Hypercontractivity of Spherical Averages in Hamming Space

Consider the linear space of functions on the binary hypercube and the linear operator S-delta acting by averaging a function over a Hamming sphere of radius delta n around every point. It is shown that this operator has a dimension-independent bound on the norm L-p -> L-2 with p = 1+( 1 - 2 delta)(2). This result evidently parallels a classical estimate of Bonami and Gross for L-p -> L-q norms for the operator of convolution with a Bernoulli noise. The estimate for S-delta is harder to obtain since the latter is neither a part of a semigroup nor a tensor power. The result is shown by a detailed study of the eigenvalues of S-delta and L-p -> L-2 norms of the Fourier multiplier operators Pi(a) with symbol equal to a characteristic function of the Hamming sphere of radius a (in the notation common in boolean analysis Pi(a)f = f(=a), where f(=a) is a degree-a component of function f). A sample application of the result is given: Any set A subset of F-n(2) with the property that A vertical bar A contains a large portion of some Hamming sphere (counted with multiplicity) must have cardinality a constant multiple of 2(n).