  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 $\Pi_a$ 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 $A\subset \FF_2^n$ with the property that $A+A$ contains a large portion of
some Hamming sphere (counted with multiplicity) must have cardinality a
constant multiple of $2^n$.
