Hypercontractive inequalities for the second norm of highly concentrated functions, and Mrs. Gerber's-type inequalities for the second Renyi entropy
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Let T_ϵ, 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 Renyi entropy of T_ϵ 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 ℓ_2 norm of T_ϵ f which take into account the ratio between ℓ_q and ℓ_1 norms of f.
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