论文标题
塔格兰的影响不平等重新审视
Talagrand's influence inequality revisited
论文作者
论文摘要
令$ \ mathscr {c} _n = \ { - 1,1 \}^n $是配备均匀概率度量$σ_n$的离散超立方体。 Talagrand的影响不平等（1994年）断言，存在（0，\ infty）$中的$ c \，以至于每一个$ n \ in \ athbb {n} $，每个函数$ f：\ m athscr {c} _n \ to \ sathbb {c} $ f to \ mathb {c} $满足$ p} \ sum_ {i = 1}^n \ frac {\ | \ | \ partial_if \ | _ {l_2（σ_n）}^2} {1+ \ log \ big \ big（\ | \ | \ partial_if \ | _ {l_2（l_2（n n）}（σ_n）}/\ partial_i f \ | _ {l_1（σ_n）} \ big）}。$$在这项工作中，我们通过谐波分析和随机分析技术对此进行了系统的研究，并将应用于公制嵌入。我们证明，在必要的假设下，Talagrand的不平等扩展到了额外的双对数因素，即在目标空间具有Rademacher 2型的必要假设下，如果目标空间可以省略目标空间，则可以省略该目标空间的函数。这些是Talagrand影响不平等的第一个矢量值扩展。 We also obtain a joint strengthening of results of Bakry-Meyer (1982) and Naor-Schechtman (2002) on the action of negative powers of the hypercube Laplacian on functions $f:\mathscr{C}_n\to E$, whose target space $E$ has nontrivial Rademacher type via a new vector-valued version of Meyer's multiplier theorem (1984).受Talagrand的影响不平等的启发，我们引入了一种名为Talagrand类型的新公制不变性，并将其估算为带有开处方的Rademacher或Martingale类型的Banach空间，Gromov双曲线群，并简单地连接了Riemannian pinch的负率。最后，我们证明talagrand类型是对HyperCube $ \ Mathscr {C} _n $的非线性商的Bi-Lipschitz嵌入性的阻碍，从而为这些有限指标得出了新的非限制性结果。
Let $\mathscr{C}_n=\{-1,1\}^n$ be the discrete hypercube equipped with the uniform probability measure $σ_n$. Talagrand's influence inequality (1994) asserts that there exists $C\in(0,\infty)$ such that for every $n\in\mathbb{N}$, every function $f:\mathscr{C}_n\to\mathbb{C}$ satisfies $$\mathrm{Var}_{σ_n}(f) \leq C \sum_{i=1}^n \frac{\|\partial_if\|_{L_2(σ_n)}^2}{1+\log\big(\|\partial_if\|_{L_2(σ_n)}/\|\partial_i f\|_{L_1(σ_n)}\big)}.$$ In this work, we undertake a systematic investigation of this and related inequalities via harmonic analytic and stochastic techniques and derive applications to metric embeddings. We prove that Talagrand's inequality extends, up to an additional doubly logarithmic factor, to Banach space-valued functions under the necessary assumption that the target space has Rademacher type 2 and that this doubly logarithmic term can be omitted if the target space admits an equivalent 2-uniformly smooth norm. These are the first vector-valued extensions of Talagrand's influence inequality. We also obtain a joint strengthening of results of Bakry-Meyer (1982) and Naor-Schechtman (2002) on the action of negative powers of the hypercube Laplacian on functions $f:\mathscr{C}_n\to E$, whose target space $E$ has nontrivial Rademacher type via a new vector-valued version of Meyer's multiplier theorem (1984). Inspired by Talagrand's influence inequality, we introduce a new metric invariant called Talagrand type and estimate it for Banach spaces with prescribed Rademacher or martingale type, Gromov hyperbolic groups and simply connected Riemannian manifolds of pinched negative curvature. Finally, we prove that Talagrand type is an obstruction to the bi-Lipschitz embeddability of nonlinear quotients of the hypercube $\mathscr{C}_n$, thus deriving new nonembeddability results for these finite metrics.
