Quantitative affine approximation for UMD targets

Tuomas Hytönen, Sean Li, Assaf Naor

Research output: Contribution to journalArticlepeer-review

5 Scopus citations


It is shown here that if (Y,||Y) is a Banach space in which martingale differences are unconditional (a UMD Banach space) then there exists c = c(Y) ∈ (0,∞) with the following property. For every n ∈ ℕ and ε ∈ (0,1=2], if (X;| · | is an n-dimensional normed space with unit ball BX and f: BX → Y is a 1-Lipschitz function then there exists an affine mapping Λ: X → Y and a sub-ball B* = y+ρBX ⊆ BX of radius ρ > exp(-(1=ε)cn such that | f Λ(x)-L(x)|Y ≤ εr for all x ∈ B*. This estimate on the macroscopic scale of affine approximability of vector-valued Lipschitz functions is an asymptotic improvement (as n→∞) over the best previously known bound even when X is ℝn equipped with the Euclidean norm and Y is a Hilbert space.).

Original languageEnglish (US)
Pages (from-to)1-48
Number of pages48
JournalDiscrete Analysis
Issue number2016
StatePublished - 2016

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics


  • Littlewood- Paley theory
  • Quantitative differentiation
  • Unconditional martingale differences


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