Tail spaces estimates on Hamming cube and Bernstein–Markov inequality

Abstract

This note contains some estimates for tail spaces and some Bernstein–Markov inequalities for Banach space valued functions on Hamming cube. We use analytic paraproduct operator for tail spaces. For Bernstein–Markov inequalities the novelty is in getting rid of some irritating logarithms and in proving Bernstein–Markov inequalities for |∇f|X rather than for Δ1/2f for X-valued polynomials on Hamming cube. There is an interesting difference for giving the estimates for |∇f|X rather than for Δ1/2f as their comparison is equivalent to estimates of Riesz transforms on Hamming cube.