Hypercontractive Inequalities for the Second Norm of Highly Concentrated Functions, and Mrs. Gerber’s-Type Inequalities for the Second Rényi Entropy

Abstract

Let , , be the noise operator acting on functions on the boolean cube . Let f be a distribution on and let . We prove tight Mrs. Gerber-type results for the second Rényi entropy of which take into account the value of the Rényi entropy of f. For a general function f on we prove tight hypercontractive inequalities for the norm of which take into account the ratio between and norms of f.