Weighted Estimates For Doob Maximal Operators

Martingale spaces and harmonic analysis are connected by the dyadic techniques(martingale methods).Dyadic techniques have been widely used in harmonic analysis.By establishing a dyadic system in Euclidean spaces,the problem naturally transformed into the corresponding problem in dyadic martingale spaces,thus we can use the theory of martingale spaces to study the problem.In the past two decades,the dyadic methods have played a great role in harmonic analysis,especially in the breakthrough of weighted theory.The study of weighted inequalities in martingale spaces has a long history.The theory of Ap was introduced by Izumisawa and Kazamaki,under the premise of attaching regularity conditions to martingales,they made it clear that Doob maximal operators satisfy strong-type inequalities.Long used test conditions to characterize the two-weight strong-type inequalities of Doob maximal operators.Two-weight theory in harmonic analysis shows that Orlicz bump conditions,mixed conditions and entropy bumps are easier to detect than the test conditions.In martingale spaces,the Luxemburg norm in the bump conditions is not suitable for conditional expectation,so we study the mixed weighted conditions and the entropy bumps in the martingale spaces.In the(discrete)martingale spaces,corresponding to the sparse family in harmonic analysis,Tanaka and Terasawa construct a family of principal sets.In the martingale spaces with a continuous-time parameter,sparse domination of differential subordinate martingales begins to appear.This paper studies the weighted theory in martingale spaces by using principal sets and sparse domination.First,we study the weighted theory in filtered measure spaces,find the conditional sparsity property of the principal sets and characterize the boundedness of Doob maximal operators by the construction of the principal sets.Second,we study the sparse domination in an martingale setting with a continuous time parameter and prove that the domination has the property of conditional sparsity.With this property,we present amixed estimation ApαArβ which improves the sharp weighted Lp estimation and give entropy bumps of the maximal function in this setting.This article includes three chapters.The first chapter is the preface,which introduces the weighted theory in dyadic harmonic analysis and martingale theory,as well as the main contents we study.In the second chapter,we introduce Doob maximal operators on filtered spaces and weighted inequalities for Doob maximal operators.In the third chapter,we study the weighted estimation by continuous-time sparse domination in martingale spaces,which proves that the domination has the property of conditional sparsity.In addition,we study the related weighted problems.