Hypercontractivity of Spherical Averages in Hamming Space

Abstract

© 2019 Society for Industrial and Applied Mathematics 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 Lp \rightarrow L2 with p = 1 + (1 - 2\delta )2. This result evidently parallels a classical estimate of Bonami and Gross for Lp \rightarrow Lq 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 Lp \rightarrow L2 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 af = 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 \BbbFn2 with the property that A + A contains a large portion of some Hamming sphere (counted with multiplicity) must have cardinality a constant multiple of 2n