Decouplings and applications - math.umn.eduCiprian Demeter, IU Bloomington Decouplings and...

Decouplings and applications

Ciprian Demeter, IU Bloomington

April 27, 2018

Ciprian Demeter, IU Bloomington Decouplings and applications

Let Ξ be a collection of frequency points ξ on some curved,compact manifold S of diameter ∼ 1 in Rn (e.g. the unit sphereSn−1)

Let BR = B(c ,R) be a ball with radius R ≥ 1. Let also aξ ∈ C becomplex coefficients.

Notation: e(z) = e2πiz .

Question

How to estimate‖∑ξ∈Ξ

aξe(ξ · x)‖Lp(BR)


The case of tiny balls: spatial scale ∼ 1

Since the Fourier transform∑

ξ aξδξ of∑ξ

aξe(ξ · x)

is supported inside a ball of radius ∼ 1, the Uncertainty Principleproves that the exponential sum is essentially constant on each ballB(c , 1) of radius ∼ 1

|∑ξ

aξe(ξ · x)| ∼ |∑ξ

aξe(ξ · c)|, x ∈ B(c , 1).

The estimate via the triangle inequality

|∑ξ∈Ξ

aξe(ξ · c)| ≤∑ξ∈Ξ

|aξ|

is lossy, unless |Ξ| = O(1).


The case of medium size balls:

spatial scale R ∼ (freq. separation)−1

A particular case of interest is when the points ξ are separated by∼ 1

R .

The L2 estimate (in fact the equivalence)

|BR |−12 ‖

∑ξ∈Ξ

aξe(ξ · x)‖L2(BR) ∼ ‖aξ‖l2

is a rather immediate consequence of L2 orthogonality.


The case of medium size balls continued

The value p = 2(n+1)n−1 is special (Stein-Tomas exponent).

If ξ are 1R separated on the paraboloid Pn−1, the sphere Sn−1 (or in

general, on a hypersurface with nonzero Gaussian curvature) then

‖∑ξ∈Ξ

aξe(ξ · x)‖L

2(n+1)n−1 (BR)

. Rn−1

2 ‖aξ‖2.

This is a discrete equivalent reformulation of the Stein-Tomasrestriction theorem (the sphere) and of the Strichartz estimate forthe Schrodinger equation (the paraboloid).

The exponent n−12 is sharp. The proof uses TT ∗ and the decay of

Fourier transform of surface measure.


The case of medium size ball continued:

Let us continue to assume the points ξ are separated by ∼ 1R , on

Sn−1 or Pn−1. Another important p is the restriction indexp = 2n

n−1 .

The estimate at p = 2nn−1

‖∑ξ∈Ξ

aξe(ξ · x)‖Lp(BR) .ε Rn−1+ε‖aξ‖∞

is known in two dimensions, but open in higher dimensions. Ifproved, it would imply (among other things) that Kakeya sets inRn have full Hausdorff dimension.

Conclusion. The medium ball case is fully understood (for all p) if‖aξ‖l2 is used on the right, and mostly open when ‖aξ‖l∞ is used.


The case of ”large” balls: spatial scale ∼ (freq. separation)−2

Let us now assume that the frequencies ξ are 1R1/2 -separated. We

will be interested in determining the sharp exponents αp,n suchthat

‖∑ξ∈Ξ

aξe(ξ · x)‖Lp(BR) . Rαp,n‖aξ‖l2 .

Note that the exponential sum has more space to oscillate than inthe medium scale case, so it is expected that further cancellationswould occur.

Of particular interest is to find the range 2 ≤ p ≤ pc such that thefollowing reverse Holder holds

|BR |−1/p‖∑ξ∈Ξ

aξe(ξ·x)‖Lp(BR) . ‖aξ‖l2 ∼ |BR |−12 ‖

∑ξ∈Ξ

aξe(ξ·x)‖L2(BR)


The importance of the large spatial scale

Say we want to understand the size of

‖N∑

n=1

ane(nx1 + n2x2)‖Lp([0,1]2,dx1dx2).

A change of variables reduces this to the integral

‖N∑

n=1

ane(x · ( nN,n2

N2))‖Lp([0,N]×[0,N2],dx),

or, via periodicity to the integral on the large ”ball ”[0,N2]× [0,N2]

‖N∑

n=1

ane(x · ( nN,n2

N2))‖Lp([0,N2]×[0,N2],dx).

Note that the points ( nN ,

n2

N2 ) ∈ P1 are 1N -separated.


It turns out that

|BR |−1/p‖∑ξ∈Ξ

aξe(ξ · x)‖Lp(BR) .ε Rε‖aξ‖l2 (1)

holds true for each 2 ≤ p ≤ 2(n+1)n−1 and all 1√

R-separated points ξ

on Sn−1, Pn−1.However, (1) is false if the large ball BR is replaced with themedium size ball B√R .

(1) gives a Strichartz estimate in the (quasi)-periodic case of the(ir)rational torus. It will follow from the more general decouplingtheory we will introduce next.


Let (fj)Nj=1 be N elements in a normed space (X , ‖ · ‖X ). In this

generality the triangle inequality

‖N∑j=1

fj‖X ≤N∑j=1

‖fj‖X

is the best estimate available for the norm of the sum of fj . Whencombined with the Cauchy–Schwarz inequality, it leads to

‖N∑j=1

fj‖X ≤ N12 (

N∑j=1

‖fj‖2X )1/2.

Choosing X = L1(Rn) and positive functions fj with equal L1

norms shows that the inequality above can be sharp.


However, if X is a Hilbert space and if fj are pairwise orthogonalthen we have a stronger inequality (in fact an equality)

‖N∑j=1

fj‖X ≤ (N∑j=1

‖fj‖2X )1/2.

An example which is ubiquitous in Fourier analysis is whenX = L2(Rn), and the fj are functions whose Fourier transforms aredisjointly supported.

It is natural to ask whether there is an analogous phenomenon inLp(Rn) when p 6= 2, in the absence of Hilbert space orthogonality.


Given a smooth function ψ : U → Rn−d we define thed-dimensional manifold in Rn

M =Mψ = {(ξ, ψ(ξ)) : ξ ∈ U}.

Examples of hypersurfaces (d = n − 1) include the paraboloid

Pn−1 = {(ξ1, . . . , ξn−1, ξ21 + . . .+ ξ2

n−1) : |ξi | < 1},

the hemispheres

Sn−1± = {(ξ,±

√1− |ξ|2), |ξ| < 1},

the truncated cone

Conen−1 = {(ξ, |ξ|) : 1 < |ξ| < 2}.

A ”nice ” curve (d = n − 1) is the moment curve

Γn = {(t, t2, . . . , tn) : t ∈ [0, 1]}.


Denote by NM(δ) the δ-neighborhood of M. Let Θδ = Θδ(M)be a partition of NM(δ) into box-like sets θ of thicknes δ. Foreach θ let PθF be the Fourier restriction of F to θ

PθF = (F1θ)ˇ or PθF = F1θ.

If F is supported inside NM(δ) then

F =∑θ∈Θδ

PθF

Problem (l2 decoupling for manifolds)

Find the range 2 ≤ p ≤ pc such that the following l2 decouplingholds for each F with Fourier transform supported in NM(δ)

‖F‖Lp(B(0,δ−1)) .ε δ−ε(

∑θ∈Θδ

‖PθF‖2Lp(B(0,δ−1)))1/2.


Bourgain’s observation (2011): To get from...

Theorem (decoupling theorem for M)

‖F‖Lp(B(0,δ−1)) .ε δ−ε(

∑θ

‖PθF‖2Lp(B(0,δ−1)))1/2.

...to the exponential sum estimate (reverse Holder)

Theorem (Exponential sum estimate for M)

For each θ ∈ Θδ let ξθ ∈ θ ∩M and aθ ∈ C. Then

|B(0, δ−1)|−1/p‖∑θ∈Θδ

aθe(ξθ · x)‖Lp(B(0,δ−1)) .ε δ−ε(

∑θ

|aθ|2)1/2

simply use (a smooth approximation of)

F (ξ) =∑θ

aθδξθ


The first decoupling was proved for the cone by T. Wolff (2000).The following is the first sharp (i.e. full range) decoupling(Bourgain covered the smaller range 2 ≤ p ≤ 2n

n−1 in 2011)

Theorem (Bourgain, D. 2014)

For δ ≤ 1, let Θδ be a partition of the δ-neighborhood N (δ) ofPn−1 into essentially rectangular boxes of dimensionsδ1/2 × . . .× δ1/2 × δ.Let F be supported in N (δ). If 2 ≤ p ≤ 2(n+1)

n−1 we have

‖F‖Lp(B(0,δ−1)) .ε δ−ε(

∑θ∈Θδ

‖PθF‖2Lp(B(0,δ−1)))1/2.

The critical exponent is the Stein–Tomas index p = 2(n+1)n−1 , and

this is sharp. The same result holds for all hypersurfaces withnonzero Gaussian curvature.


In the following we will try to analyze the theorem for the parabolaP1, at the critical index pc = 6. We will call R = δ−1.

Theorem (Decoupling for the parabola)

For δ ≤ 1, let ΘR−1 be a partition of the 1R -neighborhood N (R−1)

of P1 into rectangles of dimensions R−1/2 × R−1. We have

‖F‖L6([−R,R]2) .ε Rε(

∑θ∈ΘR−1

‖PθF‖2L6([−R,R]2))1/2.

On the next slides we will explore some extremizers of thisinequality using wave packets.


Wave packet decomposition on [−R,R]2

For each θ ∈ ΘR−1 we write

PθF (x) =∑T∈Tθ

wTWT (x), x ∈ [−R,R]2

We assume all (R1/2,R)-rectangles T ∈ Tθ are inside, and in facttile [−R,R]2. They are dual to θ. Moreover

WT (x) ≈ 1T (x)e(ξT · x),

for some ξT ∈ θ ∩ P1. Also, wT ∈ C. Letting T = ∪θ∈ΘR−1Tθ wecall

F (x) =∑T∈T

wTWT (x)

the wave packet decomposition of F on [−R,R]2.


Assume T contains U . R1/2 rectangles for each direction.Assume also that there is a collection S of roughly U squares Swith unit side length inside [−R,R]2, so that each T intersectsexactly one S . One way to realize this is to place all S along ahorizontal line segment at distance ∼ R

U from each other and toplace a T centered at each S , for each possible direction. Call TSthose T ∈ T that intersect S , and recall TS are pairwise disjoint.

Figure U = 2 complete bushes


We choose all weights wT of magnitude roughly 1. Moreover, foreach S ∈ S with center cS we can make sure that the phases ofwT are coordinated so that

|∑T∈TS

wTWT (cS)| & R1/2.

This can be achieved due to our assumption that the collectionsTS are pairwise disjoint. The same inequality will in fact hold withcS replaced with x satisfying |x − cS | � 1. Also, the contributionfrom those T outside TS will be negligeable. To conclude, we canmake sure that

|∑T∈T

wTWT (x)| & R1/2

holds on a set of area roughly U. A simple computation shows that

‖F‖L6 & U1/6R1/2

and(∑θ

‖PθF‖2L6)1/2 ∼ U1/6R1/2.


Key elements of the proof

1. Bilinearization: the (linear) decoupling that we are trying toprove has an equivalent bilinear reformulation. This will allow us totake advantage of the bilinear Kakeya inequality.

2. Bilinear Kakeya inequality

3. L2 local orthogonality

4. Parabolic rescaling and the fact that decouplings can beiterated

5. Multi-scale argument, induction on scales


Theorem (Bilinear Kakeya)

Consider two families T1,T2 consisting of rectangles T in R2

having the following properties(i) each T has the short side of length R1/2 and the long side oflength equal to R pointing in the direction of the unit vector vT

(ii) vT1 ∧ vT2 ≥ 1100 for each Ti ∈ Ti .

We have the following inequality∫R2

∑T∈T1

1T∑T∈T2

1T . R|T1||T2|.

The implicit constant will not depend on R,Ti .


Lemma (Iteration)

Let Θ be a collection consisting of boxes θ. Assume each θ ispartitioned into smaller boxes θ1. Call Θ1 the collection of allthese smaller boxes. Assume that for each θ

‖PθF‖p ≤ D1(∑θ1⊂θ‖Pθ1F‖

2p)1/2.

Assume also that

‖F‖p ≤ D2(∑θ∈Θ

‖PθF‖2p)1/2,

whenever F is supported inside the union of all θ.Then, for such an F we also have

‖F‖p ≤ D1D2(∑θ1∈Θ1

‖Pθ1F‖2p)1/2.

The proof is an application of Minkowski’s inequality.Ciprian Demeter, IU Bloomington Decouplings and applications

Fix R = 22s . For 1 ≤ i ≤ s we denote by Θi the partition ofNP1(R−2−i+1

) into nearly rectangular tubes θi of length ∼ R−2−i.

Let Dec(R) be the smallest constant such that for eachF : R2 → C with Fourier transform supported on N (R−1) we have

‖F‖L6([−R,R]2) ≤ Dec(R)(∑θ1∈Θ1

‖Pθ1F‖2L6([−R,R]2))

12 .

We need to prove that Dec(R) .ε Rε.


Fix two pairwise disjoint intervals I1, I2 ⊂ [−1, 1], and denote by

Θi (Ij) those θi ∈ Θi that lie inside NIj (R−2−i+1

).

Let BilDec(R) be the smallest constant such that for eachFj : R2 → C with Fourier support inside NIj (R

−1) we have

‖(F1F2)12 ‖L6([−R,R]2) ≤

BilDec(R)(∑

θ1⊂Θ1(I1)

‖Pθ1F1‖2L6([−R,R]2))

14 (

∑θ1⊂Θ1(I2)

‖Pθ1F2‖2L6([−R,R]2))

14 .

One of the key ingredients of our proof is the fact that

Dec(R) ≈ BilDec(R).


For the rest of the argument we fix F1,F2 with Fourier transformssupported in NI1(R−1), NI2(R−1) and write F = F1 + F2. Let usintroduce the quantity IR(F1,F2) defined as follows

‖(F1F2)12 ‖L6([−R,R]2)

(∑

θ1⊂Θ1(I1) ‖Pθ1F1‖2L6([−R,R]2)

)14 (∑

θ1⊂Θ1(I2) ‖Pθ1F2‖2L6([−R,R]2)

)14

.

The equivalence Dec(R) ≈ BilDec(R) allows us reformulate ourgoal in the form of the following inequality

IR(F1,F2) .ε Rε.

We will seek various estimates for IR(F1,F2), via wave packetdecompositions at multiple scales and repeated pigeonholing.There will be many parameters at each scale, that reflect the localproperties of the two functions.


Let us denote [−R,R]2 by S1 and let S2 = S2(S1) be the partitionof S1 into squares S2 of side length R1/2. For 1 ≤ j ≤ 2 we use awave packet representation of Fj at scale R on [−R,R]2

Fj =∑T1∈Tj

wT1WT1 .

To recap, each T1 is a rectangle with dimensions R,R1/2, which isroughly dual to some θ1 ∈ Θ1(Ij). Also, WT1 has Fourier transformsupported inside θ1(T1) and

|WT1(x)| ≈ 1T1(x).


Proposition (Pigeonholing at scale R)

There are positive integers M1,N1, U1,V1, β1, γ1 and real numbersg1, h1 > 0, a collection S∗2 ⊂ S2 of R1/2-squares S2 and families ofrectangles T ∗1 ⊂ T1, T ∗2 ⊂ T2 such that

• (uniform weight) For each T1 ∈ T ∗1 we have |wT1 | ∼ g1 and foreach T1 ∈ T ∗2 we have |wT1 | ∼ h1.

• (uniform number of rectangles per direction) There is a family of∼ M1 tubes in Θ1(I1) such that all rectangles T1 ∈ T ∗1 are dual tosome θ1 in this family, with ∼ U1 rectangles for each such θ1. Inparticular, the size of T ∗1 is ∼ M1U1.Similarly, there is a family of ∼ N1 tubes in Θ1(I2) such that allrectangles T1 ∈ T ∗2,S1

are dual to some θ1 in this family, with ∼ V1

rectangles for each such θ1. In particular, the size of T ∗2,S1is

∼ N1V1.


Proposition (Pigeonholing at scale R, continued)

• (uniform number of rectangles per square) Each S2 ∈ S∗2intersects ∼ M1

β1rectangles from T ∗1 and ∼ N1

γ1rectangles from T ∗2 .

Moreover

‖(F1F2)12 ‖L6(S1) / ‖(F

(1)1 F

(1)2 )

12 ‖L6(∪S2∈S∗2

S2),

where for 1 ≤ j ≤ 2

F(1)j =

∑T1∈T ∗j

wT1WT1 .


Recall we aim to prove an O(Rε) estimate for

‖(F1F2)12 ‖L6([−R,R]2)

(∑

θ1⊂Θ1(I1) ‖Pθ1F1‖2L6([−R,R]2)

)14 (∑

θ1⊂Θ1(I2) ‖Pθ1F2‖2L6([−R,R]2)

)14

.

After just one round of pigeonholing we have a satisfactory (in factdefinitive) estimate for the denominator.

Proposition

We have

(∑

θ1⊂Θ1(I1)

‖Pθ1F1‖2L6([−R,R]2))

14 (

∑θ1⊂Θ1(I2)

‖Pθ1F2‖2L6([−R,R]2))

14

& R1/4(M1N1)1/4(g1h1)1/2(U1V1)1/12.


On the other hand, the estimate for

‖(F1F2)12 ‖L6([−R,R]2)

is still difficult at this point, since each x ∈ [−R,R]2 receivescontribution from many tubes.

Recall that the first round of pigeonholing produced a collection S∗2of significant squares S2 with side length R1/2, where a big

fraction of ‖(F1F2)12 ‖L6([−R,R]2) is concentrated.

We would like to repeat the wave packet decomposition on eachS2 ∈ S2. Then we will further repeat this until we reach scale ∼ 1.

But first we need to estimate the size of S∗2 .


Proposition (Counting squares via bilinear Kakeya)

We have|S∗2 | . β1γ1U1V1

Proof.

We apply the bilinear Kakeya inequality. More precisely we havethe following inequality for the fat rectangles 10T1∫

R2

(∑

T1∈T ∗1

110T1)(∑

T1∈T ∗2

110T1) . RM1N1U1V1.

Recall also that each S2 ∈ S∗2 is contained in ∼ M1β1

fat rectangles

10T1 with T1 ∈ T ∗1 and in ∼ N1γ1

fat rectangles 10T1 with T1 ∈ T ∗2 .With this observation, the bound on |S∗2 | follows immediately.


On each S2 writeFj =

∑T2∈Tj,S2

wT2WT2 ,

Each T2 is a rectangle with dimensions R1/2,R1/4, which isroughly dual to some θ2 ∈ Θ2(Ij). Also, WT1 has Fourier transformsupported inside θ2 and

|WT2(x)| ≈ 1T2(x).

Wave packets at two scales


Proposition (Pigeonholing at scale R1/2)

We may refine the collection S∗2 to get a smaller collection S∗∗2 ofsquares of side length R1/2, so that for each S2 ∈ S∗∗2 the followinghold.There are positive integers (independent of S2) M2,N2,U2,V2, β2, γ2 and real numbers g2, h2 > 0, a collection S∗3 (S2) ofR1/4-squares S3 inside S2 and two families of rectanglesT ∗1,S2

⊂ T1,S2 , T ∗2,S2⊂ T2,S2 such that

• (uniform weight) For each T2 ∈ T ∗1,S2we have |wT2 | ∼ g2 and for

each T2 ∈ T ∗2,S2we have |wT2 | ∼ h2.

• (uniform number of rectangles per direction) There is a family of∼ M2 tubes θ2 in Θ2(I1) such that each T2 ∈ T ∗1,S2

is dual to sucha θ2, with ∼ U2 rectangles for each such θ2. In particular, the sizeof T ∗1,S2

is ∼ M2U2. family of ∼ N2 tubes θ2 ∈ Θ2(I2) such thateach T2 ∈ T ∗2,S2

is dual to such a θ2, with ∼ V2 rectangles for eachsuch θ2. In particular, the size of T ∗2,S2

is ∼ N2V2.


Proposition (Pigeonholing at scale R1/2, continued)

• (uniform number of rectangles per square) Each S3 ∈ S∗3 (S2)intersects ∼ M2

β2rectangles from T ∗1,S2

and ∼ N2γ2

rectangles fromT ∗2,S2

.

Moreover

‖(F1F2)12 ‖L6(S1) / ‖(F

(2)1 F

(2)2 )

12 ‖L6(∪S3∈S∗3

S3), (2)

where for 1 ≤ j ≤ 2

F(2)j =

∑S2∈S∗∗2

∑T2∈T ∗j,S2

wT2WT2

andS∗3 = {S3 ∈ S∗3 (S2) : S2 ∈ S∗∗2 }.


To find the relation between the parameters M1,N1, . . . andM2,N2 we use local L2 orthogonality.

Proposition (Local orthogonality)

We have

U2g22R

34M2 . Rg2

1

M1

β1,

V2h22R

34N2 . Rh2

1

N1

γ1

Recall that on each S2 ∈ S∗∗2 we have two ways of representing

F(1)1 , corresponding to the scale R and R1/2, namely

F(1)1 (x) ≈

∑T1∈T ∗1T1∩S2 6=∅

wT1WT1(x), x ∈ S2

andF

(1)1 (x) ≈

∑T2∈T1,S2

wT2WT2(x), x ∈ S2.


We can repeat the previous argument to count the significantsquares S3 inside each significant square S2.

Proposition (Counting squares via bilinear Kakeya)

We have|S∗3 | . β2γ2U2V2|S∗2 |

. (β2γ2U2V2)(β1γ1U1V1)


We have so far done wave packet decompositions at two scales: Rand R1/2. Recall that R = 22s . Iterate this s times until we reachscale 2. Both F1 and F2 will be concentrated on squares Ss+1 ofside length ∼ 1.

‖(F1F2)12 ‖L6([−R,R]2) / ‖(F

(s)1 F

(s)2 )

12 ‖L6(∪Ss+1∈S∗s+1

Ss+1),

Key achievement: There will be only O(1) wave packetscontributing to both F1 and F2:

F(s)1 ≈

∑Ts

gsWTs ,

F(s)2 ≈

∑Ts

hsWTs

where each Ts is now a unit square. At this tiny scale we may write

|F s1 | ∼ |gs |, |F s

2 | ∼ |hs | on ∪Ss+1∈S∗s+1Ss+1


Combining

‖(F1F2)12 ‖L6([−R,R]2) / ‖(F

(s)1 F

(s)2 )

12 ‖L6(∪Ss+1∈S∗s+1

Ss+1),

with|F s

1 | ∼ |gs |, |F s2 | ∼ |hs | on ∪Ss+1∈S∗s+1

Ss+1

and (via s applications of bilinear Kakeya)

|S∗s+1| /s∏

k=1

(βkγkUkVk)

leads to a good estimate

‖(F1F2)12 ‖L6([−R,R]2) / (gshs)1/2

s∏k=1

(βkγkUkVk)1/6


Conclusion: The three estimates

‖(F1F2)12 ‖L6([−R,R]2) / (gshs)1/2

s∏k=1

(βkγkUkVk)1/6

(∑

θ1⊂Θ1(I1)

‖Pθ1F1‖2L6([−R,R]2))

14 (

∑θ1⊂Θ1(I2)

‖Pθ1F2‖2L6([−R,R]2))

14

& R1/4(M1N1)1/4(g1h1)1/2(U1V1)1/12

and local L2 orthogonality at each scale

gkhkgk−1hk−1

. R2−k 1

(βk−1γk−1)1/2

1

(UkVk)1/2(Mk−1Nk−1

MkNk)1/2

lead to the desired estimate

Dec(R) . Rε.


We have so far established the optimal decoupling theory for thefollowing manifolds

• Hypersurfaces in Rn with nonzero Gaussian curvature.Applications: Optimal Strichartz estimates for Shrodingerequation on both rational and irrational tori in all dimensions,improved Lp estimates for the eigenfunctions of the Laplacian onthe torus, Schrodinger maximal function (Du, Guth, Li)

• The cone (zero Gaussian curvature) in all dimensions.Applications: progress on Sogge’s “local smoothing conjecture forthe wave equation” ; the boundedness properties of the Bergmanprojection in tube domains over full light cones, averages ofRiemann zeta on short intervals (Bourgain-Watt)

• Two dimensional surfaces in R4. Application: Bourgain usedthis to improve the estimate in the Lindelof hypothesis for thegrowth of Riemann zeta on the critical line


• (with Larry, too) Curves with torsion in all dimensions, such as

Γn = {(t, t2, . . . , tn) : t ∈ [0, 1]}.

Application: Vinogradov’s Mean Value Theorem.

• Two dimensional manifolds in Rn and beyond, work byBourgain-Guo-D. and more recently by Guo-Zhang, completelysolving the higher dimensional analogs of the Vinogradov systems.


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