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

TL;DR

This paper provides new estimates for tail spaces on the Hamming cube using analytic paraproducts and establishes Bernstein--Markov inequalities for Banach space valued functions, improving previous bounds by removing logarithmic factors.

Contribution

It introduces novel tail space estimates and Bernstein--Markov inequalities for Banach space valued functions on the Hamming cube, avoiding logarithmic complications and focusing on gradient norms.

Findings

Derived tail space estimates on Hamming cube

Established Bernstein--Markov inequalities for Banach space functions

Improved bounds by eliminating logarithmic factors

Abstract

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.