Martingale Hardy spaces and the dyadic derivative

Abstract

It is shown that the maximal operator of the one-dimensional dyadic derivative of the dyadic integral is bounded from the dyadic Hardy-Lorentz spaceHp,q toLp,q (1/2<p<∞, 0<q≤∞) and is of weak type (L1,L1). We define the twodimensional dyadic hybrid Hardy spaceH1‖ and verify that the corresponding maximal operator of a two-dimensional function is of weak type (H1‖,L1). As a consequence, we obtain that the dyadic integral of a two-dimensional functionfeH1‖⊃LlogL is dyadically differentiable and its derivative is a.e.f.