Optimal range of Haar martingale transforms and its applications

TL;DR

This paper establishes sharp distribution function estimates for Haar martingale transforms and identifies their optimal symmetric space ranges, advancing understanding of martingale inequalities and harmonic analysis applications.

Contribution

It provides the first sharp estimate for the distribution function of Haar martingale transforms and determines their optimal Banach symmetric range spaces.

Findings

Sharp distribution function estimate for Haar martingale transforms

Identification of the optimal symmetric space range for these transforms

Applications to Haar basis projections in symmetric function spaces

Abstract

Let $(\mathcal{F}_n)_{n\ge 0}$ be the standard dyadic filtration on $[0,1]$. Let $\mathbb{E}_{\mathcal{F}_n}$ be the conditional expectation from $ L_1=L_1[0,1]$ onto $\mathcal{F} _n$, $n\ge 0$, and let $\mathbb{E}_{\mathcal{F} _{-1}} =0$. We present the sharp estimate for the distribution function of the martingale transform $T$ defined by \begin{align*} Tf=\sum_{m=0}^\infty \left( \mathbb{E}_{\mathcal{F}_{2m}} f-\mathbb{E}_{\mathcal{F}_{2m-1}}f \right), ~f\in L_1, \end{align*} in terms of the classical Calder\'{o}n operator. As an application, for a given symmetric function space $E$ on $[0,1]$, we identify the symmetric space $\mathcal{S}_E$, the optimal Banach symmetric range of martingale transforms/Haar basis projections acting on $E$.