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.).
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