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

Niv Levhari; Alex Samorodnitsky

doi:10.3390/e24101376

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

Entropy (Basel). 2022 Sep 27;24(10):1376. doi: 10.3390/e24101376.

Authors

Niv Levhari^{1

2}, Alex Samorodnitsky¹

Affiliations

¹ School of Engineering and Computer Science, The Hebrew University of Jerusalem, Jerusalem 9103401, Israel.
² School of Mathematical Sciences, Tel Aviv University, Tel Aviv 6997801, Israel.

Abstract

Let Tϵ, 0≤ϵ≤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 Rényi entropy of Tϵf which take into account the value of the qth Rényi entropy of f. For a general function f on {0,1}n we prove tight hypercontractive inequalities for the ℓ2 norm of Tϵf which take into account the ratio between ℓq and ℓ1 norms of f.

Keywords: Mrs. Gerber’s inequality; Rényi entropy; entropy; hypercontractivity.