The endpoint multilinear Kakeya theorem via the - PowerPoint PPT Presentation

. . . . . . . . . . . . . . . . The endpoint multilinear Kakeya theorem via the Borsuk–Ulam/Lusternik–Schnirelmann theorem After L. Guth, A. Carbery, I. Valdimarsson Pavel Zorin-Kranich University of Bonn Kopp, October 2017 . . . . . . . . . . . . . . . . . . . . . . . . 1

. . . . . . . . . . . . . . . . Main theorem Definition Theorem (Multilinear Kakeya/perturbed Loomis–Whitney) dx Here and later implicit constants depend only on n . . . . . . . . . . . . . . 2 . . . . . . . . . . . A 1 -tube T ⊂ R n is the 1 -neighborhood of a straight doubly infinite line in the direction e ( T ) ∈ S n − 1 . Let T 1 , · · · , T n be families of 1 -tubes in R n such that e ( T j ) is close to the basis vector e j for T j ∈ T j . 1/( n − 1)   ∫  ∑ ∑ χ T 1 ( x ) · · · χ T n ( x )  R n T 1 ∈T 1 T n ∈T n ≲ ( # T · · · # T n ) 1/( n − 1) .

Q M Q S j Q S j Q # j . j . . . . . . . . Main theorem, discrete version Equivalent formulation: for every M Then j . with there exist S j with G Q M Q n j n Q . . . . . . . . . . . . . . . . . . . 3 . . . . . . . . . . . Let Q denote the lattice of dyadic cubes of unit size. Let also ) 1/( n − 1) ( ∏ # { T j ∈ T j | T j ∩ Q ̸ = ∅} G ( Q ) = . ∑ ∏ ( # T j ) 1/( n − 1) . G ( Q ) ≲ Q ∈Q

. . . . . . . . . . . . . . . . Main theorem, discrete version j Then j j Q . . . . . . . . . . . . . 3 . . . . . . . . . . . Let Q denote the lattice of dyadic cubes of unit size. Let also ) 1/( n − 1) ( ∏ # { T j ∈ T j | T j ∩ Q ̸ = ∅} G ( Q ) = . ∑ ∏ ( # T j ) 1/( n − 1) . G ( Q ) ≲ Q ∈Q Equivalent formulation: for every M : Q → R + with ∑ Q M ( Q ) = 1 there exist S j : Q → R + with G ( Q ) M ( Q ) 1/( n − 1) ≲ ∏ S j ( Q ) 1/( n − 1) , ∑ S j ( Q ) ≲ # T j .

. . . . . . . . . . . . Q . Q Q n (hypothesis) n Q (Hölder) n (hypothesis) Not much happened, but cross-interaction and self-interaction are separated. . Proof of equivalence (needed direction) . . . . . . . . . . . . . . 4 . . . . . . . . . . . . Let G := ∑ Q G ( Q ) and M ( Q ) = G ( Q )/ G . Then ) n /( n − 1) ( G − 1/ n ∑ G = G ( Q ) G ( Q ) ( n − 1)/ n M ( Q ) 1/ n ) n /( n − 1) ( ∑ = S j ( Q ) 1/ n ) n /( n − 1) ( ∑ ∏ ≲ j =1 ) 1/( n − 1) ∏ ( ∑ ≤ S j ( Q ) j =1 ) 1/( n − 1) . ∏ ( ≲ # T j j =1

S j Q T j p Q e T j . . . . . . . . . . . . . . Ansatz for tubes Theorem j Wlog M compactly supported. Will find polynomial p of degree and set Add this to handout! . . . . . . . . . . . . . . 5 . . . . . . . . . . . . ∑ S j ( Q ) = S j ( Q , T j ) T ∈T j For every function M : Q → R + with ∑ M = 1 there exist S j : Q × T j → R + with ∏ S j ( Q , T j ) if T j ∩ Q ̸ = ∅ , M ( Q ) ≲ ∑ S j ( Q , T j ) ≲ 1 for each T j ∈ T j . Q ∈Q : T j ∩ Q ̸ = ∅

. . . . . . . . . . . . . . . . Ansatz for tubes Theorem j and set Add this to handout! . . . . . . . . . . . . . 5 . . . . . . . . . . . ∑ S j ( Q ) = S j ( Q , T j ) T ∈T j For every function M : Q → R + with ∑ M = 1 there exist S j : Q × T j → R + with ∏ S j ( Q , T j ) if T j ∩ Q ̸ = ∅ , M ( Q ) ≲ ∑ S j ( Q , T j ) ≲ 1 for each T j ∈ T j . Q ∈Q : T j ∩ Q ̸ = ∅ Wlog M compactly supported. Will find polynomial p of degree ≲ λ S j ( Q , T j ) := λ − 1 s p , Q ( e ( T j )) .

p Q e T N x d Z p T . . . . . . . . . . . mollified version instead to apply a topological result) Directional surface area (small lie: this is not a continuous function of p , so one has to use a . Guth’s tube estimate: Q T Q e T n x deg p This takes care of the self-interaction term. . . . . . . . . . . . . . . . . . 6 . . . . . . . . . . . Let Z p be the zero set of the polynomial p . Let ∫ |⟨ v , N x ⟩| d H n − 1 ( x ) , s p , Q ( v ) := N x normal unit vector . Z p ∩ Q

. . . . . . . . . . . . . . . . Directional surface area (small lie: this is not a continuous function of p , so one has to use a mollified version instead to apply a topological result) Guth’s tube estimate: T This takes care of the self-interaction term. . . . . . . . . . . . . . 6 . . . . . . . . . . . Let Z p be the zero set of the polynomial p . Let ∫ |⟨ v , N x ⟩| d H n − 1 ( x ) , s p , Q ( v ) := N x normal unit vector . Z p ∩ Q ∫ ∑ |⟨ e ( T ) , N x ⟩| d H n − 1 ( x ) ≲ deg p . s p , Q ( e ( T )) ≤ Z p ∩ ˜ Q ∈Q , T ∩ Q ̸ = ∅

. . . . . . . . . . . . . . . . Cross-interaction term j j by transversality Want to find polynomial p with . . . . . . . . . . . . . 7 . . . . . . . . . . . Let now T j ∈ T j be tubes and Q ∈ Q . Then ∏ S ( Q , T j ) = λ − n ∏ s p , Q ( e ( T j )) ) − 1 ∼ λ − n ( vol conv (0 , e ( T j )/ s p , Q ( e ( T j ))) ) − 1 ≳ λ − n ( vol B s p , Q , where B is the unit ball of the norm s . ) − 1 ( λ n M ( Q ) ≲ vol B s p , Q

. . . . . . . . . . . . . . . . Visibility Want to find polynomial p with Notice that Q is approximately the dimension of the space of polynomials of degree . . . . . . . . . . . . . . . . . . . 8 . . . . . ) − 1 ( ˜ M ( Q ) := λ n M ( Q ) ≲ =: Vis p , Q . vol B s p , Q Small lie: we pretend that B s p , Q ⊂ B for all Q with M ( Q ) ̸ = 0 . For this we need λ to be large enough; this is how we choose λ . ∑ ˜ M ( Q ) = λ n ≲ λ in n variables. Let P ∗ be the unit sphere in this space.

. . . . . . . . . . . . . . . . Naive approach to maximizing visibility Imagine for the moment that the directional surface area is isotropic, In this case: 2. Use the polynomial ham sandwich theorem to find p of degree isoperimetric inequality. . . . . . . . . . . . . . . . . . . . 9 . . . . . i.e. s p , Q ( v ) ∼ s p , Q ∥ v ∥ for all p , Q , where s p , Q is the usual surface area. 1. Fit into each cube Q approximately ˜ M ( Q ) disjoint balls of measure ˜ M ( Q ) − 1 . ≲ λ that bisects all these balls. 3. In each ball Z p has surface area at least ˜ M ( Q ) − ( n − 1)/ n by the 4. Summing up gives s p , Q ≳ ˜ M ( Q ) 1/ n , hence B s p , Q ⊂ ˜ M ( Q ) − 1/ n B , hence Vis p , Q ≳ ˜ M ( Q ) . Problem: s p , Q not isotropic.

. . . . . . . . . . . . . . . . Numerology of adapted ellipsoids case: 3. By the polynomial ham sandwich theorem there exists p of some j (this is an affine invariant formulation of the isoperimetric inequality). . . . . . . . . . . . . . 10 . . . . . . . . . . . Suppose now that B s p , Q basically does not depend on p , e.g. in the sense that its John ellipsoid E Q ⊂ B does not depend on p . In this 1. We do not have to worry about the cubes with vol E Q ≲ ˜ M ( Q ) − 1 . 2. For the cubes with vol E Q ≫ ˜ M ( Q ) − 1 a fixed positive proportion can be covered by ≤ ˜ M ( Q ) disjoint copies of η E Q for some small absolute constant η . degree ≲ λ that bisects all these copies. 4. Let v 1 , . . . , v n be principal axes of E Q . In each copy E ′ of η E Q the surface Z p has area at least vol ( η E Q ) in the direction η v j for 5. Adding these contributions we s p , Q ( η v j ) ≳ 1 – contradiction for small enough η .

Q M Q closed sets that are disjoint from their antipodes. The John ellipsoid E p Q of p Q depends continuously on p . Using the on each of which E p Q is locally . pair of antipodal points. B Q We claim that the sets The topological input Theorem (Lusternik, Schnirelmann, 1930) Vis p Q . . . . . p . To see this we will write M Q do not cover . Q B Q as the union of geometry of the space of all ellipsoids we split the sphere into O n symmetric closed subsets constant. Let B Q p Vis p Q M Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 If S N is covered by N + 1 closed sets, then one of these sets contains a

John ellipsoid E p Q of p Q depends continuously on p . Using the on each of which E p Q is locally . pair of antipodal points. . . . . . . . . The topological input Theorem (Lusternik, Schnirelmann, 1930) The We claim that the sets . geometry of the space of all ellipsoids we split the sphere into O n symmetric closed subsets constant. Let B Q p Vis p Q M Q . . . . . . . . . . . . . . . . . 11 . . . . . . . . . . . . . If S N is covered by N + 1 closed sets, then one of these sets contains a B ( Q ) = { p ∈ P ∗ | 1 ≤ Vis p , Q ≤ ˜ M ( Q ) } do not cover P ∗ . To see this we will write ∪ Q B ( Q ) as the union of Q ˜ ≲ ∑ M ( Q ) closed sets that are disjoint from their antipodes.