Optimal Range of Haar Martingale Transforms and Its Applications

Abstract

Let (Fn)n≥0 be the standard dyadic filtration on [0, 1). Let EFn be the conditional expectation from L1 = L1 [0, 1) onto Fn, n ≥ 0, and let EF-1 = 0. We present the sharp estimate for the distribution function of the martingale transform T defined by Tf = ∞Σ m=0 (EF2mf-EF2m-1f), f ϵ L1, 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 SE, the optimal Banach symmetric range of martingale transforms/Haar basis projections acting on E. © 2024 Elsevier B.V., All rights reserved.