Tail spaces estimates on Hamming cube and Bernstein--Markov inequality

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 $\mathrm{\mid }\mathrm{\nabla }f{\mathrm{\mid }}_X$ rather than for ${\mathrm{\Delta }}^{1\mathrm{/}2}f$ for $X$ valued polynomials on Hamming cube.