The dyadic derivative and cesàro mean of banach-valued martingales

Abstract

In this article, the Banach space X and the martingales with values in it are considered. It is shown that the maximal operators of the one-dimensional dyadic derivative of the dyadic integral and Cesàro means are bounded from the dyadic Hardy- Lorentz space ▪ ra ( X) to L ra ( X) when X is isomorphic to a p-uniformly smooth space (1 < p ≤ 2). And it is also bounded from H ra( X) to L ra ( X) (0 < r < ∞,0 < a ≤ ∞) when X has Radon-Nikodym property. In addition, some weak-type inequalities are given.