### Abstract

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 B_{X} and f: B_{X} → Y is a 1-Lipschitz function then there exists an affine mapping Λ: X → Y and a sub-ball B* = y+ρB_{X} ⊆ B_{X} 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 ℝ^{n} equipped with the Euclidean norm and Y is a Hilbert space.).

Original language | English (US) |
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Pages (from-to) | 1-48 |

Number of pages | 48 |

Journal | Discrete Analysis |

Volume | 6 |

Issue number | 2016 |

DOIs | |

State | Published - Jan 1 2016 |

### All Science Journal Classification (ASJC) codes

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

### Keywords

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

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## Cite this

*Discrete Analysis*,

*6*(2016), 1-48. https://doi.org/10.19086/da.614