Optimal Range of Haar Martingale Transforms and Its Applications

Abstract

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 $E$.