• Corpus ID: 225041291

# Talagrand's influence inequality revisited

@article{CorderoErausquin2020TalagrandsII,
title={Talagrand's influence inequality revisited},
author={Dario Cordero-Erausquin and Alexandros Eskenazis},
journal={arXiv: Functional Analysis},
year={2020}
}
• Published 22 October 2020
• Mathematics
• arXiv: Functional Analysis
Let $\mathscr{C}_n=\{-1,1\}^n$ be the discrete hypercube equipped with the uniform probability measure $\sigma_n$. Talagrand's influence inequality (1994) asserts that there exists $C\in(0,\infty)$ such that for every $n\in\mathbb{N}$, every function $f:\mathscr{C}_n\to\mathbb{C}$ satisfies $$\mathrm{Var}_{\sigma_n}(f) \leq C \sum_{i=1}^n \frac{\|\partial_if\|_{L_2(\sigma_n)}^2}{1+\log\big(\|\partial_if\|_{L_2(\sigma_n)}/\|\partial_i f\|_{L_1(\sigma_n)}\big)}.$$ In this work, we undertake a…
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