Quantitative affine approximation for UMD targets

Hytönen, Tuomas; Li, Sean

doi:10.19086/da.614

Cited by 3 publications

(2 citation statements)

References 56 publications

(123 reference statements)

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“…Since in our setting X is finite dimensional (dim.X/ D n > 2), such a qualitative statement is vacuous without its quantitative counterpart (1.89). The first inequality in (1.89) can be deduced from [260] (together with the computation of the implicit dependence on p in [260] that was carried out in [131,Lemma 32]). The second inequality in (1.89) follows from an examination of the proof in [259].…”

Section: Intersection With a Euclidean Ballmentioning

confidence: 99%

Extension, Separation and Isomorphic Reverse Isoperimetry

Naor

2024

Memoirs of the European Mathematical Society

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The content of this volume is licensed under the CC BY 4.0 license, with the exception of the logo and branding of the European Mathematical Society and EMS Press, and where otherwise noted. Bibliographic information published by the Deutsche NationalbibliothekThe Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de.

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Section: Intersection With a Euclidean Ballmentioning

confidence: 99%

Extension, Separation and Isomorphic Reverse Isoperimetry

Naor

2024

Memoirs of the European Mathematical Society

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“…We describe the Dorronsoro-type theorem we will use. This is due to Hytönen, Li and Naor [HLN16]. For a Lipschitz function f : R d → R n , let f Lip denote its Lipschitz constant, and for B ⊆ R d and 1 < p < ∞, set…”

Section: Families Of Cubesmentioning

confidence: 99%

A d-dimensional analyst’s travelling salesman theorem for subsets of Hilbert space

Hyde¹

2022

Math. Ann.

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We are interested in quantitative rectifiability results for subsets of infinite dimensional Hilbert space H. We prove a version of Azzam and Schul's d-dimensional Analyst's Travelling Salesman Theorem in this setting by showing for any lower d-regularwhere β d (E) give a measure of the curvature of E and the error term is related to the theory of uniform rectifiability (a quantitative version of rectifiability introduced by David and Semmes).To do this, we show how to modify the Reifenberg Parametrization Theorem of David and Toro so that it holds in Hilbert space. As a corollary, we show that a set E ⊆ H is uniformly rectifiable if and only if it satisfies the so-called Bilateral Weak Geometric Lemma, meaning that E is bi-laterally well approximated by planes at most scales and locations. Contents 1. Introduction 1 2. Preliminaries 5 3. Reifenberg parametrisations in Hilbert space 12 4. Estimating curvature for Reifenberg flat sets 16 5. Estimating curvature for general sets 24 6. Estimating measure 33 7. Bilateral Weak Geometric Lemma 39 Appendix A. Parameterisation in Hilbert space 44 Appendix B. Constants 55 Appendix C. Reduction to Euclidean space 57 References 62

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Euclidean Structures and Operator Theory in Banach Spaces

Kalton¹,

Lorist²,

Weis³

2023

Memoirs of the AMS

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We present a general method to extend results on Hilbert space operators to the Banach space setting by representing certain sets of Banach space operators Γ \Gamma on a Hilbert space. Our assumption on Γ \Gamma is expressed in terms of α \alpha -boundedness for a Euclidean structure α \alpha on the underlying Banach space X X . This notion is originally motivated by R \mathcal {R} - or γ \gamma -boundedness of sets of operators, but for example any operator ideal from the Euclidean space ℓ n 2 \ell ^2_n to X X defines such a structure. Therefore, our method is quite flexible. Conversely we show that Γ \Gamma has to be α \alpha -bounded for some Euclidean structure α \alpha to be representable on a Hilbert space. By choosing the Euclidean structure α \alpha accordingly, we get a unified and more general approach to the Kwapień–Maurey factorization theorem and the factorization theory of Maurey, Nikišin and Rubio de Francia. This leads to an improved version of the Banach function space-valued extension theorem of Rubio de Francia and a quantitative proof of the boundedness of the lattice Hardy–Littlewood maximal operator. Furthermore, we use these Euclidean structures to build vector-valued function spaces. These enjoy the nice property that any bounded operator on L 2 L^2 extends to a bounded operator on these vector-valued function spaces, which is in stark contrast to the extension problem for Bochner spaces. With these spaces we define an interpolation method, which has formulations modelled after both the real and the complex interpolation method. Using our representation theorem, we prove a transference principle for sectorial operators on a Banach space, enabling us to extend Hilbert space results for sectorial operators to the Banach space setting. We for example extend and refine the known theory based on R \mathcal {R} -boundedness for the joint and operator-valued H ∞ H^\infty -calculus. Moreover, we extend the classical characterization of the boundedness of the H ∞ H^\infty -calculus on Hilbert spaces in terms of B I P BIP , square functions and dilations to the Banach space setting. Furthermore we establish, via the H ∞ H^\infty -calculus, a version of Littlewood–Paley theory and associated spaces of fractional smoothness for a rather large class of sectorial operators. Our abstract setup allows us to reduce assumptions on the geometry of X X , such as (co)type and UMD. We conclude with some sophisticated counterexamples for sectorial operators, with as a highlight the construction of a sectorial operator of angle 0 0 on a closed subspace of L p L^p for 1 > p > ∞ 1>p>\infty with a bounded H ∞ H^\infty -calculus with optimal angle ω H ∞ ( A ) > 0 \omega _{H^\infty }(A) >0 .

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Quantitative affine approximation for UMD targets

Cited by 3 publications

References 56 publications

Extension, Separation and Isomorphic Reverse Isoperimetry

Extension, Separation and Isomorphic Reverse Isoperimetry

A d-dimensional analyst’s travelling salesman theorem for subsets of Hilbert space

Euclidean Structures and Operator Theory in Banach Spaces

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