UNIVERSITY
OF WROCAW
 
Main Page
Contents
Online First
General Information
Instructions for authors


VOLUMES
43.2 43.1 42.2 42.1 41.2 41.1 40.2
40.1 39.2 39.1 38.2 38.1 37.2 37.1
36.2 36.1 35.2 35.1 34.2 34.1 33.2
33.1 32.2 32.1 31.2 31.1 30.2 30.1
29.2 29.1 28.2 28.1 27.2 27.1 26.2
26.1 25.2 25.1 24.2 24.1 23.2 23.1
22.2 22.1 21.2 21.1 20.2 20.1 19.2
19.1 18.2 18.1 17.2 17.1 16.2 16.1
15 14.2 14.1 13.2 13.1 12.2 12.1
11.2 11.1 10.2 10.1 9.2 9.1 8
7.2 7.1 6.2 6.1 5.2 5.1 4.2
4.1 3.2 3.1 2.2 2.1 1.2 1.1
 
 
WROCAW UNIVERSITY
OF SCIENCE AND
TECHNOLOGY

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,ℙ). Fix k ∈ ℤ+ and let {𝒢ij}1 ≤ i ≤ n, 1 ≤ j ≤ k be a reference family of sub-σ-fields of such that {𝒢ij}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.

Download:        Full text