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

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 $f$.

References in corpus (2)

• A modified logarithmic Sobolev inequality for the Hamming cube and some applications
• Strong Brascamp-Lieb Inequalities