Discrete logarithmic Sobolev inequalities in Banach spaces

Abstract

Let ${\mathcal{C}}_n=\{-1,1{\}}^n$ be the discrete hypercube equipped with the uniform probability measure ${\sigma }_n$. We prove that if $(E,\Vert \cdot {\Vert }_E)$ is a Banach space of finite cotype and $p\in [1,\infty )$, then every function $f:{\mathcal{C}}_n\to E$ satisfies the dimension-free vector-valued $L_p$ logarithmic Sobolev inequality $$\Vert f-\mathbb{E}f{\Vert }_{L_p(\log L{)}^{p/2}(E)}\le {\mathsf{K}}_p(E){\left({\int }_{{\mathcal{C}}_n}\Vert \sum _{i=1}^n{\delta }_i{\partial }_{i}f{\Vert }_{L_p(E)}^{p}d{\sigma }_n(\delta )\right)}^{1/p}.$$ 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.

References in corpus (2)

• Hypercontractive measures, Talagrand's inequality, and influences
• On sharp isoperimetric inequalities on the hypercube