The Dual And Atomic Decomposition Of Variable Lebesgue Weighted Tent Spaces

In the 1970s,R.R.Coifman and G.Weiss introduced the concept of homogeneous space in order to generalize various functional Spaces and singular integral operator theories of Euclidean space to more general metric Spaces,and they started type space increases and analysis research,and obtained many important results tent space is a kind of contact area of the integral and Carleson measure function space,R.R.Coifiman,Y.Meyer,E.M.Stein and others for the first time in 1985 in the thesis put forward the definition of tent space in Euclidean Space,the space is also given the duality theory and atomic decomposition theorem,the theory and the theorem in harmonic analysis and partial differential equations,there are many critical application.In this paper,under the research background that the bottom space is homogeneous space,combined with the theory of variable Lebesgue space and tent space theory,the dual and atomic decomposition of the variable weighted tent space are established.In the first part,we introduce some duality properties of tent space with constant index and weighted tent space with constant index are introduced,and then we proved the duality properties of variable tent spaceTp(·),q(?)and variable Lebesgue weighted tent spaceTsp(·),q(?).In the second part,we introduce the variable tent space Tp(·),2(?)atomic features and then transition to the space after weighting,as well as the general constant change 2 to the general situation,proved that the tent space Tsp(·),2(?)and the tent space Tp(·),q(?)atomic decomposition theorem still stood.