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

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 $\ell2$ norm of $T{\epsilon} f$ which take into account the ratio between $\ellq$ and $\ell1$ norms of $f$.