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 $L_p\to L_2$ with $p = 1+(1-2\delta)^2$. This result evidently parallels a classical estimate of Bonami and Gross for $L_p\to L_q$ 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 $L_p\to L_2$ norms of the Fourier multiplier operators ${\mathrm{\Pi }}_a$ with symbol equal to a characteristic function of the Hamming sphere of radius $a$ (in the notation common in boolean analysis ${\mathrm{\Pi }}_{a}f=f^{=a}$, where $f^{=a}$ is a degree-$a$ component of function $f$). A sample application of the result is given: Any set $A\subset {\text{FF}}_2^n$ with the property that $A+A$ contains a large portion of some Hamming sphere (counted with multiplicity) must have cardinality a constant multiple of $2^n$.