Optimal Range of Haar Martingale Transforms and Its Applications

Abstract

Let (F-n)(n >= 0 )be the standard dyadic filtration on [0, 1). Let E-Fn be the conditional expectation from L-1 = L1 [0, 1) onto .F-n , n > 0, and let EF-1 = 0. We present the sharp estimate for the distribution function of the martingale transform T defined by Tf = Sigma(infinity)(m=0)(E(F2m)f - E(F2m-1)f), f is an element of L-1, in terms of the classical Calderon operator. As an application, for a given symmetric function space E on [0, 1), we identify the symmetric space S-E , the optimal Banach symmetric range of martingale transforms/Haar basis projections acting on E.