Talagrand’s influence inequality revisited

Let $\mathscr{C}_n=\{-1,1\}^n$ be the discrete hypercube equipped with uniform probability measure $\sigma_n$. Talagrand's influence inequality (1994) asserts that there exists $C\in(0,\infty)$ such for every $n\in\mathbb{N}$, 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 systematic investigation of and related inequalities via harmonic analytic stochastic techniques derive applications to metric embeddings. We prove extends, up an additional doubly logarithmic factor, Banach space-valued functions under necessary assumption target space has Rademacher type 2 term can omitted if admits equivalent 2-uniformly smooth norm. These are first vector-valued extensions inequality. also obtain joint strengthening results Bakry-Meyer (1982) Naor-Schechtman (2002) on action negative powers Laplacian $f:\mathscr{C}_n\to E$, whose $E$ nontrivial new version Meyer's multiplier theorem (1984). Inspired by inequality, introduce invariant called Talagrand estimate it spaces prescribed or martingale type, Gromov hyperbolic groups simply connected Riemannian manifolds pinched curvature. Finally, is obstruction bi-Lipschitz embeddability nonlinear quotients $\mathscr{C}_n$, thus deriving nonembeddability these finite metrics.