## 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

### Citations:

Zbl 1344.46018; Zbl 0908.47027
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### References:

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