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 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 $E$.