Optimal range of Haar martingale transforms and its applications

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 Calderón 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 $E$.