Discrete logarithmic Sobolev inequalities in Banach spaces

Abstract

Let $\mathscr{C}_n=\{-1,1\}^n$ be the discrete hypercube equipped with the uniform probability measure $\sigma_n$. We prove that if $(E,\|\cdot\|_E)$ is a Banach space of finite cotype and $p\in[1,\infty)$, then every function $f:\mathscr{C}_n\to E$ satisfies the dimension-free vector-valued $L_p$ logarithmic Sobolev inequality $$\|f-\mathbb{E} f\|_{L_p(\log L)^{p/2}(E)} \leq \mathsf{K}_p(E) \left( \int_{\mathscr{C}_n} \Big\| \sum_{i=1}^n \delta_i \partial_i f\Big\|_{L_p(E)}^p \, d\sigma_n(\delta)\right)^{1/p}.$$ The finite cotype assumption is necessary for the conclusion to hold. This estimate is the hypercube counterpart of a result of Ledoux (1988) in Gauss space and the optimal vector-valued version of a deep inequality of Talagrand (1994). As an application, we use such vector-valued $L_p$ logarithmic Sobolev inequalities to derive new lower bounds for the bi-Lipschitz distortion of nonlinear quotients of the Hamming cube into Banach spaces with prescribed Rademacher type.