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.