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

Oscillatory 1 Philip T. Gressman, University of Pennsylvania multilinear Joint work with Ellen Urheim Radon-like transforms In honor of the 90th birthday of Guido Weiss

2 Problem Statement This falls within the framework of Christ, Li, Tao, and Thiele when ρ is affine Let ρ and Φ be real-valued functions on linear. some neighborhood U of the origin in R 2 d and suppose that ∇ ρ is nonvanishing there. Let M { x ∈ U | ρ ( x ) = 0 } and let σ be := Similar objects arise in recent work of Lebesgue measure on M . The main object: Christ on "best of the best" decay rates. 2 d ∫ Here we sacrifice ∏ e iλ Φ( x ) I λ ( f 1 , . . . , f 2 d ) := f j ( x j ) dσ ( x ) . some decay for M stability. j =1 We wish to understand the asymptotic behav- Our methods are in ior of the norm as λ → ∞ when f j ∈ L p j ( R ) . Of many ways classical but there are a few particular concern: interesting deviations. • Obtaining sharp decay in excess of | λ | − 1 / 2 • Stability results

3 Theorem. Suppose that there is a positive con- The condition is a hybrid of the typical stant c such that for every x ∈ U and every Hessian condition for τ, τ ) ∈ R 2 such that ˜ Φ and rotational τ 2 + τ 2 = 1 , the indices (˜ curvature condition { 1 , . . . , 2 d } may be partitioned into two sets for ρ . Think of it as an inhomogeneous { i 1 , . . . , i d } and { j 1 , . . . , j d } such that FIO. No faster decay in λ is possible for these ∂ i 1 ρ ( x ) ∂ 2 ∂ 2 � �  i 1 j 1 (˜ τ Φ( x ) + τρ ( x )) · · · i 1 j d (˜ τ Φ( x ) + τρ ( x ))  � � exponents. In some . . . ... . . . � � . . . �   � cases we know faster det ≥ c �   � ∂ i d ρ ( x ) ∂ 2 τ Φ( x ) + τρ ( x )) · · · ∂ 2 i d j 1 (˜ i d j d (˜ τ Φ( x ) + τρ ( x )) � � decay is impossible   � � 0 ∂ j 1 ρ ( x ) · · · ∂ j d ρ ( x ) � � for any exponents. Then for all f 1 , . . . , f 2 d ∈ L 2 ( R ) , The switching around 2 d of indices { i 1 , . . . , i d } , | I λ ( f 1 , . . . , f 2 d ) | ≲ | λ | − d − 1 ∏ || f j || 2 . 2 { j 1 , . . . , j d } comes from multilinearity j =1 and yields nontrivial Proof: well-chosen wave packets. Hörmander results when linear objects must at low freq and Radon at high freq. degenerate.

4 Sharpness and Stability of For multilinearity of order 2 d , the analogous best 2 d | I λ ( f 1 , . . . , f 2 d ) | ≲ | λ | − d − 1 ∏ Christ operator has || f j || 2 : 2 decay like | λ | − d +1 . It's essentially an j =1 iterated Fourier • It's relatively easy to show that no better transform. decay is faster on products of L 2 ( R ) by us- Related to Christ's ing standard Knapp-type arguments. "Best of the Best": it seems quite hard to tell what the best • The hypotheses are stable, so remain true possible stable decay for sufficiently small smooth perturbations rate is, but no better √ than | λ | − d + d − O (1) of Φ and ρ . This is quite different than (still a huge gap). overdetermined CLTT case: Algebra is hard and incomplete: How low 2 d ∫ does the rank go ∏ e iλ Φ( x ) f j ( x j ) dσ ( x ) generically for real x 1 + ··· + x 2 d = ϵ Φ linear combinations j =1 of two symmetric has no decay in λ for any ϵ ̸ = 0 . matrices and diagonal matrices?

5 Main Decomposition Oscillatory integral operators 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 • Divide frequency space into tiles such that extremes and almost diagonalizes the tile containing ξ has diameter ∼ | ξ | 1 / 2 . operator. Kernel decay is so good • Resize innermost boxes so that none is there's no need for TT ∗ . smaller than | λ | 1 / 2 .

6 Technical Details For our purposes, this continuous decomposition is easiest to work with. There exists a map f ( x ) �→ V f ( x, ξ ) such that ∫ f ( x ) = R d × R d V f ( y, ξ ) φ ξ ( x − y ) dydξ, Can the formula can be discretized? Work of Hernández, Labate, Weiss, and || V f || L 2 ( R d × R d ) ≈ || f || L 2 ( R d ) , Wilson (ACHA 2004) suggest no tight frames will exist. | α | 2 + 1 � ≲ (max {| ξ | , | λ |} ) � ∂ α � x ( e − 2 πix · ξ φ ξ ( x )) � 4 , Even without λ correction, such a decomposition nearly diagonalizes φ ξ ( x ) = 0 when | x | ≳ (max {| ξ | , | λ |} ) − 1 Radon-like averages 2 . over hypersurfaces with nonvanishing rotational curvature.

7 Substitute in the expansion: The vector fields X i are orthonormal and span the tangent 2 d ∫ space of ∏ e iλ Φ( x ) I λ ( f 1 , . . . , f 2 d ) := f j ( x j ) dσ ( x ) M = { y : ρ ( y ) = 0 } M at every point. j =1 2 d ∫ ∏ = R 2 d × R 2 d I ( y, ξ ) V f j ( y j , ξ j ) dydξ We use a minor j =1 variation on the usual integration by For convenience, let r j := (max {| λ | , | ξ j |} ) − 1 / 2 . parts technique for I ( y, ξ ) is supported on the region | ρ ( y ) | ≲ stationary phase to make things a little max j r j . Stationary phase gives: cleaner. − N ( 2 d ) 1  2  If the r j ̸≈ r j ′ then 2 d  1 + (min j r j ) 2 1 ∑ ∏ | X i ( λ Φ + 2 πξ · x ) | y | 2 r − 1 |I ( y, ξ ) | ≲ r | ξ | + | ξ ′ | ≳ | λ | . As long 2  j 0 j max j r j as M is transverse to i =1 j =1 all coordinate for every j 0 ∈ { 1 , . . . , 2 d } . directions, the phase is highly nonstationary and |I ( y, ξ ) | is very small.

8 Having good control For convenience, let r := (max {| λ | , | ξ |} ) − 1 / 2 . on | τ | + | λ | is important because it − N ( 2 d ) 1  2  will come up as a ∑ | X i ( λ Φ + 2 πξ · x ) | y | 2 r d − 1 χ | ρ ( y ) | ≲ r |I ( y, ξ ) | ≲  1 + r  Jacobian i =1 determinant of some change of variables. Now manually add another derivative in the direction of ∇ ρ ( y ) to make life easier: So far, there's nothing special ∫ (1 + r | λ ∇ Φ( y ) + τ ∇ ρ ( y ) + 2 πξ ) | ) − N r d χ | ρ ( y ) | ≲ r dτ |I ( y, ξ ) | ≲ about multilinearity; the analysis proceeds similarly Again, we may assume | τ | + | λ | ≈ | ξ | + | λ | ≈ r − 2 for fewer functions of higher dimensions. as otherwise the phase is nonstationary. The quantity to the ∏ 2 d r d χ | ρ ( y ) | ≲ r j =1 | V f j ( y j , ξ j ) | ∫ left is similar to (1 + r |∇ ( λ Φ( y ) + τρ ( y )) + 2 πξ | ) N dydξdτ things you'd get from TT ∗ but it has better E localization and where E := {| τ | + | λ | ≈ | ξ | + | λ | ≈ r − 2 } . Minor works better with decomposition after nuisance that r depends on ξ . the fact.

9 Good news: freezing ∏ 2 d r d χ | ρ ( y ) | ≲ r j =1 | V f j ( y j , ξ j ) | a ξ j essentially ∫ (1 + r |∇ ( λ Φ( y ) + τρ ( y )) + 2 πξ | ) N dydξdτ freezes r , so it behaves like a E constant even Now interpolate: Put half of the V f j ∈ L ∞ ( R × though it isn't. R ) and the others in L 1 ( R × R ) . Reduces to estimating an integral Bad News: For fixed choice of i 1 , . . . , j d , r d χ | ρ ( y ) | ≲ r dy j 1 · · · dy j d dτ ∫ there are rarely any instances where the (1 + r | P i 1 ··· i d ( ∇ ( λ Φ( y ) + τρ ( y )) + 2 πξ ) | ) N change of variables E is nonsingular. where i 1 , . . . , i d , j 1 , . . . , j d are distinct. This in- tegral can be understood via change of vars: Good News: Because our estimate is ( y j 1 , . . . , y j d , τ ) �→ pointwise, we can chop up the domain ( ∂ i 1 ( λ Φ( y ) + τρ ( y )) , as we wish and do the interpolation . . . , ∂ i d ( λ Φ( y ) + τρ ( y )) , ρ ( y )) different ways on different pieces.

10 Jacobian determinant of No need for Φ , ρ to be polynomials: noncompactness of domain in τ can be ( τ, y j 1 , . . . , y j d ) �→ ( ∂ i 1 ( λ Φ( y ) + τρ ( y )) , . . . , ∂ i d ( λ Φ( y ) + τρ ( y )) , ρ ( y )) handled by scaling is properties of the system of equations. ∂ i 1 ρ ( y ) ∂ 2 ∂ 2  i 1 j 1 ( λ Φ( y ) + τρ ( y )) · · · i 1 j d ( λ Φ( y ) + τρ ( y ))  . . . ... . . . . . .   det  .   ∂ i d ρ ( y ) ∂ 2 i d j 1 ( λ Φ( y ) + τρ ( y )) · · · ∂ 2 i d j d ( λ Φ( y ) + τρ ( y )) Special cases: τ = 0  0 ∂ j 1 ρ ( y ) · · · ∂ j d ρ ( y ) implies infinitesimal CLTT-type Det is homogeneous of degree ( d − 1) in the nondegeneracy. λ = 0 is rotational pair ( λ, τ ) . For even d and any fixed y , it must curvature. vanish along some line in ( λ, τ ) . BUT if y = ( s, t, u, v ) ∈ ( R d/ 2 ) 4 , then It should be possible to study the singular Φ( s, t, u, v ) = s · t + u · v linear operator as ρ ( s, t, u, v ) = − → well. The simplest 1 · ( s + t + u + v ) + s · u + t · v singularities are like has one partition with determinant cτ d − 1 and inhomogeneous folds and may not one with c ′ λ d − 1 , so it's all good. need new machinery.