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Quantitative affine approximation for UMD targets

- Mathematics
- University of Helsinki
- More about Tuomas Hytönen

- Mathematics
- University of Chicago
- More about Sean Li

- Mathematics
- Princeton University (NJ)
- More about Assaf Naor

### Editorial introduction

Quantitative affine approximation for UMD targets, Discrete Analysis 2016:6, 48 pp.

Let Y be a Banach space. A *martingale difference sequence* in Y is a sequence of Y-valued random variables such that E[di|d1,…,di−1]=0 for every i (and E(d1)=0). Given 1<p<∞, we say that Y satisfies the UMDp property if there is a constant Cp such that

E‖n∑i=1ϵidi‖p≤CppE‖n∑i=1di‖p

for every martingale difference sequence d1,…,dn and every choice of signs ϵ1,…,ϵn∈{−1,1}. An *unconditional martingale difference* space, or UMD space for short, is a Banach space Y that satisfies the UMDp property for some 1<p<∞, which can be shown to imply that it satisfies the UMDp property for all such p. It is easy to show that when Y is a Hilbert space, it has the UMDp property with a constant of 1. In the context of stochastic analysis, UMD spaces can be thought of as spaces that have many of the good properties of Hilbert spaces.

The question considered in this paper concerns a kind of differentiability property of Lipschitz functions. Let X be an n-dimensional normed space with unit ball BX and let Y be a UMD Banach space. Given a 1-Lipschitz function f:BX→Y and ϵ>0, the aim is to find a sub-ball B′⊂B of radius ρ, with ρ as large as possible, and an affine function Λ:B′→Y, such that ‖f(x)−Λ(x)‖≤ϵρ for every x∈B′. Importantly, the lower bound on ρ must be independent of the function f.

The paper obtains a lower bound of exp(−(1/ϵ)cn) for a constant c that depends on Y only. This improves on the previously best known bound even when Y is a Hilbert space. In the other direction, the best known upper bound (on how large ρ is in the worst case) is singly exponential in n, so there is still a large and very interesting gap. There are several other attractive open problems in the paper.

The proof of the main result of the paper is not easy, but it contains many new ideas in the geometry of Banach spaces and in harmonic analysis, some of which are interesting results in their own right.