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Bourgain Discretization Using Lebesgue-Bochner Spaces

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Bourgain Discretization Using Lebesgue-Bochner Spaces Bourgain discretization using Lebesgue-Bochner spaces Mikhail I. Ostrovskii and Beata Randrianantoanina This paper is dedicated to the memory of our friend Joe Diestel (1943–2017). Lebesgue- Bochner spaces were one of the main passions of Joe. He started to work in this direction in his Ph.D. thesis [Die68], and devoted to Lebesgue-Bochner spaces a large part of his most popular, classical, Dunford-Schwartz-style monograph [DU77], joint with Jerry Uhl. Abstract We study the Lebesgue-Bochner discretization property of Banach spaces Y , which ensures that the Bourgain discretization modulus for Y has a good lower estimate. We prove that there exist spaces that do not have the Lebesgue-Bochner discretization property, and we give a class of examples of spaces that enjoy this property. 1 Introduction We denote by cY (X) the greatest lower bound of distortions of bilipschitz embeddings of a metric space (X,dX ) into a metric space (Y,dY ), that is, the greatest lower bound of the numbers C for which there are a map f : X Y and a real number r> 0 such that → u,v X rd (u,v) d (f(u),f(v)) rCd (u,v). ∀ ∈ X ≤ Y ≤ X See [Mat02], [Nao18], and [Ost13] for background on this notion. Let X be a finite- dimensional Banach space and Y be an infinite-dimensional Banach space. Definition 1.1. For ε (0, 1) let δ → (ε) be the supremum of those δ (0, 1) for which ∈ X, Y ∈ every δ-net in B satisfies c ( ) (1 ε)c (X). The function δ → (ε) is called the Nδ X Y Nδ ≥ − Y X, Y Bourgain discretization modulus for embeddings of X into Y . It is not immediate that the discretization modulus is defined for any ε (0, 1), but ∈ this can be derived using the argument of [Rib76] and [HM82] (see [GNS12, Introduction]). Giving a new proof of the Ribe theorem [Rib76], Bourgain proved the following remarkable result [Bou87] (we state it in a stronger form which was proved in [GNS12]): 2010 Mathematics Subject Classification. Primary: 46B85; Secondary: 46B06, 46B07, 46E40. Keywords. Bourgain discretization theorem, distortion of an embedding, Lebesgue-Bochner space 1 Theorem 1.2 (Bourgain’s discretization theorem). There exists C (0, ) such that for ∈ ∞ every two Banach spaces X,Y with dim X = n< and dim Y = , and every ε (0, 1), ∞ ∞ ∈ we have Cn −(cY (X)/ε) δ → (ε) e . (1) X, Y ≥ Bourgain’s discretization theorem and the described below result of [GNS12] on im- proved estimates in the case of Lp spaces have important consequences for quantitative estimates of L1-distortion of the metric space consisting of finite subsets (of equal car- dinality) in the plane with the minimum weight matching distance, see [NS07, Theorem 1.2]. The proof of Bourgain’s discretization theorem was clarified and simplified in [Beg99] and [GNS12] (see also its presentation in [Ost13, Section 9.2]). Different approaches to proving Bourgain’s discretization theorem in special cases were found in [LN13], [HLN16], and [HN16+]. However these approaches do not improve the order of estimates for the discretization modulus. On the other hand the paper [GNS12] contains a proof with much better estimates in the case where Y = Lp. The approach of [GNS12] is based on the following result (whose proof uses methods of [JMS09]; origins of this approach can be found in [GK03]). Theorem 1.3 ([GNS12, Theorem 1.3]). There exists a universal constant κ (0, ) with ∈ ∞ the following property. Assume that δ, ε (0, 1) and D [1, ) satisfy δ κε2/(n2D). ∈ ∈ ∞ ≤ Let X,Y be Banach spaces with dim X = n< , and let be a δ-net in B . Assume that ∞ Nδ X c ( ) D. Then there exists a separable probability space (Ω,ν), a finite dimensional Y Nδ ≤ linear subspace Z Y , and a linear operator T : X L∞(ν,Z) satisfying ⊆ → 1 ε x X, − x Tx Tx (1 + ε) x . (2) ∀ ∈ D k kX ≤ k kL1(ν,Z) ≤ k kL∞(ν,Z) ≤ k kX Remark 1.4. The measure ν in the proof of [GNS12, Theorem 1.3] is atomless, it is the normalized Lebesgue measure on the unit ball of X. As is noted in [GNS12], since (Ω,ν) is a probability measure, we have k · kL1(ν,Z) ≤ k · kLp(ν,Z) ≤ k · kL∞(ν,Z) for every p [1, ]. Therefore (2) implies that X admits an embedding into Lp(ν,Z) ∈ ∞D(1 + ε) with distortion . Since, by the well-known Carath´eodory theorem, L (ν,L ) is ≤ 1 ε p p isometric to L (see [Lac74,− 14]), we get that if Z is a subspace of L , then L (ν,Z) is also p § p p a subspace of Lp. As explained in [GNS12], it follows that the Bourgain’s discretization modulus for the case of Y = Lp satisfies a much better estimate κε2 δX,→L (ε) p ≥ n5/2 (since for all spaces X,Y and all δ > 0, c ( ) √n, see [GNS12]). Y Nδ ≤ To generalize this approach to a wider class of spaces, it is natural to introduce the following definition. 2 Definition 1.5. We say that a Banach space Y has the Lebesgue-Bochner discretization property if there exists a function f : [1, ) [1, ) such that for any separable proba- ∞ → ∞ bility measure µ, for any C 1 and any finite dimensional subspace Z Y , if W is any ≥ ⊂ finite-dimensional subspace of L∞(µ,Z) such that for all w W ∈ w C w , (3) k kL∞(µ,Z) ≤ k kL1(µ,Z) then W is f(C)-embeddable into Y . Lemma 1.6. In Definition 1.5 it suffices to require that the function f exists in the case where µ is the Lebesgue measure on [0, 1]. A function f which works for some separa- ble atomless probability measure space works for any other separable atomless probability measure space. Proof. The second statement follows immediately from the Carath´eodory theorem [Lac74, 14] stating that all separable atomless probability measure spaces are isomorphic. § To prove the first statement assume that f is a function which works for any separable atomless probability measure space, and let (Ω, Σ,µ) be a separable probability measure space with atoms a M , where M is either a positive integer or . Let Ω=Ω˜ a M . { i}i=1 ∞ \{ i}i=1 For each i let (Ωi, Σi,µi) be a separable atomless measure space satisfying µi(Ωi)= µ(ai). We create a separable atomless probability measure space as the disjoint union M Ω˜ Ωi ∪ ! i[=1 with the natural σ-algebra and measure, we denote this measure by ν. For a subspace W L∞(µ,Z) satisfying (3) we introduce a subspace W˜ L∞(ν,Z) ⊂ ⊂ as the set of all vectors obtained in the following way: for each w W we introduce ∈ w˜ L∞(ν,Z) by ∈ w(t) if t Ω˜, w˜(t)= ∈ (w(ai) if t Ωi. ∈ It is easy to see that W˜ is isometric to W , and that w˜ C w˜ . (4) k kL∞(ν,Z) ≤ k kL1(ν,Z) Thus, the function f which can be used for the atomless measure ν, can also be used for µ. We need the following generalization of the Bourgain discretization modulus: Given an increasing function g : R+ R+, we define δg (ε) as the supremum of δ such that → X,→Y for all δ-nets δ of BX , N 1+ ε c (X) g c ( ) . (5) Y ≤ 1 ε Y Nδ − The following is a corollary of Theorem 1.3. 3 Corollary 1.7. Let Y be a Banach space with the Lebesgue-Bochner discretization prop- erty. Then δg (ε) κε2/(n5/2), X,→Y ≥ where κ is the constant of Theorem 1.3, g(t) := tf(t) and f is the function of Defini- tion, 1.5 corresponding to a separable atomless probability measure space. Proof. Let δ κε2/(n5/2) and be a δ-net in an n-dimensional Banach space X. By ≤ Nδ Theorem 1.3, there exists a finite dimensional subspace Z Y and a finite-dimensional ⊂ subspace W L∞(ν,Z) (the image of the operator T ) such that W satisfies (3) with ⊂ C = 1+ε c ( ). Thus by the Lebesgue-Bochner discretization property of Y , c (W ) 1−ε Y Nδ Y ≤ f( 1+ε c ( )), and we obtain 1−ε Y Nδ 1+ ε 1+ ε c (X) c ( )f c ( ) . Y ≤ 1 ε Y Nδ 1 ε Y Nδ − − Problem 1. Characterize Banach spaces with the Lebesgue-Bochner discretization prop- erty. At the meeting of the Simons Foundation (New York City, February 20, 2015) Assaf Naor mentioned that at that time no examples of Banach spaces which do not have the Lebesgue-Bochner discretization property were known, although people who were working on this (Assaf Naor and Gideon Schechtman) believed that such examples should exist. We note that the argument which uses the Fubini and Carath´eodory theorems to show that Lp(Lp) is isometric to Lp (for suitable measure spaces) fails for other function spaces, even in a certain ‘isomorphic’ form (see [BBS02, Appendix]). For some spaces a very strong opposite of the situation in the Lp-case happens: Raynaud [Ray89] proved that when Lϕ([0, 1],µ) is an Orlicz space that is not isomorphic to some Lp and does not contain c or ` , then for any r [1, ), the space ` (L ) (and thus also L ([0, 1],µ,L )) 0 1 ∈ ∞ r ϕ ϕ ϕ not only does not embed in Lϕ([0, 1],µ), but is not even crudely finitely representable in it.
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