Nonlinear Schrödinger Evolutions
from Low Regularity Initial Data
J. Colliander
University of Toronto
Warwick, June 2009
1 Cubic NLS on R2
2 High-Low Fourier Truncation
3 Bilinear Strichartz Estimate
4 The I -Method of Almost Conservation
1. Cubic NLS Initial Value Problem on R2
1. Cubic NLS Initial Value Problem on R2
We consider the initial value problems:
(i∂t + ∆)u = ±|u|2 u
u(0, x) = u0 (x).
(NLS3± (R2 ))
The + case is called defocusing; − is focusing. NLS3± is ubiquitous
in physics. The solution has a dilation symmetry
u λ (τ, y ) = λ−1 u(λ−2 τ, λ−1 y ).
which is invariant in L2 (R2 ). This problem is L2 -critical.
(This talk mostly addresses the defocusing case.)
Time Invariant Quantities
Z
|u(t, x)|2 dx.
Z
Momentum = 2=
u(t)∇u(t)dx.
R2
Z
1
1
Energy = H[u(t)] =
|∇u(t)|2 dx± |u(t)|4 dx.
2 R2
2
Mass =
Rd
Hamiltonian
kinetic
potential
Mass is L2 ; Momentum is close to H 1/2 ; Energy involves H 1 .
Dynamics on a sphere in L2 ; focusing/defocusing energy.
Local conservation laws express how quantity is conserved:
e.g., ∂t |u|2 = ∇ · 2=(u∇u). Frequency Localizations?
Linear Schrödinger Propagator and Estimates
The solution of the linear Schrödinger initial value problem
(i∂t + ∆)u = 0
(LS(Rd ))
u(0, x) = u0 (x).
is denoted u(t, x) = e it∆ u0 . The solution can be given explicitly
Fourier Multiplier Representation:
Z
2
it∆
e u0 (x) = cπ
e ix·ξ e −it|ξ| ub0 (ξ)dξ.
Rd
Convolution Representation:
e it∆ u0 (x) = cπ1
1
(it)d/2
Z
Rd
ei
|x−y |2
4t
u0 (y )dy .
Linear Schrödinger Propagator and Estimates
The solution of the linear Schrödinger initial value problem
(i∂t + ∆)u = 0
(LS(Rd ))
u(0, x) = u0 (x).
is denoted u(t, x) = e it∆ u0 . The solution can be given explicitly
Fourier Multiplier Representation:
Z
2
it∆
e u0 (x) = cπ
e ix·ξ e −it|ξ| ub0 (ξ)dξ.
Rd
Convolution Representation:
e it∆ u0 (x) = cπ1
1
(it)d/2
Modulus 1 Multiplier
Z
Rd
ei
|x−y |2
4t
u0 (y )dy .
Linear Schrödinger Propagator
The solution of the linear Schrödinger initial value problem
(i∂t + ∆)u = 0
(LS(Rd ))
u(0, x) = u0 (x).
is denoted u(t, x) = e it∆ u0 . The solution can be given explicitly
Fourier Multiplier Representation:
Z
2
it∆
e u0 (x) = cπ
e ix·ξ e −it|ξ| ub0 (ξ)dξ.
Rd
Convolution Representation:
e it∆ u0 (x) = cπ1
1
(it)d/2
Z
ei
|x−y |2
4t
u0 (y )dy .
Rd
Time Decay
Estimates for Linear Schrödinger Propagator
Fourier Multiplier Representation =⇒ Unitary in H s :
kDxs e it∆ u0 kL2x = kDxs u0 kL2x .
Convolution Representation =⇒ Dispersive estimate:
ke it∆ u0 kL2x∞ ≤
C
t d/2
ku0 kL1x .
Spacetime estimates? Strichartz estimates hold, for example,
ke it∆ u0 kL4 (Rt ×R2x ) ≤ C ku0 kL2 (R2x ) .
(Strichartz estimates linked to Fourier restriction phenomena.)
Local-in-time theory for NLS3± (R2 )
∀ u0 ∈ L2 (R2 ) ∃ Tlwp (u0 ) determined by
ke it∆ u0 kL4tx ([0,Tlwp ]×R2 ) <
1
such that
100
∃ unique u ∈ C ([0, Tlwp ]; L2 ) ∩ L4tx ([0, Tlwp ] × R2 ) solving
NLS3+ (R2 ).
−2
∀ u0 ∈ H s (R2 ), s > 0, Tlwp ∼ ku0 kH ss and regularity persists:
u ∈ C ([0, Tlwp ]; H s (R2 )).
Define the maximal forward existence time T ∗ (u0 ) by
kukL4tx ([0,T ∗ −δ]×R2 ) < ∞
for all δ > 0 but diverges to ∞ as δ & 0.
∃ small data scattering threshold µ0 > 0
ku0 kL2 < µ0 =⇒ kukL4tx (R×R2 ) < 2µ0 .
Global-in-time theory?
What is the ultimate fate of the local-in-time solutions?
L2 -critical Scattering Conjecture:
L2 3 u0 7−→ u solving NLS3+ (R2 ) is global-in-time and
kukL4t,x < A(u0 ) < ∞.
Moreover, ∃ u± ∈ L2 (R2 ) such that
lim ke ±it∆ u± − u(t)kL2 (R2 ) = 0.
t→±∞
Same statement for focusing NLS3− (R2 ) if ku0 kL2 < kQkL2 .
Remarks:
Known for small data ku0 kL2 (R2 ) < µ0 .
Known for large radial data [Killip-Tao-Visan 07].
NLS3± (R2 ): Present Status for General Data
regularity
s > 23
s > 47
s > 12
s ≥ 12
s > 25
idea
high/low frequency decomposition
H(Iu)
resonant cut of 2nd energy
H(Iu) & Interaction Morawetz
H(Iu) & Interaction I -Morawetz
1
3
resonant cut & I -Morawetz
s>
reference
[Bourgain98]
[CKSTT02]
[CKSTT07]
[Fang-Grillakis05]
[CGTz07]
[C-Roy08]
s > 0?
Morawetz-based arguments are only for defocusing case.
Focusing results assume ku0 kL2 < kQkL2 .
Unify theory of focusing-under-ground-state and defocusing?
2. Bourgain’s High-Low Fourier Truncation
2. Bourgain’s High-Low Fourier Truncation
IMRN International Mathematics Research Notices
1998, No. 5
Refinements of Strichartz’ Inequality
and Applications to 2D-NLS
with Critical Nonlinearity
J. Bourgain
Summary
Consider the 2D IVP
!
iut + ∆u + λ|u|2 u = 0
(†)
u(0) = ϕ ∈ L2 (R2 ).
The theory on the Cauchy problem asserts a unique maximal solution
∗
2
2
4
∗
4
2
2. Bourgain’s High-Low Fourier Truncation
Consider the Cauchy problem for defocusing cubic NLS on R2 :
(i∂t + ∆)u = +|u|2 u
(NLS3+ (R2 ))
u(0, x) = φ0 (x).
We describe the first result to give global well-posedness below H 1 .
NLS3+ (R2 ) is GWP in H s for s >
2
3
[Bourgain 98].
First use of Bilinear Strichartz estimate was in this proof.
Proof cuts solution into low and high frequency parts.
For u0 ∈ H s , s > 32 , Proof gives (and crucially exploits),
u(t) − e it∆ φ0 ∈ H 1 (R2x ).
Setting up; Decomposing Data
Fix a large target time T .
Let N = N(T ) be large to be determined.
Decompose the initial data:
φ0 = φlow + φhigh
where
Z
φlow (x) =
c0 (ξ)dξ.
e ix·ξ φ
|ξ|<N
Our plan is to evolve:
φ0 = φlow + φhigh
u(t) = ulow (t) + uhigh (t).
Setting up; Decomposing Data
Low Frequency Data Size:
Kinetic Energy:
k∇φlow k2L2 =
Z
c0 (ξ)|2 dx
|ξ|2 |φ
|ξ|<N
Z
=
c0 (ξ)|2 dx
|ξ|2(1−s) |ξ|2s |φ
|ξ|<N
2(1−s)
=N
kφ0 k2H s ≤ C0 N 2(1−s) .
1/2
1/2
Potential Energy: kφlow kL4x ≤ kφlow kL2 k∇φlow kL2
=⇒ H[φlow ] ≤ CN 2(1−s) .
High Frequency Data Size:
kφhigh kL2 ≤ C0 N −s , kφhigh kH s ≤ C0 .
LWP of Low Frequency Evolution along NLS
The NLS Cauchy Problem for the low frequency data
(i∂t + ∆)ulow = +|ulow |2 ulow
ulow (0, x) = φlow (x)
is well-posed on [0, Tlwp ] with Tlwp ∼ kφlow k−2
∼ N −2(1−s) .
H1
We obtain, as a consequence of the local theory, that
kulow kL4
[0,Tlwp ],x
≤
1
.
100
LWP of High Frequency Evolution along DE
The NLS Cauchy Problem for the low frequency data
(i∂t + ∆)uhigh = +2|ulow |2 uhigh + similar + |uhigh |2 uhigh
uhigh (0, x) = φhigh (x)
is also well-posed on [0, Tlwp ].
Remark: The LWP lifetime of NLS evolution of ulow AND the
LWP lifetime of the DE evolution of uhigh are controlled by
kulow (0)kH 1 .
Extra Smoothing of Nonlinear Duhamel Term
The high frequency evolution may be written
uhigh (t) = e it∆ uhigh + w .
The local theory gives kw (t)kL2 . N −s . Moreover, due to
smoothing (obtained via bilinear Strichartz), we have that
w ∈ H 1 , kw (t)kH 1 . N 1−2s+ .
Let’s postpone the proof of (SMOOTH!).
(SMOOTH!)
Nonlinear High Frequency Term Hiding Step!
∀ t ∈ [0, Tlwp ], we have
u(t) = ulow (t) + e it∆ φhigh + w (t).
At time Tlwp , we define data for the progressive sheme:
u(Tlwp ) = ulow (Tlwp ) + w (Tlwp ) + e iTlwp ∆ φhigh .
(2)
(2)
u(t) = ulow (t) + uhigh (t)
for t > Tlwp .
(2)
Hamiltonian Increment: φlow (0) 7−→ ulow (Tlwp )
The Hamiltonian increment due to w (Tlwp ) being added to low
frequency evolution can be calcluated. Indeed, by Taylor
expansion, using the bound (SMOOTH!) and energy conservation
of ulow evolution, we have using
(2)
H[ulow (Tlwp )] = H[ulow (0)] + (H[ulow (Tlwp ) + w (Tlwp )] − H[ulow (Tlwp )])
∼ N 2(1−s) + N 2−3s+ ∼ N 2(1−s) .
Moreover, we can accumulate N s increments of size N 2−3s+ before
we double the size N 2(1−s) of the Hamiltonian. During the
iteration, Hamiltonian of “low frequency” pieces remains of size
. N 2(1−s) so the LWP steps are of uniform size N −2(1−s) . We
advance the solution on a time interval of size:
N s N −2(1−s) = N −2+3s .
For s > 32 , we can choose N to go past target time T .
How do we prove (SMOOTH!)?
Bourgain’s Bilinear Strichartz Estimate: For (dyadic) N ≤ L
ke
it∆
fL e
it∆
gN kL2t,x ≤
N
2−1
2
1
L2
kfL kL2x kgN kL2x .
Corollary
For s ≥
1
2
kDxs (u1 u2 )kL2
[0,δ],x
≤C (ku1 kX s,1/2+ ku2 kX 0,1/2+
[0,δ]
[0,δ]
+ ku1 kX 1/2,1/2+ ku2 kX s−1/2,1/2+ ).
[0,δ]
[0,δ]
Thus, the Bilinear Estimate allows us move half a derivative off the
high frequency part and instead onto of the low frequency part.
Treatment of a typical term in w
Using the controls we have on ulow , uhigh from the local theory
on [0, Tlwp ], we want to prove for
Z
w=
t
0
e i(t−t )∆ |ulow |2 uhigh (t 0 )dt 0
0
that supt∈[0,Tlwp ] k∇w kL2 < N 1−2S+ .
By Sobolev embedding, we have
kw kL∞
[0,Tlwp ]
H1
≤ kw kX 1,1/2+ .
[0,Tlwp ]
Rt
0
The mapping f 7−→ 0 e i(t−t )∆ is formally
f 7−→ (i∂t + ∆)−1 f which, due to time localization, is
essentially b
f 7−→ hτ + |ξ|2 ib
f . It suffices to control
2
kDx |ulow | uhigh kX 0,−1/2+ . Proceed by duality....
Treatment of a typical term in w
kw kL∞
[0,Tlwp ]
H1
≤
hg , Dx (|ulow |2 uhigh )i.
sup
kg kX 0,1/2− ≤1
. suphgDx ulow , ulow uhigh i + suphgulow , Dx (ulow uhigh )i
g
g
1/2
1/2
= easier + suphDx (gulow ), Dx (ulow uhigh i.
g
The corollary and the available bounds then give (SMOOTH!).
3. Bourgain’s Bilinear Strichartz Estimate
3. Bourgain’s Bilinear Strichartz Estimate
Recall the Strichartz estimate for (i∂t + ∆) on R2 :
ke it∆ u0 kL4 (Rt ×R2x ) ≤ C ku0 kL2 (R2x ) .
We can view this trivially as a bilinear estimate by writing
ke it∆ u0 e it∆ v0 kL2 (Rt ×R2x ) ≤ C ku0 kL2 (R2x ) kv0 kL2 (R2x ) .
Bourgain refined this trivial bilinear estimate for functions
having certain Fourier support properties.
Bourgain’s Bilinear Strichartz Estimate
Shrinks Constant
Theorem
For (dyadic) N ≤ L and for x ∈ R2 ,
1
ke
it∆
fL e
it∆
gN kL2t,x ≤
N2
1
L2
kfL kL2x kgN kL2x .
Here spt (fbL ) ⊂ {|ξ| ∼ L}, gN similar.
q
Observe that NL 1 when N L.
M
|ξ|∼M
Then, for M1 ≤ M2 , the following inequality holds:
3. Bourgain’s Proof "
!(eit∆ ψ1 )(eit∆ ψ2 )!L2 (R2 ×R) ≤ C
Proof.
M1
M2
#1/2
(112)
!ψ1 !2 !ψ2 !2 .
B98:IMRN
Since the standard Strichartz inequality yields (112) without the
"
#1
M1 2
-factor,
M2
we may assume M2 ' M1 .
Writing
(eit∆ ψ1 )(eit∆ ψ2 ) =
!
!1 (ξ1 )ψ
!2 (ξ2 )ei[(ξ1 +ξ2 ).x+(|ξ1 |2 +|ξ2 |2 )t] dξ1 dξ2 ,
ψ
it follows from Parseval’s identity and Cauchy-Schwarz that
!(eit∆ ψ1 )(eit∆ ψ2 )!22 =
!
$!
$2
$
$
!1 (ξ1 )ψ
!2 (ξ − ξ1 )δ0 (|ξ1 |2 + |ξ − ξ1 |2 − λ) dξ1 $
dξdλ $$ ψ
$
%
≤ !ψ1 !22 !ψ2 !22
sup mes(1) [ξ1 | |ξ1 | ∼ M1
λ,|ξ|∼M2
2
2
&
and |ξ1 | + |ξ − ξ1 | = λ]
<C
M1
.
M2
Proof Based on Change of Variables
Ideas from (Kenig-Ponce-Vega); see [C-Delort-Kenig-Staffilani].
Recall the Fourier multiplier representation of the propagator:
Z
2
it∆
e f (x) =cπ
e ix·ξ e −it|ξ| b
f (ξ)dξ
2
ZR
=cπ
e i(x·ξ+tτ ) δ0 (τ + |ξ|2 )b
f (ξ)dτ dξ.
R1+2
spacetime inverse
Fourier transform
With f = fL and g = gN , we wish to estimate
ke it∆ f e it∆ g kL2t,x = kF[e it∆ f e it∆ g ]kL2 .
τ,ξ
Using Fourier tranform property, F(ab) = b
a∗b
b, we find....
Fourier Manipulations; Dirac Evaluations
We wish to estimate (in L2τ,ξ ) the expression
Z
δ0 (τ1 + |ξ1 |2 )b
f (ξ1 )δ0 (τ2 + |ξ2 |2 )b
g (ξ2 ).
τ = τ1 + τ 2
ξ = ξ1 + ξ2
Evaluating the δ functions, we find τj = −|ξj |2 , so
Z
b
f (ξ1 )b
g (ξ2 )
τ = −|ξ1 |2 − |ξ2 |2
ξ = ξ1 + ξ2
We proceed by duality. Let’s test this against d(τ, ξ)....
Duality Reduces Matters to Certain Integral
*
ke
it∆
f e
it∆
g kL2t,x =
sup
kdkL2
τ,ξ
Z
= sup
Z
d(τ, ξ) ,
≤1
+
b
f (ξ1 )b
g (ξ2 ) .
τ = −|ξ1 |2 − |ξ2 |2
ξ = ξ1 + ξ2
d(−|ξ1 |2 − |ξ2 |2 , ξ1 + ξ2 ) b
f (ξ1 )b
g (ξ2 )dξ1 dξ2 .
d
The preceding Fourier manipulations have reduced matters to
bounding a certain integral. Our task is to show the integral above
is bounded by
r
N
.
kf kL2 kg kL2 kdkL2 .
L
Setting Up the Change of Variables
Let’s define a change of variables motivated by the arguments of d:
u = −|ξ1 |2 − |ξ2 |2 , v = ξ1 + ξ2 .
Note that u ∈ R and v ∈ R2 . Thus, dudv is a measure in 3d
while dξ1 dξ2 is a measure in 4d.
Note also that ξ2 is the argument of g = gN so it is localized
to the smaller dyadic shell |ξ2 | ∼ N L.
Let’s denote the components of ξj ∈ R2 with superscripts:
ξj = (ξj1 , ξj2 ).
The full change of variables is the defined via
dudv dξ21 = |J| dξ11 dξ12 dξ22 dξ21 .
We have an extra variable outside the changed integral.
The Jacobian
The Jacobian matrix J is calculated as
 ∂u ∂v 1 ∂v 2 
1
1
 ∂ξ
∂u
J=
 ∂ξ21
∂u
∂ξ22
∂ξ11
∂v 1
∂ξ21
∂v 1
∂ξ22
∂ξ11

∂v 2 
∂ξ21  .
∂v 2
∂ξ22
The explicit forms for u, v permit calculating
|J| = 2|ξ11 − ξ12 |.
Since |ξ1 | ∼ L, we may assume by rotation that |J| ∼ L.
Changing Variables
Our task: Estimate, for |ξ1 | ∼ L, |ξ2 | ∼ N, the integral
Z
Z
d(−|ξ1 |2 − |ξ2 |2 , ξ1 + ξ2 ) b
f (ξ1 )b
g (ξ2 )dξ11 dξ12 dξ21 dξ22 .
|ξ22 |.N ξ1 ,ξ21
We insert the Jacobian and reexpress inner integration as
Z
b
f (ξ1 )b
g (ξ2 )
|J|dξ11 dξ12 dξ21 .
d(−|ξ1 |2 − |ξ2 |2 , ξ1 + ξ2 )
|J|
ξ1 ,ξ21
Changing variables, we observe this equals
Z
d(u, v )H(u, v ; ξ22 )|J|dudv
u,v
where
H(u, v ; ξ22 ) =
b
f (ξ1 )b
g (ξ2 )
.
|J|
Cauchy-Schwarz; Jacobian Remnant
We apply Cauchy-Schwarz in u, v to bound by
Z
kdkL2
|H(u, v ; ξ22 )|2 dudv
1/2
.
u,v
We drop kdkL2 ≤ 1 by duality and change variables back. We get
1/2
2
Z b
g (ξ2 ) 
f (ξ1 )b

|J|dξ11 dξ12 dξ21  .

|J|

ξ1 ,ξ22
One factor of the Jacobian denominator remains! We gain L−1/2 .
We still have the extra outside integration....
Trivial Cauchy-Schwarz on Extra Integral
Recalling what we must control, using what we have obtained....
1/2
2
Z b
g (ξ2 ) 
f (ξ1 )b

|J|dξ11 dξ12 dξ21  dξ22 .

|J|

Z
|ξ22 |.N
ξ1 ,ξ22
Apply Cauchy-Schwarz in ξ22 and pay the penalty in the numerator
of N 1/2 .
We gain over the trivial bilinear estimate by the factor
s
r
(measure of extra support)
N
=
.
|J|
L
4. The I -Method of Almost Conservation
4. The I -Method of Almost Conservation
−2/s
Let H s 3 u0 7−→ u solve NLS for t ∈ [0, Tlwp ], Tlwp ∼ ku0 kH s .
Consider two ingredients (to be defined):
A smoothing operator I = IN : H s 7−→ H 1 . The NLS
evolution u0 7−→ u induces a smooth reference evolution
H 1 3 Iu0 7−→ Iu solving I (NLS) equation on [0, Tlwp ].
e [Iu] built using the reference evolution.
A modified energy E
We postpone how we actually choose these objects.
e = H[Iu]
First Version of the I -method: E
For s < 1, N 1 define smooth monotone m : R2ξ → R+ s.t.
(
1
m(ξ) = s−1
|ξ|
N
for |ξ| < N
for |ξ| > 2N.
d
The associated Fourier multiplier operator, (Iu)(ξ)
= m(ξ)b
u (ξ),
s
1
satisfies I : H → H . Note that, pointwise in time, we have
kukH s . kIukH 1 . N 1−s kukH s .
e [Iu(t)] = H[Iu(t)]. Other choices of E
e are mentioned later.
Set E
AC Law Decay and Sobolev GWP index
1
Modified LWP. Initial v0 s.t. k∇Iv0 kL2 ∼ 1 has Tlwp ∼ 1.
2
Goal. ∀ u0 ∈ H s , ∀ T > 0, construct u : [0, T ] × R2 → C.
3
4
5
⇐⇒ Dilated Goal. Construct u λ : [0, λ2 T ] × R2 → C.
Rescale Data. kI ∇u0λ kL2 . N 1−s λ−s ku0 kH s ∼ 1 provided we
1−s
choose λ = λ(N) ∼ N s ⇐⇒ N 1−s λ−s ∼ 1.
Almost Conservation Law. kI ∇u(t)kL2 . H[Iu(t)] and
sup
H[Iu(t)] ≤ H[Iu(0)] + N −α .
t∈[0,Tlwp ]
6
Delay of Data Doubling. Iterate modified LWP N α steps
with Tlwp ∼ 1. We obtain rescaled solution for t ∈ [0, N α ].
λ2 (N)T < N α ⇐⇒ T < N α+
2(s−1)
s
so s >
2
suffices.
2+α
e = H[Iu]
First Version of the I -method: E
A Fourier analysis established the almost conservation property of
e = H[Iu] with α = 3 which led to...
E
2
Theorem (CKSTT 02)
NLS3+ (R2 ) is globally well-posed for data in H s (R2 ) for
4
7
< s < 1.
Moreover, ku(t)kH s . htiβ(s) for appropriate β(s).
The smoothing property u(t) − e it∆ u0 ∈ H 1 is not obtained.
Same result for NLS3− (R2 ) if ku0 kL2 < kQkL2 . Here Q is the
ground state (unique positive solution of −Q + ∆Q = −Q 3 ).
Fourier analysis leading to α =
frequency interactions.
3
2
in fact gives α = 2 for most
Almost Conservation Law for H[Iu]
Proposition
Given s > 47 , N 1, and initial data φ0 ∈ C0∞ (R2 ) with
E (IN u0 ) ≤ 1, then there exists a Tlwp ∼ 1 so that the solution
u(t, x) ∈ C ([0, Tlwp ], H s (R2 ))
of NLS3+ (R2 ) satisfies
3
E (IN u)(t) = E (IN u)(0) + O(N − 2 + ),
for all t ∈ [0, Tlwp ].
Ideas in the Proof of Almost Conservation
Standard Energy Conservation Calculation:
Z
cancellation
∂t H(u) = <
ut (|u|2 u − ∆u)dx
2
ZR
=<
ut (|u|2 u − ∆u − iut )dx = 0.
R2
For the smoothed reference evolution, we imitate....
Z
∂t H(Iu) = <
Iut (|Iu|2 Iu − ∆Iu −i Iut )dx
2
R
commutator!
Z
2
2
=<
Iut (|Iu| Iu − I (|u| u))dx 6= 0.
R2
The increment in modified energy involves a commutator,
Z tZ
H(Iu)(t) − H(Iu)(0) = <
Iut (|Iu|2 Iu − I (|u|2 u))dxdt.
0
R2
Littlewood-Paley, Case-by-Case, (Bi)linear Strichartz, Xs,b ....
Remarks
The almost conservation property
sup
e [Iu(t)] ≤ E
e [Iu0 ] + N −α
E
t∈[0,Tlwp ]
leads to GWP for
2
.
2+α
The I -method is a subcritical method. To prove the Scattering
Conjecture at s = 0 via the I -method would require α = +∞.
s > sα =
The I -method localizes the conserved density in frequency.
Similar ideas appear in recent critical scattering results.
There is a multilinear corrections algorithm for defining other
e which yield a better AC property.
choices of E