Strichartz estimates for Schr\"odinger equations on irrational tori in two and three dimensions

0.2 MB·

by  Nils Strunk

STRICHARTZ ESTIMATES FOR SCHRÖDINGER
EQUATIONS ON IRRATIONAL TORI IN TWO AND
THREE DIMENSIONS
4
1
NILS STRUNK
0
2

n Abstract. InthispaperweprovenewmultilinearStrichartzesti-
u mates,whichareobtainedbyusingtechniquesofBourgain. These
J
estimatesleadtonewcriticalwell-posednessresultsforthenonlin-

6 earSchrödingerequationonirrationaltoriintwoandthreedimen-

sionswithsmallinitialdata. Inthreedimensions,thisincludesthe
]
P energy critical case. This extends recent work of Guo–Oh–Wang.
A
.
h
t
a 1. Introduction
m
Theaimofthispaper istoobtainnewcriticalwell-posedness results for
[

the nonlinear Schrödinger equation (NLS) posed on two-dimensional
2
v and three-dimensional irrational tori. For d,k ∈ N we consider the
0 following Cauchy problem of the NLS
8
0 i∂ u−∆u = ±|u|2ku
6 (cid:26) t u| = φ ∈ Hs(Td), (1)
. t=0 α
1
0 where we follow the notation of [3] and denote the flat irrational torus
4 by
1
d
: 1
v Td = R/(α T), < α < C, j = 1,...,d. (2)
i α j C j
X Yj=1
r By a change of spatial variables, (1) is equivalent to the following non-
a
linear Schrödinger equation on the rational torus Td = Rd/Zd:
i∂ u−∆u = ±|u|2ku
t (3)
(cid:26) u| = φ ∈ Hs(Td),
t=0
where the Laplace operator ∆ is defined via
∆f(n) = −4π2Q(n)f(n),
with n = (n ,...,n ) ∈ Zd and Q(n) = α n2 + ... + α n2. In the
1 d c b1 1 d d
present paper, we study (3).
2010 Mathematics Subject Classification. Primary 35Q55; Secondary 35B33.
The author was supported by the German Research Foundation, Collaborative
Research Center 701.
1
2 N. STRUNK
The (scaling-)critical Sobolev index is given by
d 1
s = − . (4)
c
2 k
For strong solutions u: (−T,T) ×Td → C one easily verifies that the
energy is conserved, i.e.
1 1
E u(t) = |∇u(t,x)|2dx± |u(t,x)|2k+2dx = E(φ),
(cid:0) (cid:1) 2 ZTd 2k +2 ZTd
(5)
and so is the L2-mass, i.e.
1
M u(t) = |u(t,x)|2dx = M(φ). (6)
(cid:0) (cid:1) 2 ZTd
Hence, the problem with k = 2 in dimension d = 3 is called energy-
critical.
Several authors studied critical well-posedness for NLS on flat ra-
tional tori [5, 6, 8, 7, 3]. However, to our knowledge the first critical
well-posedness results for the NLS on irrational tori have been recently
obtained by Guo–Oh–Wang [3, Theorem 1.7]. They proved critical
well-posedness for small data in the cases
(i) d = 2: k ≥ 6, (ii) d = 3: k ≥ 3, (iii) d ≥ 4: k ≥ 2.
Furthermore, they considered the energy critical case d = 3 and k = 2
on partially irrational tori, where two periods are the same [3, Appen-
dix B]. In the present paper we are going to extend the known results
in two and three dimensions to
(i) d = 2: k ≥ 3, (ii) d = 3: k = 2. (7)
We use an approach similar to Bourgain’s linear Strichartz estimate
for three-dimensional irrational tori [2]. Guo–Oh–Wang applied this
idea to obtain well-posedness in several dimensions [3]. However, in
contrast to [3], we use mixed Lp((−T,T),Lq(Td)) spaces to improve
the trilinear Strichartz estimate [3, Proposition 5.7], leading to the
corresponding result in the energy critical case d = 3 and k = 2. In
addition to that, we use a refined trilinear Strichartz estimate in two
dimensions (Lemma 3.4) to treat the two dimensional case.
In this paper, we will focus on the multilinear Strichartz estimates
in Proposition 3.3 and Proposition 4.1. These propositions serve as
a replacement for Proposition 5.7 in [3], which implies critical well-
posedness results by standard arguments, cf. [3, Section 5], [5, Sec-
tion 4], and the references therein: Define appropriate iteration spaces
that go back to Herr–Tataru–Tzvetkov [5, Definition 2.6–2.7], in which
STRICHARTZ ESTIMATES ON IRRATIONAL TORI 3
one may control the Duhamel term, cf. [5, Proposition 4.7] and [3,
Proposition 5.6]. Finally, a fixed-point argument proves local well-
posedness [5, Proof of Theorem 1.1]. Global well-posedness for small
data follows essentially from the conservation laws (5), (6), see e.g. [5,
Proof of Theorem 1.2]. Hence our results lead to:
Theorem 1.1. Let s be defined by (4) and let d,k ∈ N satisfy (7).
c
Then, for all s ≥ s the initial value problem (1) is locally well-posed in
c
Hs(Td), and globally well-posed in Hs(Td), if the initial value is small
α α
in Hsc(Td).
α
A moreprecise formulationof the theorem may be foundfor instance
in [5, Theorem 1.1 and Theorem 1.2].
The paper is organized as follows: In Section 2 we introduce some
basic notation that will be used later on. The two and three dimen-
sionalcases areconsidered inSection3 andSection4, respectively. The
paper is not self-contained. We rely on results from [1, 2, 3, 4].
2. Notation
The following notations are quite standard, see e.g. [5]. We will write
A . B, if there exists a harmless constant c > 0 such that A ≤ cB.
Define the spatial Fourier coefficients
f(n) := e−2πix·nf(x)dx, n ∈ Zd.
Z
[0,1]d
b
We fix a non-negative, even function ψ ∈ C∞((−2,2)) with ψ(s) = 1
0
for |s| ≤ 1 to define a partition of unity: for a dyadic number N ≥ 1,
we set
|ξ| 2|ξ|
ψ (ξ) := ψ −ψ for N ≥ 2, ψ (ξ) := ψ(|ξ|). (8)
N 1
(cid:18)N (cid:19) (cid:18) N (cid:19)
We also define the frequency localization operators P : L2(Td) →
N
L2(Td) as the Fourier multiplier with symbol ψ . Moreover, we de-
N
fine P := P .
≤N 1≤M≤N M
More genePrally, given a set S ⊆ Zd, we define P to be the Fourier
S
multiplieroperatorwithsymbolχ ,whereχ denotesthecharacteristic
S S
function of S.
For N,M ≥ 1 we define the collection of rectangular sets
R := C ⊆ Zd : ∃z ∈ Zd, O orthogonal d×d-Matrix s.t.
N,M
(cid:8) OC +z ⊆ [−N,N]d−1 ×[−M,M] .
Furthermore, we set C := R . (cid:9)
N N,N
4 N. STRUNK
For s ∈ R, we define the Sobolev space Hs(Td) = (1−∆)−2sL2(Td)
endowed with the norm
kfk2Hs(Td) = hni2s|f(n)|2L2(Td), where hxi = (1+|x|2)12.
nX∈Zd
b
We denote the linear Schrödinger evolution by
(eit∆f)(x) = f(n)e2πi(n·x+Q(n)t).
nX∈Zd
b
3. The two dimensional case
The following lemma may be seen as a variant of the Hausdorff–Young
inequality and was used by several authors, e.g. [2, 3]. This lemma is
one of our most important ingredients, hence, we are going to prove it.
Lemma 3.1. Let p ≥ 2 and r ≥ 1. There exists a compact interval
I ⊆ R such that for all bounded S ⊆ Zd, f: S → R, and S defined by
k
S := n ∈ S : |f(n)−k| ≤ r , k ∈ Z,
k
(cid:8) (cid:9)
the following estimate holds true
k#Skkℓpk . (cid:13)(cid:13)Xn∈Se2πif(n)t(cid:13)(cid:13)Ltp−p1(I).
(cid:13) (cid:13)
(cid:13) (cid:13)
Proof. The proof may be found in [2, (1.1.8’)–(1.1.9)]. However, we
are going to spell out some details. Let η: R → R be a continuous,
compactly supported function with η(τ) ≥ 0 for all τ ∈ R and η(τ) ≥ 1
for all τ ∈ [−r,r], e.g. for c > 0 large enough
b b
η(t) = cχ ∗χ (t),
[−r,r] [−r,r]
with Fourier transform η(τ) = c sin(2πrτ) 2. Define ψ: R → R by
τ
(cid:0) (cid:1)
ψb(t) = e2πif(n)tη(t),
X
n∈S
and set I := suppψ. Then, the Hausdorff–Young inequality implies
k#Skkℓpk ≤ (cid:13)(cid:13)Xn∈S η(cid:0)k −f(n)(cid:1)(cid:13)(cid:13)ℓpk = (cid:13)ψ(k)(cid:13)ℓpk . (cid:13)(cid:13)Xn∈Se2πif(n)t(cid:13)(cid:13)Ltp−p1(I).
(cid:13) b (cid:13) (cid:13)b (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13) (cid:3)
In several applications it turned out to bebeneficial to use almost or-
thogonality in time. This was first observed by Herr–Tataru–Tzvetkov
[5, Proof of Proposition 3.5] for the rational torus T3, and later also
applied for Zoll manifolds such as S3 [4, Proof of Proposition 3.6]. The
next lemma shows what this means in our setting.
STRICHARTZ ESTIMATES ON IRRATIONAL TORI 5
Lemma 3.2. Let σ > 0 and τ ⊂ R be a bounded interval. Further-
0
more, let τ ⊃ τ be an open interval. Then, for all φ ,...,φ ∈
1 0 1 2k+1
L2(Td) and dyadic numbers N ≥ ... ≥ N ≥ 1 there exist rectan-
1 2k+1
gles R ∈ R , where M = max N22,1 , such that P = P P
ℓ N2,M N1 N1 ℓ Rℓ N1
and (cid:8) (cid:9) P
2k+1 2 2k+1 2
P eit∆φ . P P eit∆φ P eit∆φ
(cid:13) Nj j(cid:13) (cid:13) Rℓ N1 1 Nj j(cid:13)
(cid:13)Yj=1 (cid:13)L2(τ0×Td) Xℓ (cid:13) Yj=2 (cid:13)L2(τ1×Td)
(cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) 2k+1 (cid:13)
+N−σ kP φ k2 .
2 Nj j L2(Td)
Yj=1
Proof. Thanks to spatial orthogonality, we may replace P eit∆φ by
N1 1
P eit∆φ , where C ∈ C . Without loss of generality, we may assume
C 1 N2
N ≫ N . Asintheproofof[5,Proposition3.5],wedefinethefollowing
1 2
partition: Let ξ ∈ Zd be the center of C. We define almost disjoint
0
N2
strips of width M = max 2,1 , which are orthogonal to ξ :
N1 0
(cid:8) (cid:9)
N
R := ξ ∈ C : ξ·ξ ∈ |ξ |Mℓ,|ξ |M(ℓ+1) ∈ R , N ∋ ℓ ≈ 1.
ℓ 0 0 0 N2,M M
(cid:8) (cid:2) (cid:1)(cid:9)
Since C = R , we have P P eit∆φ = P P eit∆φ .
ℓ ℓ C N1 1 ℓ Rℓ N1 1
Let η ∈ SC0∞(τ1) be a cutoff function satiPsfying η(t) = 1 for all t ∈ τ0.
Obviously,
2k+1 2 2k+1 2
P eit∆φ ≤ η(t) P eit∆φ . I +I ,
(cid:13) Nj j(cid:13) (cid:13) Nj j(cid:13) 1 2
(cid:13)Yj=1 (cid:13)L2(τ0×Td) (cid:13) Yj=1 (cid:13)L2(R×Td)
(cid:13) (cid:13) (cid:13) (cid:13)
wh(cid:13)ere (cid:13) (cid:13) (cid:13)
2k+1 2
I := P P eit∆φ P eit∆φ ,
1 (cid:13) Rℓ N1 1 Nj j(cid:13)
ℓ≈XN1/M(cid:13)(cid:13) Yj=2 (cid:13)(cid:13)L2(τ1×Td)
(cid:13) (cid:13)
and
2k+1
I := η(t)2P P eit∆φ P eit∆φ ,
2 (cid:28) Rℓ N1 1 Nj j
ℓ,ℓ′≈XN1/M: Yj=2
|ℓ−ℓ′|≫1
2k+1
P P eit∆φ P eit∆φ .
Rℓ′ N1 1 Nj j(cid:29)
Yj=2 L2(R×Td)
We have to show that I . N−σ 2k+1kP φ k2 . Interpreting the
2 2 j=1 Nj j L2(Td)
integration with respect to t as FQourier transform on R and taking the
6 N. STRUNK
absolute value, we get
2k+1
|I | . η2 Q(n′)−Q(n )
2 (cid:18) j j (cid:19)
ℓ,|ℓℓ′−≈XℓN′|1≫/M1:nj,nn′j1∈∈BRNXℓj,,nj′1=∈2R..ℓ.2′,k+1(cid:12)(cid:12)b(cid:12)(cid:12) Xj=1
2k+1
× φ (n ) φ (n′) ,
j j j j
Yj=1(cid:12) (cid:12)(cid:12) (cid:12)
(cid:12)b (cid:12)(cid:12)b (cid:12)
where B := suppψ with ψ defined in (8). Similar to the proof of
N N N
[5, Proposition 3.5] we get
2k+1
Q(n′)−Q(n ) = M2|ℓ−ℓ′|(ℓ+ℓ′)+O(M2ℓ)+O(N2) & N2hℓ−ℓ′i,
(cid:12) j j (cid:12) 2 2
(cid:12)Xj=1 (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
since ℓ,ℓ′ ≈ N1 and |ℓ−ℓ′| ≫ 1. Thus, for any β > 0, we may estimate
M
2k+1
η2 Q(n′)−Q(n ) . N−2βhℓ−ℓ′i−β.
(cid:18) j j (cid:19) β 2
(cid:12) (cid:12) Xj=1
(cid:12)b(cid:12)
Using Cauchy-Schwarz with respect to n , n′, j = 1,...,2k+1, yields
j j
2k+1
|I | . N−σ hℓ−ℓ′i−βkP φ k kP φ k kP φ k2 ,
2 2 Rℓ j L2 Rℓ′ j L2 Nj j L2
ℓ,ℓ′≈XN1/M: Yj=2
|ℓ−ℓ′|≫1
where σ = 2β − (2k + 1)d and L2 = L2(Td). Finally, Schur’s lemma
implies
hℓ−ℓ′i−βkP φ k kP φ k . kP φ k2 ,
Rℓ j L2 Rℓ′ j L2 N1 1 L2(Td)
ℓ,ℓ′≈XN1/M:
|ℓ−ℓ′|≫1
provided β > 1. (cid:3)
In the following let τ ⊆ [0,1] be any time interval and k ≥ 3. We
0
are going to prove the following proposition:
STRICHARTZ ESTIMATES ON IRRATIONAL TORI 7
Proposition 3.3. There exists δ > 0 such that for all φ ,...,φ ∈
1 k+1
L2(T2) and dyadic numbers N ≥ ... ≥ N ≥ 1 the following esti-
1 k+1
mate holds true
k+1
P eit∆φ
(cid:13) Nj j(cid:13)
(cid:13)Yj=1 (cid:13)L2(τ0×T2)
(cid:13) (cid:13)
(cid:13) (cid:13) δ k+1
N 1
. k+1 + kP φ k NsckP φ k .
(cid:18) N N (cid:19) N1 1 L2(T2) j Nj j L2(T2)
1 2 Yj=2
Before we start proving Proposition 3.3, we turn to the following
trilinear Strichartz estimate, which improves [3, Lemma 5.9] for d = 2,
using ideas of [2]. Combined with Lemma 3.2, this is the essence of the
proof of Proposition 3.3. Note that the implicit constants depend on
C in (2) and the local time interval τ .
0
Lemma 3.4. Let 2 < p ≤ 4. Then, for all N,M ≥ 1 with N ≥ M,
C ∈ C , C ,C ∈ C , and φ ,φ ,φ ∈ L2(T2) we have
1 N 2 3 M 1 2 3
3 3
(cid:13) PCjeit∆φj(cid:13) . M2−2p kPCjφkL2(T2).
(cid:13)Yj=1 (cid:13)Lp(τ0,L2(T2)) Yj=1
(cid:13) (cid:13)
(cid:13) (cid:13)
Proof. Thisproofisatrilinearvariantofthepoofof[2,Proposition1.1].
Hence, we omit some details and refer the reader also to e.g. [3, Proof
of Proposition 2.2]. We will write LpLq := Lp(τ ,Lq(T2)) and Lp :=
t x 0 t
Lp(τ ) for brevity.
0
The left hand side may be estimated by
2 1
2
φ (a−n−m)φ (n)φ (m)e2πi(Q(a−n−m)+Q(n)+Q(m))t
1 2 3
(cid:20) (cid:13) (cid:13) (cid:21)
aX∈Z2(cid:13)nX∈C2, (cid:13)Lpt
(cid:13)(cid:13)m∈C3 b b b (cid:13)(cid:13)
using Plancherel’s identity with respect to x and Minkowski’s inequal-
ity. Applying Hausdorff–Young (cf. [3, Lemma 2.1]) and setting c :=
j,n
|φ (n)| yields
j
b p p−1
p−1 p
k···kLpt . (cid:20) (cid:12) c1,a−n−mc2,nc3,m(cid:12) (cid:21) .
Xk∈Z(cid:12)|Q(a−n−m)+QX(n)+Q(m)−k|≤1 (cid:12)
(cid:12) 2 (cid:12)
(cid:12) (cid:12)
One easily verifies that |Q(a−n−m)+Q(n)+Q(m)−k| ≤ 1 may be
2
written as
|Q(3n−2a)+3Q(m)+2Q(a)−6k| ≤ 3,
e e
8 N. STRUNK
where n := n+m and m := n−m. Hence,
# (n,m) ∈ C ×C : |Q(a−n−m)+Q(n)+Q(m)−k| ≤ 1 . #S ,
e 2 3 e 2 ℓ
(cid:8) (cid:9)
with
S := (n,m) ∈ C ×C : |Q(n)+3Q(m)−ℓ| ≤ 4 ,
ℓ 2 3
ℓ := ⌊6k −2Q(a(cid:8))⌋e∈eZ, aned cubees C2,eC3 ∈ C3Me. This conc(cid:9)lusion and
two applications of Hölder’s estimate yield (cf. [2, (1.1.5)–(1.1.7)])
e e
p−2 1
k···kLpt . (cid:18) (#Sℓ)p−p2(cid:19) 2p (cid:18) c21,a−n−mc22,nc23,m(cid:19)2,
Xℓ∈Z nX∈C2,
m∈C3
thus we arrive at
3 p−2 3
(cid:13) PCjeit∆φj(cid:13) . (cid:18) (#Sℓ)p−p2(cid:19) 2p kPCjφjkL2(T2).
(cid:13)Yj=1 (cid:13)Lp(τ0,L2(T2)) Xℓ∈Z Yj=1
(cid:13) (cid:13)
Th(cid:13)e assumption(cid:13)p ≤ 4 ensures that p ≥ 2 and hence, by Lemma 3.1
p−2
we may estimate
p−2 2
(cid:18) (#Sℓ)p−p2(cid:19) p . (cid:13) e2πiαj(ne2j+3me2j)t(cid:13) p
Xℓ∈Z (cid:13)(cid:13)Yj=1∆nej,X∆mej≈M (cid:13)(cid:13)Lt2(I)
(cid:13) 4 (cid:13)
. e2πin2t . M4(1−1p)
(cid:13) (cid:13)
(cid:13)∆Xn≈M (cid:13)L2tp(I)
(cid:13) (cid:13)
for some compact interval I ⊆(cid:13)R, provided(cid:13)p > 2. Here, we use the no-
tation = a+cN for some a ∈ Z and harmless c ∈ N. The last
∆n≈N n=a
inequalPity goes baPck to Bourgain [1, Proposition 1.10 and Section 4].
However, we need a slight variation here, which may be found in [4,
Lemma 3.1]. This implies the desired estimate. (cid:3)
Corollary 3.5. Let p > 6.
(i) For every N ≥ 1, C ∈ C and φ ∈ L2(T2) we have
N
kPCeit∆φkLp(τ0,L6(T2)) . N32−p2kPCφkL2(T2).
(ii) Let 6 ≤ q < p. Then, for all N,M ≥ 1 with N ≥ M, R ∈
R and φ ∈ L2(T2) it holds that
N,M
kPReit∆φkLp(τ0,Lq(T2)) . N21+1q−p2M12−q3kPRφkL2(T2).
Proof. The first estimate is a direct consequence of Lemma 3.4, pro-
vided p ≤ 12. Bernstein’s inequality implies the result for p = ∞:
kPCeit∆φkL∞(τ0×T2) . NkPCφkL2(T2).
STRICHARTZ ESTIMATES ON IRRATIONAL TORI 9
For 12 < p < ∞, the desired estimate follows from Hölder’s estimate
and the estimates for p = 12 and p = ∞.
The second statement follows from (i), the estimate
kPReit∆φkL∞(τ0×T2) ≤ #(R∩Z2) 21kPRφkL2(T2) . (NM)12kPRφkL2(T2),
(cid:0) (cid:1)
which may easily be obtained by applying Cauchy-Schwarz in Fourier
space, and Hölder’s inequality. The conclusion works as follows: Set
f(t,x) := P eit∆φ(x), ε = 6p −6 > 0 and ϑ = 6 ≤ 1. Then,
R q q
k|f|ϑ|f|1−ϑkLptLqx ≤ kfkϑL6t+εL6xkfk1L−∞t,ϑx . N21+1q−p2M12−3qkPRφkL2(T2).
(cid:3)
Proof of Proposition 3.3. Thanks to Lemma 3.2 it suffices to replace
P eit∆φ by P P eit∆φ , where R ∈ R with M = max N22,1 ,
N1 1 R N1 1 N2,M N1
provided we magnify the time interval to an open interval τ ⊃(cid:8)τ . (cid:9)
1 0
Let 6 < p ,q < 8 and 3 < p ≤ 24. Furthermore, let p and q
1 1 2 5 3 2
be defined via the relations 1 = 1 + 1 + k−2 and 1 = 1 + 1 + k−2,
2 p1 p2 p3 2 q1 3 q2
respectively. By Hölder’s estimate the following holds true:
k+1
(cid:13)PRPN1eit∆φ1 PNjeit∆φj(cid:13) ≤ kPRPN1eit∆φ1kLpt1Lqx1
(cid:13) Yj=2 (cid:13)L2t,x
(cid:13) (cid:13)
(cid:13) (cid:13) k+1
×kPN2eit∆φ2PN3eit∆φ3kLpt2L3x kPNjeit∆φjkLpt3Lqx2,
Yj=4
where LrLs := Lr(τ ,Ls(T2)) and L2 := L2L2. Let f := P eit∆φ ,
t x 1 t,x t x j Nj j
j = 2,3, then we treat the bilinear term as follows:
kf2f3k2Lpt2L3x = kf22f32kLtp22Lx32 ≤ kf2f32kLrtL2xkf2kLstL6x,
where s > 6 and 2 = 1 + 1. Note that p ≤ 24 ensures that r ≤ 4. By
p2 r s 2 5
Lemma 3.4 and Corollary 3.5, we have for all η > 0
kPN2eit∆φ2PN3eit∆φ3kLpt2L3x ≤ N261+ηN367−p22−η. (9)
Corollary 3.5, (9) and Sobolev’s embedding imply
P P eit∆φ k+1P eit∆φ . M 12−q31N76−p21−q21+ηN67−p22−η
(cid:13) R N1 1 Nj j(cid:13) N 2 3
(cid:13) Yj=2 (cid:13)L2t,x (cid:16) 2(cid:17)
(cid:13) (cid:13)
(cid:13) k+1 (cid:13) k+1
1− 2 (2− 1 − 1 − 1 )
× N k−2 3 p1 p2 q1 kP P φ k kP φ k ,
j R N1 j L2x Nj j L2x
Yj=4 Yj=2
10 N. STRUNK
where L2 := L2(T2). For all 0 < ν ,ν ≪ 1, there exist δ > 0 and
x 1 2
p ,q > 6 sufficiently close to 6 as well as p > 3 sufficiently close to 3
1 1 2
such that
(i) M 21−q31 = M δ (ii) N67−p21−q21+η = N21+ν1+η
N N 2 2
(cid:16) 2(cid:17) (cid:16) 2(cid:17)
(iii) N67−p22−η = N21+ν2−η (iv) N1−k−22(23−p11−p12−q11) = N1−ν1k+−ν22,
3 3 j j
where j ∈ {4,...,k +1}. Since 1 < s < 1 for k ≥ 3, we may choose
2 c
0 < ν ,ν ,η ≪ 1 small enough to get
1 2
k+1
P P eit∆φ P eit∆φ
(cid:13) R N1 1 Nj j(cid:13)
(cid:13) Yj=2 (cid:13)L2t,x
(cid:13) (cid:13)
(cid:13) (cid:13) k+1
N 1 δ
. k+1 + kP P φ k NsckP φ k .
N N R N1 1 L2x j Nj j L2x
(cid:16) 1 2(cid:17) Yj=2
(cid:3)
4. The three dimensional case
Similar to the previous section, let τ ⊆ [0,1] be any time interval.
0
Proposition 4.1. For all ε > 0, there exists δ > 0 such that for
all φ ,φ ,φ ∈ L2(T3) and dyadic numbers N ≥ N ≥ N ≥ 1 the
1 2 3 1 2 3
following estimate holds true:
3 δ 3
P eit∆φ . N3 + 1 N34+εN45−ε kP φ k .
(cid:13) Nj j(cid:13) (cid:18)N N (cid:19) 2 3 Nj j L2(T3)
(cid:13)Yj=1 (cid:13)L2(τ0×T3) 1 2 Yj=1
(cid:13) (cid:13)
(cid:13) (cid:13)
We postpone the proof and recall a linear Strichartz estimate, which
is — up to almost orthogonality — the main ingredient for the trilinear
Strichartz estimate in Proposition 4.1. This linear estimate goes back
to Bourgain [2]. Again, note that the implicit constants depend on C
in (2) and the local time interval τ .
0
Lemma 4.2. Let p > 16.
3
(i) For every N ≥ 1, C ∈ C and φ ∈ L2(T3) we have
N
kPCeit∆φkLp(τ0,L4(T3)) . N34−p2kPCφkL2(T3).
(ii) Let 4 ≤ q < 3p. Then, for all N,M ≥ 1 with N ≥ M,
4
R ∈ R and all φ ∈ L2(T3) it holds that
N,M
kPReit∆φkLp(τ0,Lq(T3)) . N1−p2−1qM21−2qkPRφkL2(T3).