Proof of a hypercontractive estimate via entropy

Consider the probability space W = { - 1, 1}(n) with the uniform (=product) measure. Let f: W --> R be a function. Let f = Sigma f(I) X-I be its uniqu e expression as a multilinear polynomial where X-I = Pi (i is an element of I) x(i). For 1 less than or equal to m less than or equal to n let f(m) = S igma (\I\ = m) f(I)X(I). Let T-epsilon(f) = Sigmaf(I)epsilon X-\I\(I) where 0 < <epsilon> < 1 is a constant. A hypercontractive inequality, proven by Bonami and independently by Beckner, states that \T-<epsilon>(f)\(2) less than or equal to \f\(1+epsilon)2. This inequality has been used in several papers dealing with combinatorial and probabilistic problems. It is equivalent to the following inequality vi a duality For any q greater than or equal to 2 \f((m) over dot)\(q) less than or equal to (rootq-1)(m)\f((m) over dot)\2. In this paper we prove a special case with a slightly weaker constant, whic h is sufficient for most applications. We show \f((m) over dot)\4 less than or equal to c(m)\f((m) over dot)\(2) where c = (4)root 28. Our proof uses probabilistic arguments, and a general ization of Shearer's Entropy Lemma, which is of interest in its own right.