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arXiv · 2509.23351

Some remarks on Davis inequality for biparameter filtrations

Maciej Rzeszut
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Abstract

The Davis inequality $\mathbb{E} Sf\simeq \mathbb{E} f^*$ 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.

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Maciej Rzeszut. 2025-09-27. Some remarks on Davis inequality for biparameter filtrations. https://arxiv.org/abs/2509.23351

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