The Doob Maximal Inequality In Martingale Spaces With Variable Exponents

Abstract:The boundedness of Hardy-Littlewood maximal operator in variable exponent lebesgue spaces have already an systematic and perfect research results. But how to study the variable Doob maximal operator in probability space still is a problem that many scholars try to resolve.The main difficulty have two sides:The first one is the contractive inequality of conditional expectation E(｜f｜｜Fn)p(.)≤E(｜f｜p(.)Fn) is difficult to prove; And the second one is that the space Lp(.)(Ω,F,P) does’t a rearrangement invariant space. The main purpose of this thesis is to summarize the boundedness of Hardy-Littlewood maximal operator of variable exponent lebesgue spaces, and we prove the strong type estimate and the weak type estimate of Doob maximal operator.In chapter1we introduce the background and the main work of this thesis briefly.In chapter2we introduce the definitions of martingale spaces with variable exponents and Doob maximal operator, then give several lemmas we will use.In chapter3we summarize the existing results of the boundedness of Hardy-Littlewood maximal operator.In chapter4we introduce the existing results of Doob maximal operator. Chapter5is the main content. We prove the boundedness of Doob maximal operator in martingale spaces with variable exponents. Which extend the results of chapter3.