nonconcentration, lp-improving estimates, and multilinear ...gressman/slides/wisc2019.pdf0. the...

Nonconcentration, Lp-Improving Estimates, andMultilinear Kakeya

Philip T. Gressman

Department of MathematicsUniversity of Pennsylvania

13 May 2019Madison Lectures in Fourier Analysis

Philip T. Gressman N-C, Lp , and M-K 0 / 23

0. The Problem of Geometry in Fourier Analysis

• There are a number of deeply geometric operators, integrals,etc., in harmonic analysis. The structure is usually defined bysmooth spaces with measures and maps between those spaces.

• With minimal structure, it’s often not even clear what keyquantities are or how to proceed.

• Introducing artificial structures (e.g., coordinates) reduces tomore familiar settings, but doing so breaks fundamentalinvariances of the problem and risks missing importantfeatures.

• There is a third way: introduce artificial auxiliary structuresand study the group action induced by transformations ofthese structures.


An Example: An n-dimensional vector space V is called a Peanospace when it is equipped with a nontrivial alternating n-linearform [v1, . . . , vn]. It is natural to use bases with [v1, . . . , vn] = 1.

Suppose U : V × V → R is a symmetric bilinear functional on areal Peano space V such that U(v , v) ≥ 0 for all v ∈ V . Is therean easy way to detect degeneracy or nondegeneracy of U?

• Option 1: Do calculations invariant under natural choices:

detU := det[U(vi , vj)]i ,j=1,...,n.

• Option 2: Do anything, then optimize over all choices. E.g.:

inf[v1,...,vn]=1

n∑j=1

|U(vi , vi )|p1/p

, p ∈ (0,∞].

Theorem: detU = inf[v1,...,vn]=1

1

n

n∑j=1

|U(vi , vi )|p n

p

.


1. Brascamp-Lieb Constant

H,Hj : Hilbert(?) spaces of dimension d , dj , j = 1, . . . ,m;πj : Surjective linear maps H → Hj ; θj : constants in [0, 1].

Brascamp-Lieb Inequality

RBL(π, θ)

∫H

m∏j=1

(fj πj)θj ≤m∏j=1

(∫Hj

fj

)θjBennett, Carbery, Christ, and Tao (2005)

RBL(π, θ) > 0 if and only if

dimV ≤m∑j=1

θj dimπj(V ) for all V ⊂ H

with equality when V = H.

Bennett, Bez, Cowling, Flock (2016)

Fixing dimensions and θ, RBL(π, θ) is continuous in π.


RBL(π, θ)

∫H

m∏j=1

(fj πj)θj ≤m∏j=1

(∫Hj

fj

)θjLieb (1990): Gaussians extremize the inequality

RBL(π, θ) = infAj∈GL(Hj )

j=1,...,m

(det∑m

j=1 θjπ∗j A∗j Ajπj

) 12∏m

j=1 | detHjAj |θj

Change determinant to infimum of trace:

[RBL(π, θ)]2d = inf

Aj∈GL(Hj )

A∈SL(H)

d−1tr∑m

j=1 θjA∗π∗j A

∗j AjπjA∏m

j=1 | detHjAj |2θj/d

= infAj∈SL(Hj )

A∈SL(H)

tj∈(0,∞)

d−1t−2θd

m∑j=1

θj t2dj

j |||AjπjA|||2

where ||| · ||| is the Hilbert-Schmidt (sum of squares) matrix norm.Philip T. Gressman N-C, Lp , and M-K 4 / 23

Keep Going: Use AM-GM Inequality again to eliminate tj :

[RBL(π, θ)]2d = inf

Aj∈SL(Hj )

A∈SL(H)

tj∈(0,∞)

d−1t−2θd

m∑j=1

θjdjd

t2dj

j

(d

dj|||AjπjA|||2

)

=

m∏j=1

d−θj djd

j

infAj∈SL(Hj )

A∈SL(H)

m∏j=1

|||AjπjA|||2θj djd

Assuming rational θj , there exist integers N,Nj such that

θjdjd

=Nj

N, j = 1, . . . ,m,

[RBL(π, θ)]Nd =

m∏j=1

d−

Nj2

j

infAj∈SL(Hj )

A∈SL(H)

m∏j=1

|||AjπjA|||Nj .


For integers N = N1 + · · ·+ Nm,

[RBL(π,N)]Nd

∣∣∣∣∣∣∣∣∣∣∣∣m∏j=1

fj πj

∣∣∣∣∣∣∣∣∣∣∣∣Ld/N(H)

≤m∏j=1

||fj ||Ldj/Nj (Hj ).

Define ΠN : HN × HN11 × · · · × HNm

m → R by the formula

ΠN(x (1), . . . , x (N), x(1)1 , . . . , x

(N1)1 , . . . , x

(Nm)m )

:=⟨π1x

(1), x(1)1

⟩H1

· · ·⟨π1x

(N1), x(N1)1

⟩H1

· · ·⟨πmx

(N), x(Nm)m

⟩Hm

and let G := SL(H)× SL(H1)× · · · × SL(Hm). Then

[RBL(π,N)]Nd =

m∏j=1

d−

Nj2

j

infG∈G|||ρGΠN |||,

ρG is the action of G on HN × · · ·×HNmm , ||| · ||| is Hilbert-Schmidt.

A Good Question: Why did we do this lovely calculation?Philip T. Gressman N-C, Lp , and M-K 6 / 23

2. Geometric Nonconcentration Inequalities

Suppose Φ is some polynomial function from (Rn)k into Rm.|Φ(x1, . . . , xk)| measures nondegeneracy of k-point configurations.Example: if ϕ(x) := (xα)|α|≤d , then

Φ(x1, . . . , xN) := det(ϕ(x1), . . . , ϕ(xN)) = 0

iff x1, . . . , xN lie on some real algebraic variety of deg. ≤ d .

Nonconcentration Inequalities

For a given Φ and s, find the “best possible” measure µ such that

S(E ) := ess.sup(x1,...,xk )∈E k

|Φ(x1, . . . , xk)| & (µ(E ))1s ,

I(E ) :=

∫E k

|Φ(x1, . . . , xk)|dµ(x1) · · · dµ(xk) & (µ(E ))k+ 1s .

We call these inequalities “Nonconcentration Inequalities” becausethey dictate that product sets E k cannot be degenerate (asmeasured by Φ) when µ(E ) > 0.


Simple Observations

• The inequality

S(E ) := ess.supx1,...,xk∈E

|Φ(x1, . . . , xk)| & µ(E )1s

is strictly easier to prove than

I(E ) :=

∫E k

|Φ(x1, . . . , xk)|dµ(x1) · · · dµ(xk) & [µ(E )]k+ 1s

• Looking at small sets suggests that the diagonalx1 = · · · = xk = x is the important part; presumably Φ and itsderivatives through order Q − 1 vanish there for some Q ≥ 1.

• By a simple scaling argument, there is a particularly importantexponent s,

1

s=

Q

n, i.e., s =

n

Q.

One doesn’t expect either inequality for S(E ) or I(E ) to holdfor “nice” µ with larger values of s.


Basic Properties of Nonconcentration Functionals

Theorem 1 (G. 2018)

∀F S(F ) ≥ c[µ(F )]1s ⇔ ∀F I(F ) ≥ c ′[µ(F )]k+ 1

s

Theorem 2

• For s > nQ , only the zero measure satisfies the inequality.

• For s = nQ , there is a “best possible” choice of µ which comes

from a generalization of Hausdorff measure. It is possible toestimate the density (Think of relating Hausdorff andLebesgue measures on a curve.)


Basic Properties of Nonconcentration Functionals

Theorem 3: Frostman’s Lemma

Let weighted Φ-Hausdorff measure of dim. s, HsΦ(E ), be given by

lim infδ→0+

∑i

ci [S(Ei )]s

∣∣∣∣∣ χE ≤∑i

ciχEi, ci ≥ 0, diamEi ≤ δ

.

Suppose E is compact. Then HsΦ(E ) > 0 if and only if there exists

a nonzero, nonnegative Borel measure µ supported on E such that

I(F ) & [µ(F )]k+ 1s and S(F ) & [µ(F )]

1s

for all Borel sets F .

• A Good Question: What does this measure “measure”?

• Note: For fixed n there are many possible interesting values ofs because one can restrict to polynomial graphs in Rn.


Quick Proof of Theorem 3

• The proof of the Frostman Lemma (in terms of weightedΦ-Hausdorff measure) follows essentially identically the proof(due to Howroyd) found in Mattila’s book which usesHahn-Banach.

• Start with a homogeneous subadditive functional

pδ(f ) := inf

∑i

ci (S(Ei ))s∣∣∣∣∣ f ≤∑

i

ciχEi , ci ≥ 0, diam(Ei ) ≤ δ

• Extend the functional which equals pδ(χE ) > 0 on χE ; it’s got

to be a positive linear functional on continuous functions• Riesz Representation gives a measure which you (fix and)

check works out.

• What about the non-weighted generalization of Hausdorff? Inthe classical case they are comparable in every dimension (seeFederer’s book), but those arguments break down.

• In this specific case, comparability for dimension nQ follows

manually.


Upper Bounds on µ

S(E ) := ess.supx1,...,xk∈E

|Φ(x1, . . . , xk)| ≥ (µ(E ))Qn

puts obvious constraints on the size of µ. Assume µ is absolutelycontinuous with respect to Lebesgue measure.Pick a point x ∈ Rn, and let E = Br (x) as r → 0+. Recall weassume derivatives of order < Q of Φ vanish on the diagonal. LetP be the degree Q Taylor polynomial of Φ at (x , . . . , x). Then[

dµ

dx

]Q/n|B1(0)|Q/n ≤ sup

||x1||,...,||xk ||≤1|P(x1, . . . , xk)|.

We could use any coordinates with the same volume element.[dµ

dx

]Q/n|B1(0)|Q/n ≤ inf

G∈SL(n,R)|||ρGP|||

where ρG is natural action of SL(n,R) on polynomials and ||| · ||| issup norm on (B1(0))k .


3. Geometric Invariant Theory

General Algebraic Setup

• V : Finite-dimensional real vector space

• G : Real reductive algebraic group

• ρ : polynomial G-Representation on V

• ||| · ||| : Norm on V , ρ-invariant on maximal compact K < G.

Main Question

Given v ∈ V , how does one understand, compute,

infG∈G|||ρGv |||?

Key Idea

Study ρ-invariant polynomials on V .


SL(d ,R) Invariant Polynomials

Suppose M ∈ Rn×n. The Caley Ω operator is defined as

ΩM := det

∂

∂M11· · · ∂

∂M1n...

. . ....

∂∂Mn1

· · · ∂∂Mnn

.Basic Features:

• ΩM [f (AM)] = (detA) [(Ωf )(AM)].

• ΩM(detM)k = cn,k(detM)k−1, cn,k > 0 when k > 0.

Renyolds Operator

When ρ is a polynomial representation, we can explicitly write aprojection operator from polynomials on V to SL(n,R)-invariantpolynomials. For homogeneous f of fixed degree:

f 7→ c ΩkM [f ρM ]


Thm (Hilbert): The algebra of G-invariant polys is fin. gen’d.

Pf: Let I be the ideal generated by G-invariant homogeneous polys(nonconstant); there must be homogeneous G-invariant f1, . . . , fNgenerating I (unbdd N violates Noetherianity). If f is invariant,

f = ϕ1f1 + · · ·+ ϕN fN .

Apply Renyolds operator:

f = (Rϕ1)f1 + · · ·+ (RϕN)fNwhere the Rϕj are invariant and have lower degrees than f .

Computing the Infimum

If f1, . . . , fN are homogeneous and generate the algebra, then

infG∈G|||ρGv ||| ≈

N∑i=1

|fi (v)|1/ deg fi .

Proof: & is trivial; . is a compactness argument. Key point: iffi (v) = 0 for all i iff inf = 0 (aka Hilbert-Mumford Criterion).

A Good Question: Why do I care so much about the algebra?


The following are equivalent:

• RBL(π,N) > 0

• For all V ⊂ H,

dimV ≤m∑j=1

Njd

Ndjdimπj(V )

• There exists SL(H)× SL(H1)× · · · × SL(Hm)-invariantpolynomial f with f (0) = 0, f (ΠN) 6= 0.

• It is easy to prove that all SL(d ,R)-invariant polynomials ofM-linear forms must be expressible as d-linear contractions:

Ai1···id 7→∑σ∈Sd

(−1)σAσ1···σd .

In our case, these are known as “dotted bracket polynomials.”• Harder to know when two such polynomials are independent

and when you can stop looking.• To do analysis, it is sometimes hard to find easily-computable

polynomials, sometimes easier to work with the infimum.


4. Multilinear Kakeya and Lp-Improving Estimates

Consider a geometric averaging operator which integrates functionson Rn over k-dimensional algebraic submanifolds:Tf (x) :=

∫xΣ fdσ. Let ρ(x , y) = 0 be the incidence relation.

Theorem

For any nonnegative continuous functions f1, . . . , fm on Rn,

∫Rn

∫xΣ· · ·∫

xΣ[RBL(Dxρ)]

m(n−k)n

m∏j=1

fj(yj)dσ(y1) · · · dσ(ym)

nm(n−k)

dx

.m∏j=1

||fj ||n

m(n−k)

L1(Rn).

This is simply a continuous version of Zorin-Kranich’sKakeya-(Rogers)-Brascamp-Lieb inequality.


A machine to prove Lp-improving estimates I

1 Weighted Kakeya-Brascamp-Lieb is an inequality which reliestransversality of cotangent spaces. The Brascamp-Liebweight compensates for lack of transversality on the diagonal.

2 The stuff inside weighted Kakeya-Brascamp-Lieb is anonconcentration quantity. Precisely, pick m points y1, . . . , ymon xΣ (submanifold associated to x).

Φ(y1, . . . , ym) := (RBL(Dxρ(x , y1), . . . ,Dxρ(x , ym)))m(n−k)

n .

3 Curvature = Infinitesimal Transversality of CotangentSpaces

4 Extracting curvature effects reduces to provingnonconcentration inequality.

5 Exploit that BLW is effectively a polynomial in Dxρ(x , yi ).


A machine to prove Lp-improving estimates II

Benefits:

• “Good Transversality” and “Good Curvature” mean somepolynomial is nonzero. Consequently valid proofs of this sortfor example operators will automatically remain valid for theright kind of small algebraic perturbations.

• This way of packaging things avoids some seemingly verydifficult challenges posed by the method of inflation. Forexample, there are no longer any arithmetic constraints ondimension and codimension.

Challenges:

• There is potential to prove a very general weightedLp-improving inequality with this machinery, but there are anumber of additional obstacles to overcome.

• Comparing to Tao-Wright, it seems that the story is not yetfinished for multilinear Kakeya?

A Good Question: Can this be worked out in any concrete cases?


A machine to prove Lp-improving estimates III

• Example: convolution with measures on the surfacet1, . . . , tk ,

k∑

j=1

λij t2j

n−k

i=1

where k ≥ n/2 and in λij , all (n − k)× (n − k) minors ofcyclically adjacent columns are nondegenerate.

• Identify a workable degree of multilinearity and a workabletransversality polynomial. In this case, m = n copies of thesubmanifold works.

• Replace Brascamp-Lieb weight with this polynomial.• To prove the nonconcentration inequality, you must bound

transversality below in small ball limit in an effectivelyarbitrary coordinate system.

• The calculation still has high algebraic complexity despite theinitial reduction.


A nice invariant polynomial

detB1 B2

B2 B3

B3 B4 B5

B5. . .

Bk

Bk+1

Bk+2

Bk+3

. . .

Bn

A1 A2

A2 A3

A3 A4 A5

A5. . .

Ak

Ak+1

Ak+2

Ak+3

. . .An


Φ := det

• Each B1, . . . ,Bk needs n− k derivatives (let derivs act on cols;undifferentiated cols will be zero after column operations).

• In our example, the various t-coordinate functions all resideon their own rows. Differentiate with respect to the variablesthat cross the diagonal and argue that lower-priorityderivatives must always be zero (so coordinate independent).

• Conclusions: (N.B. N = m = n; Φ is degree n − k in ΠN)

• [RBL(Dxρ)]n−k ≈ infG|||ρGΠN |||n−k & |Φ(t(1), . . . , t(n))|

• supt(1),...,t(n)∈xΣ∩E

|Φ(t(1), . . . , t(n))| & |xΣ ∩ E |n−k .

• Convolution satisfies: ||TχE || 2n−kn−k

. |E |n

2n−k .


Where do things stand?

• Arbitrary dimension and codimension: Yes?

• Weighted inequalities: Some but likely not all• But even for curves, do we fully understand the implication

relationships between weighted estimates?

• Kakeya-fication of Brascamp-Lieb is only the first step?

• Better understanding of algebra and geometry is needed• There exists a contraction-type formula using Ω to compute all

invariants of fixed degree.• GIT people never tried to actually compute the infimum.• Is there a nice explicit formula for the Brascamp-Lieb constant?


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