DECOUPLING INEQUALITIES
IN HARMONIC ANALYSIS AND APPLICATIONS
1
NOTATION AND STATEMENT
S compact smooth hypersurface in Rn with positive definite second
fundamental form
Examples: sphere S d−1 , truncated paraboloid
P d−1 = {(ξ, |ξ|2) ∈
Rd−1 × R; |ξ| ≤ 1}
Sδ = δ-neighborhood of S
S
√
√
Sδ ⊂ τ where τ is a δ × · · · × δ × δ rectangular box in Rd
τ
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δ
τ
For f ∈ Lp(Rd), fτ = (fˆ|τ )ˇ is the Fourier restriction of f to τ
THEOREM (B-Demeter, 014)
Assume supp fˆ ⊂ Sδ . Then
kf kp ε δ −ε
!1
X
τ
2(d + 1)
2
2
kfτ kp
for 2 ≤ p ≤
d−1
3
2d obtained earlier (B, 013)
• Range p ≤ d−1
• Exponent p = 2(d+1)
d−1 is best possible
EXAMPLE f = 1d
Sδ . Then |f | ∼
|fτ | ∼
d+1
δ 2 1◦
X
τ
τ
d−1
(1+|x|) 2
◦
and kf kp ∼ δ
with τ the polar of τ ⇒ kfτ kp ∼
1
2 2
kfτ kp
δ
d+1
d+1 1 2p
δ 2
δ
d+3
1 d−1
4
∼
δ 4 =δ
δ
4
=
d+3
δ 4
• Original motivations
PDE Smoothing for the wave equation (T.Wolff, 2000)
Schrödinger equation on tori and irrational tori (B, 92→)
Spectral Theory Eigenfunction bounds for spheres
• Diophantine applications, new approach to mean value
theorems in Number Theory, bounds on exponential sums
5
MOMENT INEQUALITIES
FOR TRIGONOMETRIC POLYNOMIALS WITH SPECTRUM
IN CURVED HYPERSURFACES
S ⊂ Rd smooth with positive curvature
λ 1, E = Zd ∩ λS = lattice points on λS
Let
φ(x) =
X
aξ e2πix.ξ ⇒ φ is 1-periodic
ξ∈E
Apply decoupling theorem to rescaled f (y) =
P
ξ∈E
2πiy. λξ
aξ e
with δ = λ−2
⇒ kφkLp([0,1]d) λε
X
ξ∈E
1
2(d + 1)
2
2
for p ≤
|aξ |
d−1
−2
η(λ
y)
COROLLARY (S = S d−1)
Eigenfunctions φE of d-torus Td , i.e. −∆φE = EφE ,
satisfy
2(d + 1)
ε
kφkp E kφk2 for p ≤
d−1
(Known for d = 3 for arithmetical reason, new for d > 3)
7
SCHRODINGER EQUATIONS AND STRICHARTZ INEQUALITIES
Linear Schrödinger equation on Rd
iut + ∆u = 0, u(0) = φ ∈ L2(Rd)
u(t) = eit∆φ =
Z
2)
i(x.ξ+t|ξ|
φ̂(ξ)e
dξ
STRICHARTZ INEQUALITY: keit∆φkLp ≤ Ckφk2 with p = 2(d+2)
d
x,t
Local wellposedness fo Cauchy problem for NLS on Rd

iu + ∆u + αu|u|p−2 = 0
t
u(0) = φ ∈ H s(Rd)
s ≥ s0 with s0 defined by p − 2 =
4
d − 2s0
(p − 2 ≥ [s] if p 6∈ 2Z)
8
TORI AND IRRATIONAL TORI
φ∈
L2(Td)
φ(x) =
X
aξ e2πix.ξ
ξ∈Zd
eit∆φ
=
X
aξ
2)
2πi(x.ξ+t|ξ|
e
ξ∈Zd
More generally (irrational tori)
eit∆φ =
X
aξ e2πi(x.ξ+tQ(ξ)) where Q(ξ) = α1ξ12+· · ·+αdξd2, αj > 0
ξ∈Zd
PROBLEM
Which Lp -inequalities are satisfied?
(periodic Strichartz inequalities)
9
EXAMPLES (B, 92)
•d=1
P
Assume φ(x) = ξ∈Z,|ξ|≤R aξ e2πixξ
keit∆φkL6
[|x|,|t|≤1]
Rεkφk2
Moreover, taking aξ = √1 for |ξ| ≤ R, aξ = 0 for |ξ| > R,
R
keit∆φk6 ∼ (log R)1/6
(Classical Strichartz inequality fails in periodic case!)
•d = 2
keit∆φk4 Rεkφk2
Proven using arithmetical techniques (# lattice points on ellipses)
•d ≥ 3
Only partial results
Irrational tori
No satisfactory results for d ≥ 2
10
THEOREM Let Q be a positive definite quadratic form on Rd .
Then
2πi(x.ξ+tQ(ξ))
aξ e
p
L[|x|,|t|≤1]
ξ∈Zd,|ξ|≤R
X
Rε
X
|aξ |
1
2 2
.
for p ≤ 2(d+2)
d
COROLLARY Local wellposedness of NLS on Td for s > s0
Recent results of Killip-Visan
11
Proof
Define f (x0, t0) =
ξ
+t0 Q(ξ)
2πi(x0 . R
2 )
P
|ξ|≤R
ξ Q(ξ)
,
∈
2
R R
aξ e
R
S ={
−2
−2
η (R x0, R t0) .
y, Q(y) ; y ∈ Rd, |y| ≤ 1} ⊂ Rd+1
From decoupling inequality
kf kLp(B
)
2
R
2(d+1) X
Rε R p
|a
ξ|
2
1/2
and setting x0 = Rx, t0 = R2t
d+2 X
R p a
2πi(x.ξ+tQ(ξ)) kf kLp(B 2 ) =
e
ξ
p
R
L|x|<R,|t|<1
2(d+1) X
2πi(x.ξ+tQ(ξ))
p
=R
aξ e
p
L|x|<1,|t|<1
12
MOMENT INEQUALITIES FOR EIGENFUNCTIONS
(M, g) compact, smooth Riemannian manifold of dimension d without
boundary
(E = λ2)
∆ϕE = −EϕE
THEOREM (Hormander, Sogge)
kϕE kp ≤ cλδ(p)kϕE k2
where


1
1
d
−
1


−


 2
2 p
δ(p) =


1
1
1



−
−
d

2
p
Estimates are sharp for M = S d
2
2(d + 1)
d−1
if
2≤p≤
if
2(d + 1)
≤p≤∞
d−1
13
FLAT TORUS
Td
CONJECTURE
kϕE kp ≤ cpkϕE k2
2d
if p <
d−2
and
d)
−
( d−2
2
p
kϕE kp ≤ cpλ
kϕE k2
2d
if p >
d−2
14
THEOREM
(B–Demeter, 014)
kϕE kp λεkϕ
2(d + 1)
E k2 if d ≥ 3 and p ≤
d−1
(∗)
2(d − 1)
d−3
(∗∗)
and
d
( d−2
2 −p)
kϕE kp ≤ Cpλ
kϕE k2 if d ≥ 4 and p >
(∗∗) follows from (∗) combined with distributional inequality
2 d−1
d−3
t
|[ϕ > t]| 2 +ε
λ d−3
for t >
d−1
λ 4
(B, 093)
(Hardy-Littlewood + Kloosterman)
15
ESTIMATES ON
2
T
E = sum of 2 squares
multiplicity E = number N of representations of E as sum of
two squares
E=
Y e
p α
α
⇒N =4
Y
(1 + eα)
On average, N ∼ (log E)1/2
THEOREM (Zygmund–Cook) kϕE k4 ≤ CkϕE k2
• P2
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P1 •
•
16
Estimating the L6 -norm ⇔
number of solutions of P1 + P2 + P3 = P4 + P5 + P6 with
Pi ∈ E = {P ∈ Z + iZ; |P |2 = E}
COMBINATORIAL APPROACH
ANALYTICAL APPROACH:
Use of incidence geometry
Use of the theory of elliptic curves
(conditional)
17
THEOREM (Bombieri–B, 012)
• kϕE k6 1 +ε
N 12 kϕ
E k2
• For all p < ∞, kϕE kp ≤ CkϕE k2 for ‘most’ E
• Conditional to GRH and Birch, Swinnerton-Dyer conjecture
kϕE k6 N εkϕE k2 for ‘most’ smooth numbers E
18
THE COMBINATORIAL APPROACH
Consider equation P1 + P2 + P3 = (A, B) with Pj = (xj , yj ) ∈ E
Set u = x1 + x2, v = y1 + y2 establishing correspondence between
E × E and P = E + E
P
• 2
P1
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•
•
Introduce curves CA,B : (A − u)2 + (B − v)2 = E
√
⇒ family C of circles of same radius E (pseudo-line system)
Need to bound
X
A,B
|CA,B ∩ P|2
USE OF SZEMEREDI-TROTTER THEOREM
FOR PSEUDO-LINE SYSTEMS
I(P, C) = number of incidences between point P and curves C
≤ C(|P|2/3|C|2/3 + |P| + |C|)
⇒ bound N 7/2
E = 65, |P| = 112, |C| = 372
REMARK Erdös unit distance conjecture would imply N 3+ε
PROBLEM Establish uniform Lp -bound for some p > 2
for higher dimensional toral eigenfunctions
Important to PDE’s
Control for Schrodinger operators on tori
THEOREM (B-Burq-Zworski, 012)
Let d ≤ 3, V ∈ L∞(Td), Ω ⊂ Td, Ω 6= φ, an open set. Let
T > 0. There is a constant C = C(Ω, T, V ) such that for
any u0 ∈ L2(Td)
ku0 kL2(Td) ≤ Ckeit(∆+V )u0 kL2(Ω×[0,T ])
(uses Zygmund’s inequality)
More regular V : Anantharaman-Macia (011)
INGREDIENTS IN THE PROOF OF DECOUPLING THEOREM
• Wave packet decomposition
• Parabolic rescaling
• Multilinear harmonic analysis
d=2
bilinear square function inequalities
d≥3
Bennett, Carbery, Tao
(Cordoba 70’s)
(06)
• Multi-scale bootstrap
22
BENNETT - CARBERRY - TAO THEOREM
S ⊂ Rd compact smooth hypersurface
S1, . . . , Sd ⊂ S disjoint patches in transversal position
|ν(ξ1) ∧ · · · ∧ ν(ξd)| > c for all ξ1 ∈ S1, . . . , ξd ∈ Sd
...
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ν2
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ν1
S2
S
S1
Let µ1, . . . , µd be measures on S , µj σ = surface measure and
suppµj ⊂ Sj
dµj = ϕj dσ, ϕj ∈ L2(σ)
Take large ball BR ⊂ Rd . Then
d
d
Y
1
Y
2d
ε
d
|µ̂j (x)| R
kϕj kL2(σ) with p =
p
d−1
L (BR )
j=1
j=1
MEAN VALUE THEOREMS AND DIOPHANTINE RESULTS
(FIRST APPLICATIONS)
(1) Around Hua’s inequality
6
Z 1Z 1 X
3
e(nx+n y) dxdy = I6(N ) < CN 3(log N )c
0 0 1≤n≤N
(Hua, 1947)
Arithmetically, I6(N ) is the number of integral points n1, . . . , n6 ≤ N on
the Segre cubic

x + x + x = x + x + x
1
2
3
4
5
6
x3 + x3 + x3 = x3 + x3 + x3
1
2
3
4
5
6
Vaughan-Wooley (95)
De La Bretèche (07)
I6(N ) = 6N 3+U (N ), U (N ) = O N 2(log N )5
Precise asymptotic for U (N )
Applications of decoupling inequality to S = {(t, t3), t ∼ 1} ⊂ R2
6
n + n 3 6
x
y dxdy N 3+ε
e
N
N
n∼N
X
2
B(N )
Z
By change of variables and periodicity
Z 1Z 1 X
3 6
n3 6
n
N 3+ε
e nx+ y =
e
nN
x+
y
0 0 n∼N
0 0 n∼N
N
N
Z 1Z 1 X
which means that


n
1 + n2 + n3 − n4 − n5 − n6 = 0

|n3 + n3 + n3 − n3 − n3 − n3| < N
1
2
3
4
5
6
has at most O(N 3+ε) solutions with ni ∼ N
25
n3 6
e nx +
y dxdy N 3+ε is optimal
0 0 n∼N
N
Z 1Z 1 X
CONJECTURE
8
Z 1Z 1 X 3
I8(N ) =
e nx + n y dxdy N 4+ε
0 0 n≤N
KNOWN
Hua (1947)
I10(N ) N 6+ε
(optimal)
Wooley (014)
I9(N ) ≤ N 5+ε
(optimal)
I8(n) 13 +ε
N9
Consequence of (optimal) Vinogradov mean value theorem
Z 1Z 1Z 1X
0
0
0
e(nx + n2y +
12
3
n z) dxdydz
N 6+ε
(2) A MEAN VALUE THEOREM OF ROBERT AND SARGOS
THEOREM (Robert-Sargos, 2000)
4 6
n
e n2x + 3 y dxdy N 3+ε
0 0 n∼N
N
Z 1Z 1 X
Proof based on Poisson summation (Bombieri-Iwaniec method)
Applications to exponential sums and Weyl’s inequality
THEOREM (B-D, 014)
6
k
n
2
e n x + k−1 y dxdy N 3+ε
0 0 n∼N
N
Z 1Z 1 X
PROOF
Apply DCT with δ = N
1
−2
to
2
k 6
X
n
n
6 |x|<N 2 e
x+
y dxdy
N
N
|y|<N n∼N
Z
(∗)
This gives a bound
6
XZ 1Z 1 X
1 3
2 x + N −k+1 nk y) dxdy 3
Nε
e(n
0
0
I
n∈I
h
i
√
N
where {I} is a partition of 2 , N in intervals of size N
If I = [`, ` +
√
N ], n = ` + m, m <
√
N
X
X
e(n2x+N −k+1nk y) ∼ e m(2`x+kN −k+1`k−1y)+m2x √
n∈I
m< N
and 6th moment bounded by
3 +ε
2
N
⇒ (∗) N ε
√
N
3 1 3
N2 3
= N 3+ε
DECOUPLING FOR CURVES IN HIGHER DIMENSION
Γ ⊂ Rd non-degenerate smooth curve, i.e.
φ0(t) ∧ · · · ∧ φ(d)(t) 6= 0
with φ : [0, 1] → Γ a parametrization
Let Γ1, . . . , Γd−1 ⊂ Γ be separated arcs and (Γj )δ a δ-neighborhood of Γj
Assume supp fˆj ⊂ (Γj )δ
Apply a version of the hypersurface DCT to
S = Γ1 + · · · + Γd−1
(Note that second fundamental form may be indefinite)
THEOREM (∗) (B-D)
d−1
1 Y
d−1
|fj |
2(d+1)
L#
(BN )
j=1
where p =
2(d+1)
d−1
d−1
1
Y X
+ε
N 2(d+1)
kf
j=1
and {τ } a partition of Γδ in size
1
δ 2 -tubes,
N = δ −1
THEOREM (∗∗) (B)
Let d be even
d/2
Y
2/d
|fj | 3d
L# (BN )
j=1
1
p
2(d+1)
k
p
j,τ L (B )
N
#
1
1 +ε d/2
Y X
3d
N6
kfj,τ k6
6
L#(BN )
j=1
MEAN VALUE THEOREMS
h
i
Let d ≥ 3 and ϕ3, . . . , ϕd : 0, 1 → R be real analytic such that
W
I1, . . . , Id−1 ⊂
h
N,N
2
i
ϕ000, . . . , ϕ000
3
d
6= 0
∩ Z intervals that are ∼ N separated
THEOREM
d−1
k
k 1 k
Y X
d−1 x3 + N ϕ 4
x4 + · · · + N ϕ d
xd e kx1 + k2 x2 + N ϕ3
L2(d+1) ([0,1]d )
N
N
N
j=1 k∈I
j
1
N 2 +ε
31
COROLLARY (Bombieri-Iwaniec, 86)
Assume ϕ : [0, 1] → R real analytic, ϕ000 > 0
Then
8
Z 1Z 1Z 1 X
k
2
dx dx dx N 4+ε
x
e(kx
+
k
x
+
N
ϕ
3
1 2 3
1
2
N
0 0 0 k∼N
1
Essential ingredient in their work on ζ 2 + it
THEOREM (Bombieri-Iwaniec, 86)
9
1
ζ
56
+ it t
for |t| → ∞
2
(later improvement by Huxley, Kolesnik, Watt.... using refinements of the
method)
32
EXPONENTIAL SUMS (Bombieri-Iwaniec method)
(Huxley Area, lattice points and Exponential Sums, LMSM 1996)
GOAL
Obtaining estimates on exponential sum
m e TF
M
m∼M
X
EXAMPLE By approximate functional equation, bounding
1 + iT ) reduces to sums
ζ( 2
m
e T log
with M < T 1/2
M
m∼M
X
33
h
M,M
2
i
STEP 1 Subdivide
in shorter intervals I of size N on which
m ) can be replaced by cubic polynomial
T F (M
⇒ cubic exponential sums of the form
X
e(a1n + a2n2 + µn3)
n≤N
with a1, a2, µ depending on I and µ is small
STEP 2 Conversion to new exponential sum by Poisson summation
(stationary phase) of the form
X
e b1h + b2h2 + b3h3/2 + b4h1/2
h≤H
with b1, b2, b3, b4 depending on I .
34
STEP 3 Understanding the distribution of b1(I), b2(I), b3(I), b4(I)
when I varies (the ‘second spacing problem’)
STEP 4 Use of the large sieve to reduce to mean value problems of the form
Z 1Z 1Z 1Z 1 X 2k
1
1
Ak (H) =
e x1h+x2h2 +x3H 2 h3/2 +x4H 2 h1/2 dx
0 0 0 0 h∼H
k = 4, 5, 6
THEOREM (B) Assume W (ϕ000, ψ 000) 6= 0 and 0 < δ < ∆ < 1. Then
Z
0
1 Z 1 Z 1 Z 1
0
0
0
1 h
1 h 10
X
2
e hx1 + h x2 + ϕ
x3 + ψ
x4 dx1 dx2 dx3 dx4 δ
H
∆ H
h∼H
[δ∆3/4 H 7 + (δ + ∆)H 6 + H 5 ]H ε
COROLLARY (Huxley-Kolesnik, 1991) A5(H) H 5+ε
32
1
32 = 0, 1561 . . .
THEOREM (Huxley, 05) |ζ( 2 + it)| |t| 205 , 205
PROBLEM (Huxley)
Obtain good bound on A6(H)
THEOREM (B, 014)
A6(H) H 6+ε
This bound follows from THEOREM (∗∗) and is optimal.
Combined with the work of Huxley on second spacing problem
COROLLARY
53 +ε 53
1
ζ
+ it |t| 342 ,
= 0, 1549 . . .
2
342
36
PROGRESS TOWARDS THE LINDELÖF HYPOTHESIS
µ(σ) = inf β > 0; |ζ(σ + it)| = 0 |t|β
Bounds on µ( 1
2)
1
4
(Lindelöf, 1908)
173
1067
1
6
(Hardy-Littlewood)
35
216
= 0, 16204
(Kolesnik, 1982)
= 0, 16201 . . .
(Kolesnik, 1985)
= 0, 16214 . . .
(Kolesnik, 1973)
163
988
= 1, 1650 . . .
(Walfisz, 1924)
139
858
17
164
= 0, 1647 . . .
(Titchmarsh, 1932)
9
56
(Phillips, 1933)
17
108
= 0, 1574 . . .
(Huxley-Kolesnik, 1990)
(Titchmarsh, 1942)
89
570
= 0, 15614 . . .
(Huxley, 2002)
(Min, 1949)
32
205
= 0, 15609 . . .
(Huxley, 2005)
229
1392
19
116
15
92
= 0, 164512 . . .
= 0, 1638 . . .
= 0, 1631 . . .
= 0, 1607 . . .
(Bombieri-Iwaniec, 1986)
SKETCH OF PROOF OF DECOUPLING IN 2D
(∗) Kp(N ) CεN γ+ε best bound in inequality
kf kLp
#(BN )
≤ Kp(N )
X
τ
kfτ k2
p
L#(BN )
1
2
1 tube
for supp fˆ ⊂ Γ 1 , τ : √1 × N
N
N
There is bilinear reduction (B-G argument)
1
1
2
k(|f | |f |) 2 kLp (B )
N
#
≤ Kp(N )
2 X
Y
i=1
τ
kfτi k2
p
L#(BN )
1
4
if supp fc1 ⊂ (Γ1) 1 , supp fc2 ⊂ (Γ2) 1 with Γ1, Γ2 ⊂ Γ separated
N
N
1 tube
By parabolic rescaling, if M ≤ N and τ 0 : √1 × M
M
N
kfτ 0 kLp (B ) ≤ Kp
N
#
M
X
τ ⊂τ 0
kfτ k2
p
L#(BN )
1
2
(∗∗) From bilinear L4 -inequality, for M ≤
√
1 tubes
N , τ 0 = √1 × M
M
2 1 Y X
4
2
≤kf 1kL2 (B )kf 2kL2 (B ) =
|fτ 0 |
4
M
M
#
#
L#(BM )
i=1
τ0
1
YX
4
kfτi 0 k2
L2
# (BM )
i
τ0
Interpolation with trivial L∞ − L∞ bound (using ware packet decomposition)
1 1
YX
Y X i 2 4
4
ε
i k2
|f
|
M
kf
p
τ0
τ 0 Lp/2 (B )
L
(B
)
M
M
#
i
i
#
τ0
τ0
κ Y X
1−κ
YX
4
4
2
ε
i
i
2
M
kfτ 0 kLp (B )
kfτ kL2 (B )
M
M
#
#
τ
i
i
τ0
1 × 1 tubes
and
τ
:
with κ = p−4
p−2
M
M2
κ Y X
1 1−κ
YX
4
4
ε
i
2
i
2
M
kfτ 0 kL2 (B )
|fτ |
p
M
#
L#(BM )
τ
0
i
i
τ
Writing BN as a union of M -balls
κ Y X
1 1 1−κ
Y X
Y X i 2 4
4
4
ε
i
2
i
2
kfτ 0 kLp (B ) N
|fτ |
|fτ 0 |
P
p
N
#
L
(B
)
L#(BN )
τ
N
i
i
i
#
τ0
τ0
−s
(∗ ∗ ∗) Iterate, considering coverings {τs}1≤s≤s̄ , τs : N −2
−s+1
× N −2
tubes
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τs
⇒ Kp(N ) N εN 2
⇒ γ ≤ 2−s̄−1+ κγ
−s̄−1
τs−1
−s̄
−s̄+1
Kp(N 1−2 )κ Kp(N 1−2
1−(1−κ)s̄
κ
− 2−s̄
1−(2(1−κ))s̄
2κ−1
)κ(1−κ) . . . Kp(1)(1−κ)
s̄
4 < 1 for p > 6
2(1 − κ) = p−2
1
κ
s̄
⇒ 0 ≤ 2 − 2κ−1 1 − (2(1 − κ)) γ
Taking s̄ large enough ⇒ γ ≤ 2κ−1
κ ⇒ γ = 0 for p = 6