Quantitative affine approximation for UMD targets (Q2826234)

Given \(X=\mathbb R^n\) with a norm \(\|\cdot\|_X\) and an infinite-dimensional Banach space \(Y\), the authors introduce the notion of modulus of \(L_p\) affine approximability, denoted by \(r_p^{X\to Y}(\varepsilon)\), as the supremum of all values \(r\geq 0\) such that for every Lipschitz function \(f:B_X\to Y\) there exist \(y\in X\) and \(\rho\geq r\) such that \(y+\rho B_X\subset B_X\) and there exist \(a\in Y\) and a linear operator \(T:X\to Y\) with \(\|T\|\leq 3 \|f\|_{\mathrm{Lip}}\) such that NEWLINE\[NEWLINE\left(\frac{1}{\mathrm{vol}(x+\rho B_X)}\int_{y+\rho B_X}\|f(z)- (a+ Tz)\|^p \, dz\right)^{1/p}\leq \varepsilon \rho \|f\|_{\mathrm{Lip}}.NEWLINE\]NEWLINE This extends the case \(p=\infty\), denoted by \(r^{X\to Y}\), first introduced in a paper by \textit{S. Bates} et al. [Geom. Funct. Anal. 9, No. 6, 1092--1127 (1999; Zbl 0954.46014)] and whose best known lower bounds (using the modulus of uniform convexity \(c(Y)\)) were obtained by \textit{S. Li} and \textit{A. Naor} [Isr. J. Math. 197, 107--129 (2013; Zbl 1291.46021)] who achieved the estimate \(r^{X\to Y}(\varepsilon)\geq e^{-(n/\varepsilon)^{c(Y)n}}\).NEWLINENEWLINEIn the present paper, an asymptotically better lower estimate is obtained, provided that \(Y\) is assumed to be a UMD space. Calling \(\beta(Y)\) the best constant appearing in the UMD condition for square-integrable \(Y\)-valued martingales, the precise statement gives that there exist \(\kappa, C\geq 1\) such that for every \(\beta\geq 2\) and every UMD space \(Y\) satisfying \(\beta\geq \beta(Y)\) one has \(r_p^{X\to Y}(\varepsilon)\geq \exp{(-(\beta n)^{C\kappa \beta}/\varepsilon^{\kappa \beta}})\).NEWLINENEWLINE The use of this modulus of \(L_p\) affine approximability for \(p<\infty\) allows the authors to get some improvement for the lower bound in the case \(p=\infty\), and to deal with methods such as Littlewood-Paley theory and complex interpolation that are at their disposal in this case. As a consequence of their improvement, they also obtain a bound on \textit{J. Bourgain}'s discretization modulus in the case of UMD spaces (see [Lect. Notes Math. 1267, 157--167 (1987; Zbl 0633.46018)]). The paper leaves a number of interesting open questions concerning the geometry of Banach spaces and the asymptotics of this modulus.