Oscillatory 1 Philip T. Gressman, University of Pennsylvania - - PowerPoint PPT Presentation

Oscillatory multilinear Radon-like transforms

Philip T. Gressman, University of Pennsylvania Joint work with Ellen Urheim In honor of the 90th birthday of Guido Weiss

1

Problem Statement

Let ρ and Φ be real-valued functions on some neighborhood U of the origin in R2d and suppose that ∇ρ is nonvanishing there. Let M := {x ∈ U | ρ(x) = 0} and let σ be Lebesgue measure on M. The main object: Iλ(f1, . . . , f2d) := ∫

M

eiλΦ(x)

2d

∏

j=1

fj(xj)dσ(x). We wish to understand the asymptotic behav- ior of the norm as λ → ∞ when fj ∈ Lpj(R). Of particular concern:

• Obtaining sharp decay in excess of |λ|−1/2
• Stability results

This falls within the framework of Christ, Li, Tao, and Thiele when ρ is affine linear. Similar objects arise in recent work of Christ on "best of the best" decay rates. Here we sacrifice some decay for stability. Our methods are in many ways classical but there are a few interesting deviations.

2

• Theorem. Suppose that there is a positive con-

stant c such that for every x ∈ U and every (˜ τ, τ) ∈ R2 such that ˜ τ2 + τ2 = 1, the indices {1, . . . , 2d} may be partitioned into two sets {i1, . . . , id} and {j1, . . . , jd} such that

• det

    ∂i1ρ(x) ∂2

i1j1(˜

τΦ(x) + τρ(x)) · · · ∂2

i1jd(˜

τΦ(x) + τρ(x)) . . . . . . ... . . . ∂idρ(x) ∂2

idj1(˜

τΦ(x) + τρ(x)) · · · ∂2

idjd (˜

τΦ(x) + τρ(x)) ∂j1ρ(x) · · · ∂jdρ(x)    

• ≥ c

Then for all f1, . . . , f2d ∈ L2(R), |Iλ(f1, . . . , f2d)| ≲ |λ|−d−1

2

2d

∏

j=1

||fj||2. Proof: well-chosen wave packets. Hörmander at low freq and Radon at high freq.

The condition is a hybrid of the typical Hessian condition for Φ and rotational curvature condition for ρ. Think of it as an inhomogeneous FIO. No faster decay in λ is possible for these

• exponents. In some

cases we know faster decay is impossible for any exponents. The switching around

• f indices {i1, . . . , id},

{j1, . . . , jd} comes from multilinearity and yields nontrivial results when linear

• bjects must

degenerate.

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Sharpness and Stability of |Iλ(f1, . . . , f2d)| ≲ |λ|−d−1

2

2d

∏

j=1

||fj||2 :

• It's relatively easy to show that no better

decay is faster on products of L2(R) by us- ing standard Knapp-type arguments.

• The hypotheses are stable, so remain true

for sufficiently small smooth perturbations

• f Φ and ρ.

This is quite different than

• verdetermined CLTT case:

∫

x1+···+x2d=ϵΦ

eiλΦ(x)

2d

∏

j=1

fj(xj)dσ(x) has no decay in λ for any ϵ ̸= 0.

For multilinearity of

• rder 2d, the

analogous best Christ operator has decay like |λ|−d+1. It's essentially an iterated Fourier transform. Related to Christ's "Best of the Best": it seems quite hard to tell what the best possible stable decay rate is, but no better than |λ|−d+

√ d−O(1)

(still a huge gap). Algebra is hard and incomplete: How low does the rank go generically for real linear combinations

• f two symmetric

matrices and diagonal matrices?

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Main Decomposition

• Divide frequency space into tiles such that

tile containing ξ has diameter ∼ |ξ|1/2.

• Resize innermost boxes so that none is

smaller than |λ|1/2.

Oscillatory integral

• perators are

well-adapted to packet bases with uniform frequency resolution ∼ |λ|1/2, i.e., Gabor-like decompositions. Radon-like transforms are often studied in Littlewood-Paley- type decompositions, where frequency resolution at frequency ξ is like |ξ|. This basis falls halfway between extremes and almost diagonalizes the

• perator. Kernel

decay is so good there's no need for TT ∗.

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Technical Details

There exists a map f (x) → V f (x, ξ) such that f (x) = ∫

Rd×Rd V f (y, ξ)φξ(x − y) dydξ,

||V f ||L2(Rd×Rd) ≈ ||f ||L2(Rd),

• ∂α

x (e−2πix·ξφξ(x))

• ≲ (max{|ξ|, |λ|})

|α| 2 +1 4,

φξ(x) = 0 when |x| ≳ (max{|ξ|, |λ|})−1

2.

For our purposes, this continuous decomposition is easiest to work with. Can the formula can be discretized? Work

• f Hernández,

Labate, Weiss, and Wilson (ACHA 2004) suggest no tight frames will exist. Even without λ correction, such a decomposition nearly diagonalizes Radon-like averages

• ver hypersurfaces

with nonvanishing rotational curvature.

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Substitute in the expansion: Iλ(f1, . . . , f2d) := ∫

M

eiλΦ(x)

2d

∏

j=1

fj(xj)dσ(x) = ∫

R2d×R2d I(y, ξ) 2d

∏

j=1

V fj(yj, ξj)dydξ For convenience, let rj := (max{|λ|, |ξj|})−1/2. I(y, ξ) is supported on the region |ρ(y)| ≲ maxj rj. Stationary phase gives:

|I(y, ξ)| ≲  1 + (minj rj)2 maxj rj ( 2d ∑

i=1

|Xi(λΦ + 2πξ · x)|y|2 ) 1



−N

r −1

j0 2d

∏

j=1

r

j

for every j0 ∈ {1, . . . , 2d}.

The vector fields Xi are orthonormal and span the tangent space of M = {y : ρ(y) = 0} at every point. We use a minor variation on the usual integration by parts technique for stationary phase to make things a little cleaner. If the rj ̸≈ rj′ then |ξ|+|ξ′| ≳ |λ|. As long as M is transverse to all coordinate directions, the phase is highly nonstationary and |I(y, ξ)| is very small.

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For convenience, let r := (max{|λ|, |ξ|})−1/2.

|I(y, ξ)| ≲  1 + r ( 2d ∑

i=1

|Xi(λΦ + 2πξ · x)|y|2 ) 1



−N

r d−1χ|ρ(y)|≲r

Now manually add another derivative in the direction of ∇ρ(y) to make life easier:

|I(y, ξ)| ≲ ∫ (1 + r|λ∇Φ(y) + τ∇ρ(y) + 2πξ)|)−Nr dχ|ρ(y)|≲rdτ

Again, we may assume |τ|+|λ| ≈ |ξ|+|λ| ≈ r −2 as otherwise the phase is nonstationary. ∫

E

r dχ|ρ(y)|≲r ∏2d

j=1 |V fj(yj, ξj)|

(1 + r|∇(λΦ(y) + τρ(y)) + 2πξ|)Ndydξdτ where E := {|τ| + |λ| ≈ |ξ| + |λ| ≈ r −2}. Minor nuisance that r depends on ξ.

Having good control

• n |τ| + |λ| is

important because it will come up as a Jacobian determinant of some change of variables. So far, there's nothing special about multilinearity; the analysis proceeds similarly for fewer functions of higher dimensions. The quantity to the left is similar to things you'd get from TT ∗ but it has better localization and works better with decomposition after the fact.

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∫

E

r dχ|ρ(y)|≲r ∏2d

j=1 |V fj(yj, ξj)|

(1 + r|∇(λΦ(y) + τρ(y)) + 2πξ|)Ndydξdτ Now interpolate: Put half of the V fj ∈ L∞(R × R) and the others in L1(R × R). Reduces to estimating an integral ∫

E

r dχ|ρ(y)|≲rdyj1 · · · dyjddτ (1 + r|Pi1···id(∇(λΦ(y) + τρ(y)) + 2πξ)|)N where i1, . . . , id, j1, . . . , jd are distinct. This in- tegral can be understood via change of vars: (yj1, . . . , yjd, τ) → (∂i1(λΦ(y) + τρ(y)), . . . , ∂id(λΦ(y) + τρ(y)), ρ(y))

Good news: freezing a ξj essentially freezes r, so it behaves like a constant even though it isn't. Bad News: For fixed choice of i1, . . . , jd, there are rarely any instances where the change of variables is nonsingular. Good News: Because

• ur estimate is

pointwise, we can chop up the domain as we wish and do the interpolation different ways on different pieces.

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Jacobian determinant of

(τ, yj1, . . . , yjd) → (∂i1(λΦ(y) + τρ(y)), . . . , ∂id(λΦ(y) + τρ(y)), ρ(y))

is

det     ∂i1ρ(y) ∂2

i1j1(λΦ(y) + τρ(y)) · · ·

∂2

i1jd(λΦ(y) + τρ(y))

. . . . . . ... . . . ∂idρ(y) ∂2

idj1(λΦ(y) + τρ(y)) · · · ∂2 idjd (λΦ(y) + τρ(y))

∂j1ρ(y) · · · ∂jdρ(y)     .

Det is homogeneous of degree (d − 1) in the pair (λ, τ). For even d and any fixed y, it must vanish along some line in (λ, τ). BUT if y = (s, t, u, v) ∈ (Rd/2)4, then Φ(s, t, u, v) = s · t + u · v ρ(s, t, u, v) = − → 1 · (s + t + u + v) + s · u + t · v has one partition with determinant cτd−1 and

• ne with c′λd−1, so it's all good.

No need for Φ, ρ to be polynomials: noncompactness of domain in τ can be handled by scaling properties of the system of equations. Special cases: τ = 0 implies infinitesimal CLTT-type nondegeneracy. λ = 0 is rotational curvature. It should be possible to study the singular linear operator as

• well. The simplest

singularities are like inhomogeneous folds and may not need new machinery.

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