An L log L characterization of H1

Abstract

AbstractWe show that if Xt is a continuous martingale with X0 = 0 then the quantity supt E(Xt log(Mt+Mt−)) defines a norm on H1 martingales equivalent to the usual norm. Here Mt+ and Mt− are one-sided maximal functions. This result is a generalization of Gundy's theorem that a positive martingale is in H1 if and only if it is in L log L