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Discrete Riesz transforms and sharp metric \(X_p\) inequalities. (English) Zbl 1364.46021

In a recent paper of A. Naor and G. Schechtman [Forum Math. Pi 4, Article ID e3, 81 p. (2016; Zbl 1344.46018)], metric \(X_p\) inequalities have been introduced.
The main goal of the present paper is to prove Conjecture 1.5 of [loc. cit.], stating that in \(L_p\), the \(X_p\) inequality holds for a wide range of different values of \(m\) (Theorem 6 in the present paper):
Suppose that \(k,m,n\in \mathbb{N}\) satisfy \(k\in \{1,\dots, n\}\) and \(m\geq \sqrt{n/k}\). Suppose also that \(p\in [2,\infty)\). Then every \(f:\mathbb{Z}_{8m}^n\to L_p\) satisfies \[ \bigg(\frac{1}{\binom{n}{k}}\sum_{\mathsf{S}\subset \{1,\dots,n\}, |\mathsf{ S}|=k}\mathbb{E}\Big[\left\|f(x+4m\varepsilon_{\mathsf{S}})-f(x)\right\|_{L_p}^p\Big]\bigg)^{\frac{1}{p}} \leq \]
\[ C(p) m\bigg(\frac{k}{n}\sum_{j=1}^n\mathbb{E}\Big[\|f(x+e_j)-f(x)\|_{L_p}^p\Big]+\left(\frac{k}{n}\right)^{\frac{p}{2}}\mathbb{E}\Big[\|f(x+\varepsilon)-f(x)\|_{L_p}^p\Big]\bigg)^{\frac{1}{p}}, \] where the expectations are taken with respect to \((x,\varepsilon)\in \mathbb{Z}_{8m}^n\times \{-1,1\}^n\) chosen uniformly at random, \(\{e_j\}\) is the unit vector basis of \(\mathbb{Z}_{8m}^n\), \(\varepsilon_{\mathsf{S}}= \sum_{j\in \mathsf{S}}\varepsilon_je_j\), and \(C(p)\) is a constant which may depend only on \(p\).
As was already observed in [op. cit.], Theorem 6 has very important consequences for the theory of metric embeddings, such as sharp estimates for distortions of embeddings into \(L_p\) of grids with the \(\ell_q^n\) metric, \(2\leq q<p\) (Theorem 1 in the present paper), and the following theorem on metric embeddings of \(L_q\) (Theorem 3 in the present paper):
Suppose that \(p,q\in (2,\infty)\) satisfy \(q<p\). Then the maximal \(\theta\in (0,1]\) for which the metric space \((L_q,\|x-y\|_{L_q}^\theta)\) admits a bilipschitz embedding into \(L_p\) equals \(q/p\).
The proof of Theorem 6 stated above uses the work of F. Lust-Piquard [J. Funct. Anal. 155, No. 1, 263–285 (1998; Zbl 0908.47027)] on dimension free estimates for discrete Riesz transforms.

MSC:

46B85 Embeddings of discrete metric spaces into Banach spaces; applications in topology and computer science
30L05 Geometric embeddings of metric spaces
46B80 Nonlinear classification of Banach spaces; nonlinear quotients
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