The Dyadic Derivative, Cesàro Mean And Inequalities For Martingales

The dyadic derivative is a part of harmonic analysis in local fields. It plays avery important role in classical analysis. There are many differences between harmonicanalysis and harmonic analysis in local fields. For example, the add and muliplyingoperations are closed in R~n, but it cannot be used directly in local fields. We canconsider the Walsh system. The general add operation is not closed in [0,1). There aremany problems for this. We must define new operations that they are closed in [0,1).For many methods in classical analysis are not applying in local fields, mathematiciansdid a lot of work on harmonic analysis in local fields and achieved many importantresults.Derivatives and differentiations play a very important role in classical analysis.Many mathematicians have tried to investigate the corresponding concepts for thosefunctions defined on local field, since the Leibniz-Newton formula for derivatives can-not be used in those settings. On the basis of the idea of derivatives introduced byButzer and Wagner, the dyadic derivative and integral on Walsh system, Vilenkin groupshas undergone a flourishing development. With obtaining the dyadic analogue of theLeibniz-Newton formula, the results of the dyadic derivative and integral are enriched.On the other hand, in the course of the development of martingale theory, inequal-ities of martingale space have been a concerned research hot spot. People always studythe properties of operators via the corresponding martingale inequalities, whereby weobtain the relationship between two operators, further, the inclusions of many martin-gale spaces are established.The relation between the dyadic derivative and martingale has been inosculated.There are many results of the dyadic derivative and integral in martingale spaces.The maximal operators of the dyadic derivative and integral are bounded on real Hardy spaces, the same to Cesaro mean. This leaved behind some questions: Howabout the boundedness of other operators in local field? In the thesis, we studyωH_p~s spaces and Banach spaces. The real Hardy space is a Hilbert space which is2-uniformly smooth and 2-uniformly convex and has property of RN. Atomic decom-position plays an important role in the theory of Hardy martingale spaces in recentyears. As a powerful technique, atomic decomposition gave simple proof of somemartingale inequalities. To Banach-space valued martingale, it is tightly related tothe geometrical properties and structure of the spaces which the martingale takesvalues in. So, we take atomic decomposition inωH_p~s and Banach spaces. And itis proved that the maximal operators of dyadic derivative and integral are boundedfromωH_p~s toωL_p(1/2＜p≤∞). We considered some B-valued spaces. For example,_p(?)_r(X). The same results can be obtained in Banach spaces. The maximal operatorof Cesaro mean is also bounded on Banach spaces. In a word, these results extend thekinds of space previously.The framework of this thesis consists of the following six parts:Chapter 1 is an introduction on the background, motivation and the principle resultsof the dissertation.In chapter 2, we give several definitions of martingale,dyadic derivative and in-tergal,the smooth and convex of Banach spaces and introduce some important resultswhich related with this paper.Chapter 3 studies the boundedness of the maximal operator of dyadic derivativeand integral inωH_p~s by atomic decomposition. It is proved that the maximal operatorand the general maximal operator of dyadic derivative and integral are bounded fromωH_p~s toωL_p(1/2＜p≤∞) and are of weak type(1,1).In chapter 4, we establish the relationships between the geometrical properties of the Banach space and the dyadic derivative. We prove several inequalities using theproperty of smooth and convexity. The maximal operator of the dyadic derivative andintegral is bounded on Banach spaces and is of weak type(1,1).In chapter 5, we define two operators—the general maximal operator of Cesaromean and the general maximal operator of Cesaro mean on Banach spaces. It is provedthat these maximal operators are bounded on Banach spaces and are of weak type(1,1).It depends the geometrical properties of the Banach space.Chapter 6 studies the B-valued martingale spaces with small index. Atomic de-composition can be obtained by the geometrical properties of the Banach space. In thischapter, we prove the maximal operator of the dyadic derivative and integral is boundedon these Banach spaces by atomic decomposition. Coinstantaneous, It is proved thatthe maximal operator of Cesaro mean is bounded on these Banach spaces too.There are mainly several innovations in this dissertation: Firstly, we studied har-monic analysis in local field which connected with martingale. This method overcomedsome difficulties of operator in local field. Extend some results previously by the kernelof Walsh-Dirichlet. Secondly, we dealed with questions about small index spaces,weakHardy spaces and Lorentz martingale space by atomic decomposition. We obtained aseries of inequalities about the maximal operators. Lastly, we introduced vector functionin harmonic analysis. We found that the results have some influences by the property ofthe geometrical of Banach space. This influences make the results of classical harmonicanalysis to special case.