Proof of a hypercontractive estimate via entropy

Consider the probability spaceW={−1, 1} ⁿ with the uniform (=product) measure. Letf: W →R be a function. Letf=Σf _IX_I be its unique expression as a multilinear polynomial whereX _I=Π _i∈I x _i. For 1≤m≤n let =Σ_|I|=m f _IX_I. LetT _ɛ (f)=Σf _Iɛ^|I| X _I where 0<ɛ<1 is a constant. A hypercontractive inequality, proven by Bonami and independently by Beckner, states that This inequality has been used in several papers dealing with combinatorial and probabilistic problems. It is equivalent to the following inequality via duality: For anyq≥2 In this paper we prove a special case with a slightly weaker constant, which is sufficient for most applications. We show where . Our proof uses probabilistic arguments, and a generalization of Shearer’s Entropy Lemma, which is of interest in its own right. Supported partially by NSF Award Abstract #0071261.