Nonconcentration, L p -Improving Estimates, and Multilinear Kakeya Philip T. Gressman Department of Mathematics University of Pennsylvania 13 May 2019 Madison Lectures in Fourier Analysis N-C, L p , and M-K Philip T. Gressman 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 by smooth spaces with measures and maps between those spaces. • With minimal structure, it’s often not even clear what key quantities are or how to proceed. • Introducing artificial structures (e.g., coordinates) reduces to more familiar settings, but doing so breaks fundamental invariances of the problem and risks missing important features. • There is a third way: introduce artificial auxiliary structures and study the group action induced by transformations of these structures. N-C, L p , and M-K Philip T. Gressman 1 / 23
An Example: An n -dimensional vector space V is called a Peano space when it is equipped with a nontrivial alternating n -linear form [ v 1 , . . . , v n ]. It is natural to use bases with [ v 1 , . . . , v n ] = 1. Suppose U : V × V → R is a symmetric bilinear functional on a real Peano space V such that U ( v , v ) ≥ 0 for all v ∈ V . Is there an easy way to detect degeneracy or nondegeneracy of U ? • Option 1: Do calculations invariant under natural choices: det U := det[ U ( v i , v j )] i , j =1 ,..., n . • Option 2: Do anything, then optimize over all choices. E.g.: 1 / p n � | U ( v i , v i ) | p inf , p ∈ (0 , ∞ ] . [ v 1 ,..., v n ]=1 j =1 n p n � 1 | U ( v i , v i ) | p Theorem: det U = inf . n [ v 1 ,..., v n ]=1 j =1 N-C, L p , and M-K Philip T. Gressman 2 / 23
1. Brascamp-Lieb Constant H , H j : Hilbert(?) spaces of dimension d , d j , j = 1 , . . . , m ; π j : Surjective linear maps H → H j ; θ j : constants in [0 , 1]. Brascamp-Lieb Inequality �� � θ j � m m � � ( f j ◦ π j ) θ j ≤ RBL( π, θ ) f j H H j j =1 j =1 Bennett, Carbery, Christ, and Tao (2005) RBL( π, θ ) > 0 if and only if m � dim V ≤ θ j dim π j ( V ) for all V ⊂ H j =1 with equality when V = H . Bennett, Bez, Cowling, Flock (2016) Fixing dimensions and θ , RBL( π, θ ) is continuous in π . N-C, L p , and M-K Philip T. Gressman 3 / 23
�� � θ j � m m � � ( f j ◦ π j ) θ j ≤ RBL( π, θ ) f j H H j j =1 j =1 Lieb (1990): Gaussians extremize the inequality � � 1 det � m 2 j =1 θ j π ∗ j A ∗ j A j π j RBL( π, θ ) = inf � m j =1 | det H j A j | θ j A j ∈ GL( H j ) j =1 ,..., m Change determinant to infimum of trace: d − 1 tr � m j =1 θ j A ∗ π ∗ j A ∗ j A j π j A 2 d = [RBL( π, θ )] inf � m j =1 | det H j A j | 2 θ j / d A j ∈ GL( H j ) A ∈ SL( H ) m � 2 d − 1 t − 2 θ j ||| A j π j A ||| 2 dj = inf θ j t d A j ∈ SL( H j ) j =1 A ∈ SL( H ) t j ∈ (0 , ∞ ) where ||| · ||| is the Hilbert-Schmidt (sum of squares) matrix norm. N-C, L p , and M-K Philip T. Gressman 4 / 23
Keep Going: Use AM-GM Inequality again to eliminate t j : � d � m � 2 θ j d j 2 d − 1 t − 2 θ dj ||| A j π j A ||| 2 d = [RBL( π, θ )] inf d t d j d j A j ∈ SL( H j ) j =1 A ∈ SL( H ) t j ∈ (0 , ∞ ) m m � � θ j dj 2 θ j dj − = d d inf ||| A j π j A ||| d j A j ∈ SL( H j ) j =1 j =1 A ∈ SL( H ) Assuming rational θ j , there exist integers N , N j such that θ j d j = N j N , j = 1 , . . . , m , d m m � � Nj N − d = ||| A j π j A ||| N j . [RBL( π, θ )] d 2 inf j A j ∈ SL( H j ) j =1 j =1 A ∈ SL( H ) N-C, L p , and M-K Philip T. Gressman 5 / 23
For integers N = N 1 + · · · + N m , � � � � � � � � m m � � � � � � N � � � � [RBL( π, N )] f j ◦ π j ≤ || f j || L dj / Nj ( H j ) . d � � � � � � � � j =1 j =1 L d / N ( H ) Define Π N : H N × H N 1 × · · · × H N m → R by the formula 1 m Π N ( x (1) , . . . , x ( N ) , x (1) 1 , . . . , x ( N 1 ) , . . . , x ( N m ) ) m 1 � � � � � � π 1 x (1) , x (1) π 1 x ( N 1 ) , x ( N 1 ) π m x ( N ) , x ( N m ) := · · · · · · m 1 1 H 1 H 1 H m and let G := SL( H ) × SL( H 1 ) × · · · × SL( H m ). Then m � Nj N − inf d = [RBL( π, N )] d 2 G ∈G ||| ρ G Π N ||| , j j =1 ρ G is the action of G on H N × · · · × H N m m , ||| · ||| is Hilbert-Schmidt. A Good Question: Why did we do this lovely calculation? N-C, L p , and M-K Philip T. Gressman 6 / 23
2. Geometric Nonconcentration Inequalities Suppose Φ is some polynomial function from ( R n ) k into R m . | Φ( x 1 , . . . , x k ) | measures nondegeneracy of k -point configurations. Example: if ϕ ( x ) := ( x α ) | α |≤ d , then Φ( x 1 , . . . , x N ) := det( ϕ ( x 1 ) , . . . , ϕ ( x N )) = 0 iff x 1 , . . . , x N 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 1 s , S ( E ) := ( x 1 ,..., x k ) ∈ E k | Φ( x 1 , . . . , x k ) | � ( µ ( E )) ess.sup � E k | Φ( x 1 , . . . , x k ) | d µ ( x 1 ) · · · d µ ( x k ) � ( µ ( E )) k + 1 s . I ( E ) := We call these inequalities “Nonconcentration Inequalities” because they dictate that product sets E k cannot be degenerate (as measured by Φ) when µ ( E ) > 0. N-C, L p , and M-K Philip T. Gressman 7 / 23
Simple Observations • The inequality 1 S ( E ) := ess.sup | Φ( x 1 , . . . , x k ) | � µ ( E ) s x 1 ,..., x k ∈ E is strictly easier to prove than � E k | Φ( x 1 , . . . , x k ) | d µ ( x 1 ) · · · d µ ( x k ) � [ µ ( E )] k + 1 I ( E ) := s • Looking at small sets suggests that the diagonal x 1 = · · · = x k = x is the important part; presumably Φ and its derivatives through order Q − 1 vanish there for some Q ≥ 1. • By a simple scaling argument, there is a particularly important exponent s , 1 s = Q n , i.e., s = n Q . One doesn’t expect either inequality for S ( E ) or I ( E ) to hold for “nice” µ with larger values of s . N-C, L p , and M-K Philip T. Gressman 8 / 23
Basic Properties of Nonconcentration Functionals Theorem 1 (G. 2018) 1 s ⇔ ∀ F I ( F ) ≥ c ′ [ µ ( F )] k + 1 ∀ F S ( F ) ≥ c [ µ ( F )] s Theorem 2 • For s > n Q , only the zero measure satisfies the inequality. • For s = n Q , there is a “best possible” choice of µ which comes from a generalization of Hausdorff measure. It is possible to estimate the density (Think of relating Hausdorff and Lebesgue measures on a curve.) N-C, L p , and M-K Philip T. Gressman 9 / 23
Basic Properties of Nonconcentration Functionals Theorem 3: Frostman’s Lemma Let weighted Φ-Hausdorff measure of dim. s , � H s Φ ( E ), be given by � �� � � � � c i [ S ( E i )] s lim inf � χ E ≤ c i χ E i , c i ≥ 0 , diam E i ≤ δ . � δ → 0 + i i Suppose E is compact. Then � H s Φ ( E ) > 0 if and only if there exists a nonzero, nonnegative Borel measure µ supported on E such that I ( F ) � [ µ ( F )] k + 1 1 s and S ( F ) � [ µ ( F )] s for all Borel sets F . • A Good Question: What does this measure “measure”? • Note: For fixed n there are many possible interesting values of s because one can restrict to polynomial graphs in R n . N-C, L p , and M-K Philip T. Gressman 10 / 23
Quick Proof of Theorem 3 • The proof of the Frostman Lemma (in terms of weighted Φ-Hausdorff measure) follows essentially identically the proof (due to Howroyd) found in Mattila’s book which uses Hahn-Banach. • Start with a homogeneous subadditive functional � �� � � � � c i ( S ( E i )) s p δ ( f ) := inf � f ≤ c i χ E i , c i ≥ 0 , diam( E i ) ≤ δ � i i • Extend the functional which equals p δ ( χ E ) > 0 on χ E ; 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? In the classical case they are comparable in every dimension (see Federer’s book), but those arguments break down. • In this specific case, comparability for dimension n Q follows manually. N-C, L p , and M-K Philip T. Gressman 11 / 23
Upper Bounds on µ Q S ( E ) := ess.sup | Φ( x 1 , . . . , x k ) | ≥ ( µ ( E )) n x 1 ,..., x k ∈ E puts obvious constraints on the size of µ . Assume µ is absolutely continuous with respect to Lebesgue measure. Pick a point x ∈ R n , and let E = B r ( x ) as r → 0 + . Recall we assume derivatives of order < Q of Φ vanish on the diagonal. Let P be the degree Q Taylor polynomial of Φ at ( x , . . . , x ). Then � d µ � Q / n | B 1 (0) | Q / n ≤ sup | P ( x 1 , . . . , x k ) | . dx || x 1 || ,..., || x k ||≤ 1 We could use any coordinates with the same volume element. � d µ � Q / n | B 1 (0) | Q / n ≤ G ∈ SL( n , R ) ||| ρ G P ||| inf dx where ρ G is natural action of SL( n , R ) on polynomials and ||| · ||| is sup norm on ( B 1 (0)) k . N-C, L p , and M-K Philip T. Gressman 12 / 23
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