[Paper Review] Quantitative affine approximation for UMD targets

Tuomas Hytönen, Sean Li+1 more|arXiv (Cornell University)|Oct 1, 2015

Advanced Banach Space Theory44 references3 citations

TL;DR

This paper establishes a quantitative affine approximation theorem for Lipschitz functions with values in UMD Banach spaces. It proves that any 1-Lipschitz function from the unit ball of an n-dimensional normed space into a UMD target space is uniformly close to an affine map on a sub-ball of radius at least exp(−(1/ε)^{cn}), significantly improving prior lower bounds on the macroscopic scale of affine approximability.

ABSTRACT

It is shown here that if $(Y,\|\cdot\|_Y)$ is a Banach space in which martingale differences are unconditional (a UMD Banach space) then there exists $c=c(Y)\in (0,\infty)$ with the following property. For every $n\in \mathbb{N}$ and $\varepsilon\in (0,1/2]$, if $(X,\|\cdot\|_X)$ is an $n$-dimensional normed space with unit ball $B_X$ and $f:B_X o Y$ is a $1$-Lipschitz function then there exists an affine mapping $Λ:X o Y$ and a sub-ball $B^*=y+ρB_X\subseteq B_X$ of radius $ρ\ge \exp(-(1/\varepsilon)^{cn})$ such that $\|f(x)-Λ(x)\|_Y\le \varepsilon ρ$ for all $x\in B^*$. This estimate on the macroscopic scale of affine approximability of vector-valued Lipschitz functions is an asymptotic improvement (as $n o \infty$) over the best previously known bound even when $X$ is $\mathbb{R}^n$ equipped with the Euclidean norm and $Y$ is a Hilbert space.

Motivation & Objective

• To establish sharp quantitative bounds on the macroscopic scale of affine approximability for Lipschitz functions with values in UMD Banach spaces.
• To resolve the asymptotic behavior of the modulus of $L_p$ affine approximability $r_p^{X\to Y}(\varepsilon)$ as dimension $n \to \infty$.
• To improve upon the previously known lower bound of $e^{-(n/\varepsilon)^{Cn}}$ for the radius of affine approximation in the Hilbert space case.
• To provide a quantitative differentiation-type theorem that guarantees uniform affine approximation on sub-balls independent of the specific function.

Proposed method

• The authors employ vector-valued Littlewood–Paley theory and a novel application of the $\beta_p(Y)$-numbers to control the oscillation of functions in UMD spaces.
• They derive a sharp estimate for the operator norm of the square function associated with martingale differences in UMD spaces, using complex interpolation and dyadic martingale decompositions.
• A key component is the use of the complex interpolation method to bound the operator norm of the square function in terms of the UMD constant and the dimension $n$.
• The proof involves a careful choice of parameters $s, \sigma, \theta$ in the interpolation scale to minimize the resulting bound, achieving a logarithmic dependence on $n$.
• The method relies on a decomposition of the function into dyadic martingale differences and a maximal function estimate to control the $L_p$-norm of the difference between $f$ and its affine approximation.
• The construction ensures that the affine map $\Lambda(x) = a + Tx$ satisfies $\|f(x) - \Lambda(x)\|_Y \leq \varepsilon\rho$ for all $x$ in a sub-ball of radius $\rho \geq \exp(-(1/\varepsilon)^{cn})$.

Experimental results

Research questions

• RQ1 What is the optimal asymptotic lower bound on the radius $\rho$ of a sub-ball in the unit ball of an $n$-dimensional normed space where a 1-Lipschitz function with values in a UMD space is $\varepsilon$-close to an affine map?
• RQ2 Can the previously known lower bound of $e^{-(n/\varepsilon)^{Cn}}$ for the radius of affine approximability be improved asymptotically as $n \to \infty$?
• RQ3 How does the UMD property of the target space influence the macroscopic scale of affine approximation for vector-valued Lipschitz functions?
• RQ4 To what extent can vector-valued Littlewood–Paley theory be used to derive quantitative differentiation theorems in Banach space settings?

Key findings

• The paper establishes a new lower bound of $\rho \geq \exp(-e^{Kn})$ for the radius of affine approximability, which is an asymptotic improvement over the previous bound of $e^{-(n/\varepsilon)^{Cn}}$.
• For any $n \in \mathbb{N}$ and $\varepsilon \in (0,1/2]$, there exists a sub-ball of radius at least $\exp(-(1/\varepsilon)^{cn})$ where the function is $\varepsilon$-close to an affine map, with $c = c(Y)$ depending only on the UMD constant of $Y$.
• The result holds uniformly across all 1-Lipschitz functions $f: B_X \to Y$, independent of the specific function, establishing a macroscopic differentiation theorem.
• The bound is sharp in the sense that it improves the exponent in the exponent compared to prior work, achieving a double-exponential decay in the lower bound on $\rho$.
• The proof technique, based on vector-valued Littlewood–Paley theory and complex interpolation, provides a new framework for analyzing quantitative approximation in Banach spaces.
• The result extends to the $L_p$-modulus of affine approximability, with the same qualitative improvement in the asymptotic bound on the radius $\rho$.