16.
Consider the probability space
W={−1, 1}
n
with the uniform (=product) measure. Let
f: W →
R be a function. Let
f=Σ
f
IX
I be its unique expression as a multilinear polynomial where
X
I=Π
i∈I
x
i. For 1≤
m≤
n let
=Σ
|I|=m
f
IX
I. Let
T
ɛ
(
f)=Σ
f
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 any
q≥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.
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