PDF Exponentials rarely maximize Fourier adjoint restriction estimates on cones

Exponentials rarely maximize Fourier adjoint restriction estimates on cones

by Giuseppe Negro

Instituto Superior Técnico, Lisbon, Portugal

Modern Trends in Harmonic Analysis, ICTS Bengaluru 26 June 2023

The adjoint restriction estimates on 2–Cone and 1–Cone

ξ1

τ

ξd

τ²“|ξ|²

ξ1

τ

ξd

τ “|ξ|

µ“δpτ²´|ξ|²q µ“δpτ²´|ξ|²q1τą0

Restriction Conjecture: for all d ě2,

∥fxµ∥_LqpR^d`1q

∥f∥_Lppµq

ďC_p,d ă 8, for

$

&

%

q “ ^d`1_d´1p¹,

1

q ă ^d´1_2d .

We are interested inmaximizers of this ratio(provided the estimate holds).

The Strichartz case: p “ 2, q “ 2

^d`1_d´1

Is f_‹pτq “expp´|τ|q a maximizer?

Spatial dim. d 2–Cone 1–Cone

2 NO YES

3 YES YES

4, 6, 8, . . . NO Local 5, 7, 9, . . . Local Local Meaning that:

∥fxµ∥_LqpR^d`1q

∥f∥_L2pµq

ď

∥fx_‹µ∥_LqpR^d`1q

∥f_‹∥_L2pµq

for

#

all f PL²pµq (YES)

∥f ´f‹∥_L2pµqďCd (Local) f_‹ is not a critical point (see next slide). (NO) Due to:

YES: Foschi 2007 NO: G.N. 2018 Local: G.N. & F.Gonçalves 2019

The general p case, on the 1–Cone

Recall: we want to maximize ^∥fxµ∥_∥f^LqpR_∥ ^d`1q

Lppµq aroundf‹pτq “expp´|τ|q.

Def.: f_‹ is a critical point iff B

Bϵ

∥pf‹{`ϵfqµ∥<sub>L</sub>qpR<sup>d`1</sup>q

∥f_‹`ϵf∥_Lppµq

ˇ ˇ ˇ ˇ ˇϵ“0

“0, @f PL^ppµq.

Theorem. (G.N.& D.Oliveira e Silva & B.Stovall & J.Tautges 2023)

(i) Maximizers exist for all p (for which the estimate holds).

(ii)f_‹ is a critical point ðñ p “2.

τ

We say thatf_‹ rarely maximizesthe Fourier extension inequality.

Exactly the same happens on the paraboloid with µ“δpτ ´|ξ|²q (i) Stovall 2020, (ii) Christ–Quilodrán 2012.

Digression: Foschi 2007. First appearance of f

_‹

“ e

^´|τ|

.

Let µ“δpτ²´|ξ|²q1τą0,p“2,d“3. Writeζ “ pτ, ξq.

τ

∥xfµpt,xq∥⁴_L4pR⁴q “ ż

fpζ₁qfpζ₂qfpζ₃qfpζ₄qe^{ipt,xq¨pζ1`ζ2´ζ3´ζ4q}dµ^b4dtdx

“ ż

fpζ1qfpζ2qfpζ3qfpζ4qδpζ1`ζ2´ζ3´ζ4qdµ^b4

(Cauchy–Schwarz)ď ż

|fpζ1qfpζ2q|²δpζ1`ζ2´ζ3´ζ4qdµ^b4

“ ż

|fpζ₁qfpζ₂q|²µ˚µpζ₁`ζ₂qdµ^b2

pMiracleq “ p2πq¹²∥f∥⁴_L2pµq.

Miracle: µ˚µ“ p2πq¹² on its support.

Digression: Foschi 2007. First appearance of f

_‹

“ e

^´|τ|

.

Let µ“δpτ²´|ξ|²q1τą0,p“2,d“3. Writeζ “ pτ, ξq.

τ

∥fxµpt,xq∥⁴_L4pR⁴q“ ż

fpζ₁qfpζ₂qfpζ₃qfpζ₄qδpζ₁`ζ₂´ζ₃´ζ₄qdµ^b4

(C–S)ď ż

|fpζ1qfpζ2q|²δpζ1`ζ2´ζ3´ζ4qdµ^b4.

Why f_‹? Because (C–S) is an identity when f “f_‹. Proof:

e^{´τ1´τ2´τ3´τ4}δpζ₁`ζ<sub>2</sub>´ζ<sub>3</sub>´ζ<sub>4</sub>q “e<sup>´2τ1´2τ2</sup>δpζ<sub>1</sub>`ζ₂´ζ₃´ζ₄q.

There must be a less technical reason for the appearance of f‹!

The Penrose map R

^1`d

Ñ r´ π, π s ˆ S

^d

r “|x| t

R^1`d

θ T

´π

π π

r´π, πs ˆS^d

Equations of Penrose map (and definition of θ)

T “arctanpt`rq `arctanpt´rq, θ“arctanpt`rq ´arctanpt´rq.

θ

S^d

The Penrose map R

^1`d

Ñ r´ π, π s ˆ S

^d

We call D^1`d the range of the Penrose map - the Penrose diamond.

´π

π π

θ T

θ

Figure: The Penrose diamond (left) is a submanifold of the cylinderRˆS^d.

(Right: Taken from S. Carroll,Notes on general relativity, arXiv:gr-qc/9712019.)

Where does f

_‹

p τ q “ exp p´ |τ | q come from?

For exposition purposes, let us focus on the 2–Cone only.

upt,xq “fxµ solves B²_tu “∆u. Recall: f is defined on τ

The Penrose map...

...yields a functional transform pU0,U90q ÐÑf such that ż

R^d`1

|fxµ|^q“ ż

R^d`1

|u|^q“ ż

D^d`1

|U|^q¨ psome weightq

d`1 2pp´1qp2´pq

, (*)

where

!

B²_TU “∆_SdU´^pd´1q

2

4 U, pU,BTUq|T“0 “ pU₀,U9₀q )

. Constant initial data p1,0q correspond to f‹.

p “2: RHS of (*) appears to be rotationally invariant: we can expect a maximizer to be rotational invariant, i.e. constant.

p ‰2: the weight breaks the rotational symmetry.

Back to the case p “ 2, q “ 2

^d`1_d´1

Recall: the estimate reads ∥xfµ∥_LqpR^d`1qďC_d∥f∥_L2pµq. Is f_‹pτq “expp´|τ|q a maximizer?

Spatial dim. d 2–Cone 1–Cone

2 NO YES

3 YES YES

4, 6, 8, . . . NO Local 5, 7, 9, . . . Local Local Let d “2,3. We need to maximize

ż

D^d`1

|U|^q,whereU solves

#

B²_TU “∆_S²U´¹₄U, d “2,

B²_TU “∆_S³U´U, d “3, onRˆS^d. Theorem

f_‹ is a critical point for d “3, but not for d “2.

Proof. The nice case d “ 3, p “ 2: f

_‹

is critical

Let X denote the generic point on S³. Key Symmetry Lemma

Suppose that

B_T²U “∆_S³U ´U, onRˆS³. Then UpT `π,´Xq “ ´UpT,Xq.

Proof. We haveU “cospTa

1´∆_S3qU₀`^sinpT

?1´∆

S3q

?1´∆

S3

U9₀. Expand

U0 “

8

ÿ

ℓ“0

Yℓ, with ´∆_S³Yℓ “ℓpℓ`2qYℓ; so Yℓp´Xq “ p´1q^ℓYℓ.

Then UpT,Xq “ ÿ8

ℓ“0

cospTpℓ`1qqYℓ`. . . l

We used: cosppT`πqpℓ`1qq “ p´1q<sup>ℓ`1</sup>cospTpℓ`1qq.

Proof. The nice case d “ 3, p “ 2: f

_‹

is critical (cont.)

Key Symmetry Lemma

B_T²U “∆_S3U ´U onRˆS³.Then UpT `π, π´θq “ ´UpT, θq.

Corollary.

ż

D^1`3

|U|^q“ 1 2

ż_π

´π

ż

S³

|U|^q.

Proof. D^1`3 “AYB. By the lemma, ż

A

|U|^q “ ż

A˜

|U|^q and ż

B

|U|^q “ ż

B˜

|U|^q.l

Remark

For d “2 the symmetry lemma fails. For example, B²_TU_‹ “∆_S2U_‹´ ¹₄U_‹, pU₀,U9₀q “ p1,0q yields U_‹ “cos¹₂T, so |U_‹pT `πq|‰|U_‹pTq|.

´π

π π

θ T

A B

A˜

B˜

θ

Proof. The nice case d “ 3, p “ 2: f

_‹

is critical (end)

Recall: q “2^d`1_d´1 “4. We want: maxt∥xfµ∥_L4 : ∥f∥_L2pµq “1u.

Claim I “

ż

R^3`1

pf_‹{`ϵf_Kqµ⁴´ ż

R^1`3

fx_‹µ⁴ “opϵq for xf_‹|f_Ky_L²_pµq“0.

Proof. By Penrose, fK ” pU₀^K,U9₀^Kq andf‹ ” p1,0q.

Note B²_TU_‹ “∆_S³U_‹´U_‹,pU0,U90q “ p1,0q yieldsU_‹“cosT.

2I“ ż_π

´π

ż

S³

|U_‹`ϵU^K|⁴´ ż_π

´π

ż

S³

|U_‹|⁴

“4ϵ ż_π

´π

ż

S³

U_‹³U^K`opϵq “4ϵ ż_π

´π

pcosTq³ ˆż

S³

U^KpT,¨q

˙

`opϵq.

Now xf‹|fKy “0ñ xU^K|1y “0 at T “0. Therefore xU^K|1y “0 at all times T. Henceş

S³U^KpT,¨q ”0 and we concludeI “0.l

Final summary

We want to maximize ż

R^1`d

|fxµ|^q, which equals:

$

’’

’’

’’

’’

’’

&

’’

’’

’’

’’

’’

% ż

D^d`1

|U|^qpcosT `cosθq

d`1 2pp´1qp2´pq

, p ‰2, ż

D^d`1

|U|^q, p “2,d even,

ż₁

´1

ż

S^d

|U|^q, p “2,d odd.

Only in the latter case,

f_‹ “ expp´|τ|q is critical (and so has a chance of being a maximizer).

´π

π π

θ T

D^d`1

θ