Quantitative affine approximation for UMD targets. (English) Zbl 1362.46008
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
\[
\left(\frac{1}{\mathrm{vol}(x+\rho B_X)}\int_{y+\rho B_X}\|f(z)- (a+ Tz)\|^p \, dz\right)^{1/p}\leq \varepsilon \rho \|f\|_{\mathrm{Lip}}.
\]
This extends the case \(p=\infty\), denoted by \(r^{X\to Y}\), first introduced in a paper by 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 S. Li and A. Naor [Isr. J. Math. 197, 107–129 (2013; Zbl 1291.46021)] who achieved the estimate \(r^{X\to Y}(\varepsilon)\geq e^{-(n/\varepsilon)^{c(Y)n}}\).
In 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 \(r_p^{X\to Y}(\varepsilon)\geq \exp{(-(\beta n)^{C\kappa \beta}/\varepsilon^{\kappa \beta}})\).
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 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.
In 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 \(r_p^{X\to Y}(\varepsilon)\geq \exp{(-(\beta n)^{C\kappa \beta}/\varepsilon^{\kappa \beta}})\).
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 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.
Reviewer: Oscar Blasco (Valencia)
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