Weighted Inequalities For Very Large Operators

Weighted inequalities of function space arose naturally in Fourier analysis.Afterwards,it has been given a lot of attention due to its closely relationship with a variety of subjects,such as operator extrapolation theory,the boundary value problem of Laplace equation in Lipschitz domain,vector-valued function inequalities,nonlinear partial differential equations and integral equations,etc.However,it was until the 1970s that we had a deeper understanding for weighted theory,which is mainly benefited by relevant research by Muckenhoupt.Currently,dyadic techniques played an important role in the weighted theory,so it has received much attention.In fact,for harmonic analysis,many conclusions can only be more intuitive in dyadic system.On one hand,dyadic decomposition of Rn is a powerful stopping-time construction,which has many interesting application.On the other hand,dyadic techniques are link between weighted inequalities in harmonic analysis and weighted inequalities in martingale spaces,then some ideas in harmonic analysis become brief when viewed from probabilistic.As maximal operators control Calderon-Zygmund operators,weighted inequalities of maximal operators is one of the important topics in weighted theory.In this paper,we focus on the weighted inequalities of maximal operators and their related topics.Firstly,we characterize strong-type and weak-type inequalities with two weights for positive operators.Several mixed bounds for Doob maximal operator on the filtered measure spaces are also obtained,and our approaches are mainly based-on the construction of principal sets.Secondly,for a general dyadic grid,we give a Calderon-Zygmund type decomposition,which is the principle fact about the multilinear maximal function m on upper half-spaces.Using the decomposition,we study the boundedness of m.We obtain a natural extension to the multilinear setting of Muckenhoupt’s weak-type characterization,and also partially get characterizations of Muckenhoupt’s strong-type inequalities.Assuming the reverse Holder’s condition,we get a multilinear analogue of Sawyer’s two weight strong-type inequality and Hytonen-Perez’s mixed-type inequality.Finally,We study weighted mixed inequalities on the product spaces with different Muckenhoupt bases,and our methods are mainly based on the formalism of families of extrapolation pairs and Minkowski integral inequality.This paper includes four chapters.In the first chapter,we introduce weighted theories of dyadic harmonic analysis and martingale spaces,as well as the main contents we study.In the second chapter,positive operators and Doob maximal operators are our main research objects.At the same time,the weighted inequalities have been characterized for these operators on the filtered measure spaces.In the third chapter,we study the weighted theory on the upper half-spaces,and weighted estimates for multilinear maximal functions are also obtained.In the last chapter,we research the weighted theory of mixed norm.In addition,several weighted inequalities of mixed norm on product spaces with Muckenhoupt bases are gained.