  This note contains some estimates for tail spaces on Hamming cube. We use
analytic paraproduct operator for that purpose. We also show several types of
Bernstein--Markov inequalities for Banach space valued functions on Hamming
cube. Here the novelty is in getting rid of some irritating logarithms and in
proving Bernstein--Markov inequalities for $|\nabla f|_X$ rather than for
$\Delta^{1/2} f$ for $X$ valued polynomials on Hamming cube.
