Discrete logarithmic Sobolev inequalities in Banach spaces

Abstract

Let be the discrete hypercube equipped with the uniform probability measure . We prove that if is a Banach space of finite cotype and , then every function satisfies the dimension‐free vector‐valued logarithmic Sobolev inequality 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 (1993). As an application, we use such vector‐valued 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.