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Contents of PMS, Vol. 43, Fasc. 1,
pages 109 - 119
DOI: 10.37190/0208-4147.00132
Published online 1.9.2023
 

Doob's estimate for coherent random variables and maximal operators on trees maximal operators on trees

S. Cichomski
A. Osękowski

Abstract:

Let ξ be an integrable random variable defined on (Ω,ℱ,â„™). Fix k ∈ ℤ+ and let {ð’¢ij}1 ≤ i ≤ n, 1 ≤ j ≤ k be a reference family of sub-σ-fields of ℱ such that {ð’¢ij}1 ≤ i ≤ n is a filtration for each j ∈ {1, …, k}. In this article we explain the underlying connection between the analysis of the maximal functions of the corresponding coherent vector and basic combinatorial properties of the uncentered Hardy–Littlewood maximal operator. Following a classical approach of Grafakos, Kinnunen and Montgomery-Smith, we establish an appropriate version of Doob’s celebrated maximal estimate.

2010 AMS Mathematics Subject Classification: Primary 60E15; Secondary 60G42.

Keywords and phrases: coherent distribution, maximal operator, martingale, best constants.

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