Some remarks on Davis inequality for biparameter filtrations

Abstract

The Davis inequality $\mathbb{E}Sf\simeq \mathbb{E}{f}^{\ast }$ between $L^1$ norms of square function of a martingale and its maximal function is known for martingales indexed by linearly ordered filtrations and in some particular cases for double indexed one. We prove the $\gtrsim$ inequality for arbitrary filtrations satisfying the (F4) condition of Cairoli and Walsh and propose a method to attack the other inequality. The former is done by means of a two-parameter analogue of Davis-Garsia decomposition.