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 ${0,1}<sup>n$. Let $f$ be a distribution on ${0,1}<sup>n$ and let $q &gt; 1$. We prove tight Mrs. Gerber-type results for the second Renyi entropy of $T_{\epsilon} f$ which take into account the value of the $q<sup>{th}$ Renyi entropy of $f$. For a general function $f$ on ${0,1}<sup>n$ 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 $f$.