Hypercontractive inequalities for the second norm of highly concentrated functions, and Mrs. Gerber's-type inequalities for the second Renyi entropy

TL;DR

This paper establishes tight inequalities relating the second Renyi entropy of noise-affected functions on the boolean cube to their initial entropy, extending hypercontractive bounds by incorporating various norm ratios.

Contribution

It introduces new tight Mrs. Gerber-type inequalities for the second Renyi entropy and hypercontractive inequalities for the second norm, considering different norm ratios of functions.

Findings

Derived tight Mrs. Gerber-type inequalities for second Renyi entropy.

Established hypercontractive inequalities for the second norm with norm ratio considerations.

Extended the understanding of noise effects on functions on the boolean cube.

Abstract

Let $T_{\epsilon}$, $0 \le \epsilon \le 1/2$, be the noise operator acting on functions on the boolean cube $\{0,1\}^n$. Let $f$ be a distribution on $\{0,1\}^n$ and let $q > 1$. We prove tight Mrs. Gerber-type results for the second Renyi entropy of $T_{\epsilon} f$ which take into account the value of the $q^{th}$ Renyi entropy of $f$. For a general function $f$ on $\{0,1\}^n$ we prove tight hypercontractive inequalities for the $\ell_2$ norm of $T_{\epsilon} f$ which take into account the ratio between $\ell_q$ and $\ell_1$ norms of $f$.