Analysis demonstrates the Davis inequality holds for martingales under Cairoli and Walsh conditions.
The Davis inequality E Sf E f^* between L¹ 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 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) studied this question.
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