Proof of a hypercontractive estimate via entropy

Abstract

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$f_{\hat m}$=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

$$\left| {T_\varepsilon \left( f \right)} \right|_2 \le \left| f \right|_{1 + \varepsilon ^2 } $$

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

$$\left| {f_{\hat m} } \right|_q \le \left( {\sqrt {q - 1} } \right)^m \left| {f_{\hat m} } \right|_2 $$

In this paper we prove a special case with a slightly weaker constant, which is sufficient for most applications. We show

$$\left| {f_{\hat m} } \right|_4 \le c^m \left| {f_{\hat m} } \right|_2 $$

where$c = \sqrt[4]{{28}}$. Our proof uses probabilistic arguments, and a generalization of Shearer’s Entropy Lemma, which is of interest in its own right.