for −∞ < t < 0. Since clearly R t
0 e it
0D
αP N [(V ∗ |u| 2 )u] dt 0 ∈ V − 1 (R; L 2 x ) and differentiable, from the duality form and Lemma 2.4 it follows that
kP N N α (u)k U2
α
= ke itDαP N N α (u)k U2 = sup
= sup
kvk
V 2≤1
B(e itDαP N N α (u), v)
= sup
kvk
V 2≤1
Z
R
Z
R
d[V ∗ |u| 2 ]u(t)e −itDαP N v(t) dxdt
= sup
kvk
V 2α
≤1
Z
R
Z
R
d[V ∗ |u| 2 ]u(t)P N v(t) dxdt and similarly
kP N ∇ S N α (u)k Uα2 = sup
kvk
V 2α
≤1
Z
R
Z
R
d∇ S ([V ∗ |u| 2 ]u(t))P N v(t) dxdt . By Littlewood-Paley decomposition we get
kN α (u)k 2 Xs,1
. X
N ≥1
N 2s
sup
kvk
V 2α
≤1
X
N
1,N
2,N
3≥1
Z Z
[V ∗ P N1uP N2u]P N3u(t)P N v(t) dxdt
u]P N3u(t)P N v(t) dxdt
2
+ sup
kvk
V 2α
≤1
X
N
1,N
2,N
3≥1
Z Z
∇ S ([V ∗ P N1uP N2u]P N3u(t))P N v(t) dxdt
u]P N3u(t))P N v(t) dxdt
2
.
(3.2)
The nonlinear estimate (Proposition 4.1 below) yields that for any ε > 0
Z Z
[V ∗ P N1uP N2u]P N3u(t)P N v(t) dxdt +
u]P N3u(t)P N v(t) dxdt +
Z Z
∇ S ([V ∗ P N1uP N2u]P N3u(t))P N v(t) dxdt . (N min N med )
γ−α2
u]P N3u(t))P N v(t) dxdt . (N min N med )
γ−α2
3
Y
i=1
(kP Niuk U2
α
+ kP Ni∇ S uk U2
α), (3.3)
), (3.3)
where N min = min(N 1 , N 2 , N 3 ), N max = max(N 1 , N 2 , N 3 ) and N min ≤ N med ≤ N max . Since ∇ S = x × ∇ and hence ∇ S (vw) = (∇ S v)w + v(∇ S w), we can apply Lemma 2.8 and Proposition 4.1 in the next section with u i = u.
The self-mapping property (3.1) can be shown by exactly the same way as in [15]. But we brief on it for the reader’s convenience. Let us split the summation of RHS of (3.2) into three parts as follows:
RHS of (3.2) =: Σ 1 + Σ 2 + Σ 3 , where
Σ 1 = X
N
3∼N
, Σ 2 = X
N
3N
, Σ 3 = X
N
3N
.
For Σ 1 we use Lemma 2.10 w.r.t. N 3 , N and Cauchy-Schwarz inequality w.r.t. N 1 , N 2 together with (3.3) to get
Σ 1 ≤ X
N ≥1
N 2s
X
N
1,N
2,N
3≥1 N
3∼N
(N 1 N 2 )
γ−α2−s (N 1 N 2 ) s
3
Y
i=1
(kP Niuk U2
α
+ kP Ni∇ S uk U2
α)
)
2
. kuk 2 Xs,1
X
N
1,N
2≥1
(N 1 N 2 )
γ−α2−s (N 1 N 2 ) s
2
Y
j=1
(kP Niuk Uα2+ kP Ni(∇ S u)k Uα2
+ kP Ni(∇ S u)k Uα2
2
. kuk 6 Xs,1.
For Σ 2 and Σ 3 we use the fact that if N 3 N or N 3 N , then N . max(N 1 , N 2 ) or N 3 . max(N 1 , N 2 ), respectively. Applying Lemma 2.10 w.r.t.
max(N 1 , N 2 ), N and then Cauchy-Schwarz inequality w.r.t. min(N 1 , N 2 ), N 3 , we get
Σ 2 + Σ 3
. X
N ≥1
( X
N
3N
+ X
N
3N
)
N
max(N 1 , N 2 )
s
(min(N 1 , N 2 )N 3 )
γ−α2−s
(min(N 1 , N 2 )N 3 ) s (max(N 1 , N 2 )) s ×
3
Y
i=1
(kP Niuk Uα2+ kP Ni∇ S uk Uα2)
+ kP Ni∇ S uk Uα2)
)
! 2
. kuk 6 Xs,1.
This shows (3.1) and completes the proof of Theorem 1.2.
Remark 1. Let us define the Banach space Y sc,1 by Y s
c,1 := n
u : [0, ∞) → L 2
P N u, ∇ S P N u ∈ U α 2 (R; L 2 x ) ∀N ≥ 1 o with the norm
kuk Ys,1 = X
N ≥1
N sc(kP N uk U2
α
+ kP N (∇ S u)k U2
α).
Instead of (3.2), we need to estimate
kN α (u)k Ysc,1
. X
N ≥1
N s
sup
kvk
V 2α
≤1
X
N
1,N
2,N
3≥1
Z Z
[V ∗ P N1uP N2u]P N3u(t)P N v(t) dxdt
u]P N3u(t)P N v(t) dxdt
+ sup
kvk
V 2α
≤1
X
N
1,N
2,N
3≥1
Z Z
∇ S ([V ∗ P N1uP N2u]P N3u(t))P N v(t) dxdt
u]P N3u(t))P N v(t) dxdt
. kuk 3 Ysc,1
+
. (3.4)
The main part Σ 0 of the summation is the case N N 3 . Σ 0 = Σ 1 + Σ 2 + Σ 3 , where Σ 1 = P
N
1N
2, Σ 2 = P
N
1N
2and Σ 3 = P
N
1∼N
2. If N 1 N 2 (N 1 N 2 ), then N 2 ∼ N (N 1 ∼ N ), respectively, and hence by (3.3)
Σ 1 + Σ 2
.
X
N
1N
2N
3N ∼N
2+ X
N
1N
2N
3N ∼N
1
N sc(N 1 N 3 ) sc
3
Y
i=1
(kP Niuk U2
α
+ kP Ni(∇ S u)k U2
α
. kuk 2 Ysc,1
X
N
2≥1
X
N ∼N
2N sc(kP N2uk U2
uk U2
α
+ kP N2(∇ S u)k U2
α)
)
+ X
N
1≥1
X
N ∼N
1N sc(kP N1uk Uα2 + kP N1(∇ S u)k Uα2
uk Uα2 + kP N1(∇ S u)k Uα2
(∇ S u)k Uα2
. kuk 3 Ysc,1.
If N 1 ∼ N 2 , then N . N 1 ∼ N 2 . Therefore
Σ 3 . X
N
1∼N
2N
3N
2X
N .N
2N sc(N 1 N 3 ) sc
3
Y
i=1
(kP Niuk U2
α
+ kP Ni(∇ S u)k U2
α
) . kuk 3 Ysc,1.
4. Nonlinear estimate
Proposition 4.1. Assume that u 1 , u 2 , ∇ S u i (i = 1, 2, 3) ∈ U α 2 , v ∈ V α 2 . Let u i = P Niu i , v = P N v for N i , N ≥ 1, i = 1, 2, 3 and let u e i = u i or u i . Then for all N i , N ≥ 1 we have
(4.1)
Z Z
[V ∗ ( u f 1 f u 2 )](∇ S f u 3 )v dxdt . (N min N med )
γ−α22
Y
i=1
(ku i k U2
α
+ k∇ S u i k U2
α
)k∇ S u 3 k U2
αkvk V2
α
and
(4.2)
Z Z
[V ∗ ((∇ S u f 1 ) u f 2 )] f u 3 v dxdt . (N min N med )
γ−α2k∇ S u 1 k U2
α
3
Y
i=2
(ku i k U2
α
+ k∇ S u i k U2
α)kvk V2
α
. Proof of (4.1). We may assume that N 1 ≤ N 2 by summation symmetry. By Littlewood-Paley decomposition we have
Z Z
[V ∗ ( f u 1 f u 2 )](∇ S f u 3 )v dxdt
≤ X
M >0
Z Z
P ˙ M (V ∗ ( f u 1 f u 2 )) ¨ P M ((∇ S u f 3 )v) dxdt
=: X
M >0
N M .
We first consider the case (123): N 1 ≤ N 2 ≤ N 3 . This case can be split into two parts (i) N 1 N 2 ∼ M or N 1 ∼ N 2 ∼ M ; (ii) N 1 ∼ N 2 M .
Case (i) of (123). If d = 2, then by using the embedding lemmas (Lemma 2.3 and Lemma 2.6) and the Sobolev embedding H θ 1 (S 1 ) ,→ L ∞ θ (S 1 ), we have
N M
. M −(2−γ) k ˙ P M ( u f 1 u f 2 )k L1
t
L
∞xk ¨ P M (∇ S f u 3 v)k L∞t L
1x
. M −(2−γ) (N 1 N 2 )
2−α2N
−(2−α) 2
1 ku 1 k L
2 tL
∞xN
−(2−α) 2
2 ku 2 k L
2t
L
∞xk∇ S u 3 k L∞
t
L
2xkvk L∞
t L
2x
. (N 1 N 2 )
γ−α22
Y
i=1
(N
−(2−α) 2
i ku i k L
2t
L
∞ρL
2θ+ N
−(2−α) 2
i k∇ S u i k L
2t
L
∞ρL
2θ)k∇ S u 3 k U2
α
kvk V2
α
. Then by using Lemma 2.7 with X = L ∞ ρ L 2 θ and combining with (2.2) and (2.3), we get
N M . (N 1 N 2 )
γ−α22
Y
i=1
(ku i k U2
α
+ k∇ S u i k U2
α
)k∇ S u 3 k U2
αkvk V2
α
.
Thus since for (i) of (123) M ∼ N 2 , we get X
(i) of (123)
N M . (N 1 N 2 )
γ−α22
Y
i=1
(ku i k Uα2+ k∇ S u i k Uα2)k∇ S u 3 k Uα2kvk Vα2.
)k∇ S u 3 k Uα2kvk Vα2.
.
If d = 3, then by Bernstein’s inequality we have N M . M −(3−γ) k ˙ P M ( u f 1 u f 2 )k L1
t
L
∞xk ¨ P M (∇ S u f 3 v)k L∞
t
L
1x. M −(3−γ) M k f u 1 u f 2 k L1
t
L
3xk∇ S f u 3 vk L∞
t
L
1x.
Then the endpoint Strichartz estimate of (2.1) and Lemma 2.7 with X = L 6 x give us
N M . M γ−2 (N 1 N 2 )
2−α2Y
i=1,2
(N
−(2−α) 2
i ku i k L
2t
L
6x)k∇ S u 3 k L∞
t
L
2xkvk L∞
t
L
2x. (N 1 N 2 )
γ−α2Y
i=1,2
(N
−(2−α) 2
i ku i k L
2t
L
6x)k∇ S u 3 k L∞t L
2xkvk L∞t L
2x
L
2x. (N 1 N 2 )
γ−α22
Y
i=1
ku i k Uα2k∇ S u 3 k Uα2kvk Vα2.
kvk Vα2.
Thus since for (i) of (123) M ∼ N 2 , we get X
(i) of (123)
N M . (N 1 N 2 )
γ−α22
Y
i=1
ku i k U2
α
k∇ S u 3 k U2
αkvk V2
α
. Case (ii) of (123). At first we estimate
N M . M −(d−γ) k ˙ P M ( u f 1 u f 3 )k L1
t
L
∞xk ¨ P M (∇ S u f 3 v)k L∞
t
L
1x.
Let us choose r such that d−γ 2 < d r < d−α 2 . Applying Lemma 2.9 with p 1 =
r
2 , p 2 = r−2 r and q 1 = q = ∞ to the radial kernel of ˙ P M , we get N M . M −(d−γ) k ˙ P M ( u f 1 f u 2 )k L
1t
L
∞xk ¨ P M (∇ S u f 3 v)k L∞t L
1x
. M −(d−γ)+2dr k u f 1 u f 2 k
L
1tL
ρr2L
∞θk∇ S u f 3 vk L∞
t
L
1x.
From the Sobolev inequality (Lemma 2.11) and the embedding lemmas (Lemma 2.3 and Lemma 2.6) it follows that
N M . M −(d−γ)+2drku 1 k L2
t
L
rρL
∞θku 2 k L2
t
L
rρL
∞θk∇ S u 3 k L∞
t
L
2xkvk L∞
t
L
2x. M −(d−γ)+2dr(N 1 N 2 )
d−α2 −
dr
× Y
j=1,2
N −
d−α 2
+
dri (ku i k L
2t
L
rρL
dθ+ k∇ S u i k L2
t
L
rρL
dθ)k∇ S u 3 k L∞t L
2xkvk L∞t L
2x
L
2x. M −(d−γ)+2dr(N 1 N 2 )
d−α2 −
3r
× Y
j=1,2
(ku i k U2
α
+ k∇ S u i k U2
α
)k∇ S u 3 k L∞
t
L
2xkvk L∞
t
L
2x. Then we conclude that
X
(ii) of (123)
N (ii) M (1, 2, 3) . (N 1 N 2 )
γ−α22
Y
i=1
(ku i k Uα2 + k∇ S u i k Uα2)k∇ S u 3 k Uα2kvk Vα2.
)k∇ S u 3 k Uα2kvk Vα2.
.
We have shown the proposition in the case (123). Now let us consider the case (132): N 1 ≤ N 3 ≤ N 2 , which can be split into (i): N 3 N 2 ∼ M or N 3 ∼ N 2 ∼ M and (ii): N 3 ∼ N 2 M .
We first consider the case (i) of (132). If d = 2, then we change the role of u 2 and u 3 in this case. Applying Lemma 2.9 with p 2 = 1 to the radial kernels of ˙ P M and ¨ P M , we get
N M . M −(2−γ) k ˙ P M ( f u 1 u f 2 )k L2
t
L
2ρL
∞θk ¨ P M ((∇ S u f 3 )v)k L2
tL
2ρL
1θ
. M −(2−γ) k u f 1 u f 2 k L2
t
L
2ρL
∞θk(∇ S f u 3 )vk L2
tL
2ρL
1θ
. M −(2−γ) ku 1 k L2
t
L
∞xku 2 k L∞
t
L
2ρL
∞θk∇ S u 3 k L2
t
L
∞ρL
2θkvk L∞
t
L
2x. The Sobolev embedding H θ 1 (S 1 ) ,→ L ∞ θ (S 1 ) applied to the norms for u 1 , u 2
yields
N M . M −(2−γ) (ku 1 k L2
t
L
∞ρL
2θ+ k∇ S u 1 k L2
t
L
∞ρL
2θ) (ku 2 k L∞t L
2x+ k∇ S u 2 k L∞t L
2x)k∇ S u 3 k L2
L
2x)k∇ S u 3 k L2
t
L
∞ρL
2θkvk L∞t L
2x.
By the transfer principle for u 1 , u 3 and the embeddings U α 2 (R; L 2 x ), V α 2 (R; L 2 x ) ,→ L ∞ t L 2 x , we have that
N M . M −(2−γ) (N 1 N 3 )
2−α2(ku 1 k U2
α
+ k∇ S u 1 k U2
α) (ku 2 k U2
α
+ k∇ S u 2 k U2
α
)k∇ S u 3 k U2
αkvk V2
α
, which gives us
X
(i) of (132)
N (i) M (1, 3, 2) . (N 1 N 3 )
γ−α2Y
i=1,2
(ku i k U2
α
+ k∇ S u i k U2
α
)k∇ S u 3 k U2
αkvk V2
α
. If d = 3, then Bernstein’s inequality yields
N M . M −(3−γ) k ˙ P M ( u f 1 u f 2 )k L2
t
L
2xk e P M ((∇ S u f 3 )v)k L2
tL
2x
. M −(3−γ)+1 k u f 1 f u 2 k
L
2tL
32 x
k(∇ S u f 3 )vk
L
2tL
32 x
. M γ−2 ku 1 k L2
t
L
6xku 2 k L∞t L
2xk∇ S u 3 k L2
t
L
6xkvk L∞t L
2x.
Using the transfer principle for u 1 , u 3 associated with the endpoint Strichartz estimate of (2.1), we have that
X
(i) of (132)
N (i) M (1, 3, 2) . (N 1 N 3 )
γ−α2Y
i=1,2
ku i k U2
α
k∇ S u 3 k U2
αkvk V2
α
.
On the other hand, from (ii) and the support condition it follows that N 3 ∼ N 2 ∼ N 1 M . We perform a similar estimate to the case (ii) of (123) as follows:
X
(ii) of (132)
N M
. M −(d−γ)+2dr (N 1 N 3 )
d−α2 −
dr Y
i=1,2
(ku i k Uα2+ k∇ S u i k Uα2k∇ S u 3 k Uα2kvk Vα2
k∇ S u 3 k Uα2kvk Vα2
. (N 1 N 3 )
γ−α2Y
i=1,2
(ku i k U2
α
+ k∇ S u i k U2
α
k∇ S u 3 k U2
αkvk V2
α
.
Let us treat the final case (312): N 3 ≤ N 1 ≤ N 2 . This is split into the four parts: (N 3 ∼ N 1 ≤ N 2 ); (N 3 N 1 N 2 ); (N 3 N 1 ∼ N 2 and N 3 . M ); (M N 3 N 1 ∼ N 2 ). For simplicity we only consider the last case: M N 3 N 1 ∼ N 2 . The remaining cases can be handled similarly.
Let us take r such that d−γ 2d < 1 r < d−α 2d . Applying Lemma 2.9 with p = 2, p 1 =
2r
r+2 , p 2 = r−1 r , q 1 = r, q = r 2r
∗∗
−2 to the norm of u 1 u 2 and p = 2, p 1 = r+2 2r , p 2 =
r
r−1 , q 1 = q = r 2r
∗∗
+2 to the norm of u 3 v, respectively, we get N M . M −(d−γ) k ˙ P M ( u f 1 u f 2 )k
L
2tL
2ρL
2r∗
r∗−2 θ
k e P M ((∇ S u f 3 )v)k
L
2tL
2ρL
2r∗
r∗+2 θ
. M −(d−γ)+2drk u f 1 u f 2 k
L
2tL
r+22r ρ
L
rθk(∇ S u f 3 )vk
L
2tL
2r ρr+2
L
r∗+22r∗
θ
. M −(d−γ)+2dr(N 1 N 3 )
d−α2 −
dr Y
j=1,2
(ku i k U2
α
+ k∇ S u i k U2
α
)k∇ S u 3 k U2
αkvk V2
α
and hence X
(Last)
N M . (N 1 N 3 )
γ−α2Y
j=1,2
(ku i k U2
α
+ k∇ S u i k U2
α
)k∇ S u 3 k U2
αkvk V2
α
.
This proves (4.1).
Proof of (4.2). By Littlewood-Paley decomposition we have
Z Z
[V ∗ (∇ S u f 1 f u 2 )] f u 3 v dxdt
≤ X
M >0
Z Z
P ˙ M (V ∗ (∇ S u f 1 f u 2 )) ¨ P M ( u f 3 v) dxdt
=: X
M >0
N M .
We first consider the cases (123): N 1 ≤ N 2 ≤ N 3 and (213): N 2 ≤ N 1 ≤ N 3 .
These cases can be split into two parts, max(N 1 , N 2 ) ∼ M and N 1 ∼ N 2 M .
For both cases we use the embedding lemmas (Lemma 2.3 and Lemma 2.6),
Lemma 2.9 with p 2 = r−2 r , d−γ 2d < 1 r < d−α 2d , the Sobolev inequality Lemma 2.11, and then Lemma 2.7 with X = L r ρ L r θ∗, to obtain that
N M . M −(d−γ) k ˙ P M (∇ S f u 1 u f 2 )k L1
t
L
∞ρL
2θk ¨ P M ( u f 3 v)k L∞
t
L
1ρL
2θ. M −(d−γ)+2drk∇ S f u 1 f u 2 k
L
1tL
r
ρ2
L
2θk u f 3 vk L∞
t
L
1ρL
2θ. M −(d−γ)+2dr(N 1 N 2 )
d−α2 −
drN −
d−α 2
+
dr1 k∇ S u 1 k L
2t
L
rρL
r∗θN −
d−α 2
+
dr2 ku 2 k L
2t
L
rρL
∞θku 3 k L∞t L
2ρL
∞θ kvk L∞t L
2x
L
2x. M −(d−γ)+2dr(N 1 N 2 )
d−α2 −
drk∇ S u 1 k U2
α
3
Y
i=2
(ku i k U2
α
+ k∇ S u i k U2
α)kvk V2
α
, and hence
X
(123) or (213)
N M . (N 1 N 2 )
γ−α2k∇ S u 1 k U2 α
3
Y
i=2
(ku i k U2
α
+ k∇ S u i k U2
α)kvk V2
α
. Now we consider the cases (132): N 1 ≤ N 3 ≤ N 2 and (231): N 2 ≤ N 3 ≤ N 1 . These can be split into (i): N 3 max(N 1 , N 2 ) ∼ M or N 3 ∼ max(N 1 , N 2 ) ∼ M and (ii): N 3 ∼ N 2 ∼ N 1 M . The first case (i) can be easily treated with the Strichartz estimate (2.1). We consider (ii) separately; N 1 ≤ N 2 and N 2 ≤ N 1 .
If N 1 ≤ N 2 , then with the same r as above we estimate N M . M −(d−γ) k ˙ P M (∇ S f u 1 u f 2 )k L2
t
L
2ρL
2θk ¨ P M ( u f 3 v)k L2
tL
2ρL
2θ
. M −(d−γ)+2drk∇ S f u 1 f u 2 k
L
2tL
2r ρr+2
L
rr∗
r+r∗
θ
k u f 3 vk
L
2tL
2r ρr+2
L
2θ. M −(d−γ)+2drk∇ S u 1 k L2
t
L
rρL
r∗θku 2 k L∞t L
2ρL
rθku 3 k L2
t
L
rρL
∞θkvk L∞t L
2x
. M −(d−γ)+2dr(N 1 N 3 )
d−α2 −
drN −
d−α 2
+
dr1 k∇ S u 1 k L
2t
L
rρL
r∗θku 2 k L∞
t
L
2ρL
rθN −
d−α 2
+
dr3 ku 3 k L
2t
L
rρL
∞θkvk L∞
t
L
2x. M −(d−γ)+2dr(N 1 N 3 )
d−α2 −
drk∇ S u 1 k Uα2
3
3
Y
i=2
(ku i k Uα2 + k∇ S u i k Uα2)kvk Vα2. On the other hand, if N 2 ≤ N 1 , then
)kvk Vα2. On the other hand, if N 2 ≤ N 1 , then
N M . M −(d−γ) k ˙ P M (∇ S f u 1 u f 2 )k L2
t
L
2ρL
2θk ¨ P M ( u f 3 v)k L2
tL
2ρL
2θ
. M −(d−γ)+2drk∇ S f u 1 f u 2 k
L
2tL
2r r+2 ρ
L
2θk f u 3 vk
L
2tL
2r r+2 ρ
L
2θ. M −(d−γ)+2drk∇ S u 1 k L∞
t
L
2xku 2 k L2
t
L
rρL
∞θku 3 k L2
t
L
rρL
∞θkvk L∞
t
L
2x. M −(d−γ)+2dr(N 2 N 3 )
d−α2 −
drk∇ S u 1 k L∞
t
L
2xN −
d−α 2
+
dr2 ku 2 k L
∞t
L
rρL
∞θN −
d−α 2
+
dr3 ku 3 k L
2t
L
rρL
∞θkvk L∞
t
L
2x. M −(d−γ)+2dr(N 2 N 3 )
d−α2 −
drk∇ S u 1 k Uα2
3
3
Y
i=2
(ku i k Uα2 + k∇ S u i k Uα2)kvk V2
α. By taking d−γ 2d < 1 r < d−α 2d we get
)kvk V2
α. By taking d−γ 2d < 1 r < d−α 2d we get
X
(132) or (231)
N M . (min(N 1 , N 2 )N 3 )
γ−α2k∇ S u 1 k U2
α
3
Y
i=2
(ku i k U2
α
+ k∇ S u i k U2
α
)kvk Vα2. The final cases (312): N 3 ≤ N 1 ≤ N 2 and (321): N 3 ≤ N 2 ≤ N 1 are split into the four parts: (N 3 ∼ min(N 1 , N 2 )); (N 3 min(N 1 , N 2 ) max(N 1 , N 2 ));
(N 3 N 1 ∼ N 2 and N 3 . M ); (M N 3 N 1 ∼ N 2 ). Since each case can be handled by a similar way to the cases (132) and (231), we leave the details to the readers. This completes the proof of (4.2).
5. Proof of Theorem 1.3
In view of the summation argument in Section 3 and Remark 1, we have only to show the nonlinear estimate, Proposition 4.1 for V satisfying (H). Let us first observe that ψ M := M d−γ F −1 ( ¨ β( M · ) b V ) is integrable and its integral is independent of M . Here γ = α + or α. In fact, by the hypothesis of V
Z
|ψ M | = Z
|x|≤M
−1|ψ M | + Z
|x|>M
−1|ψ M | . M −1 k|x| d−1 ψ M k L∞
+ M k|x| d+1 ψ M k L∞
. M −1 M d−γ k∇ d−1 ξ ( ¨ β( ξ
M ) b V (ξ))k L1+ M M d−γ k∇ d+1 ξ ( ¨ β( ξ
M ) b V (ξ))k L1
. 1.
Since ˙ P M (V ∗ ( u f 1 f u 2 )) = M −(d−γ) ψ M ∗ ( ˙ P M ( u f 1 u f 2 )), k ˙ P M (V ∗ ( u f 1 u f 2 ))k Lr
ρL
r∗θ . M −(d−γ) k ˙ P M ( f u 1 u f 2 )k Lr
ρ
L
r∗θfor any 1 ≤ r, r ∗ ≤ ∞. And hence, N M . M −(d−α+) k ˙ P M (V ∗ ( u f 1 u f 2 ))k L
q
t
L
rρL
r∗θk ¨ P M ((∇ S u f 3 )v)k
L
q0tL
r0ρL
r0θ∗for M ≤ 1 and
N M . M −(d−α) k ˙ P M (V ∗ ( f u 1 u f 2 ))k Lq
t
L
rρL
r∗θk ¨ P M ((∇ S u f 3 )v)k
L
q0tL
r0ρL
r0θ∗for M > 1. Following the proof of Proposition 4.1, we actually get that X
M ≤1
N M . (N min N med )
α+−α23
Y
i=1
(ku i k U2
α
