Theorems of Duality and Characterization of Some Spaces on HEISENBERG Group
|Keywords||Heisenberg group Q_p space tent space Carleson measure Hardy-Hausdorff space duality atomic decomposition|
It is well known that the duallities and characterizations of function spacesplay an important role in harmonic analysis. In this thesis, the tent spaces onthe Siegel upper half space, which are defined in terms of Choquet integralswith respect to Hausdorff capacity on the Heisenberg group, are introduced.Some results about characterizations of p-Carleson measures and the dualityand the atomic decomposition of the new tent spaces are studied. We alsointroduce the Hardy-Hausdorff spaces on the Heisenberg group, and discuss theatomic characterization of these spaces. With the help of the Hardy-Hausdorffspaces, the new tent spaces and the p-Carleson measures, we discuss the dualitybetween Hardy-Hansdorff spaces and Q_p spaces on the Heisenberg group, andobtain that the predual of Q_p spaces on the Heisenberg group is the Hardy-Hausdorff spaces. The atomic decomposition of Q_p spaces on the Heisenberggroup is studied also. The thesis consists of the following parts.In the first chapter, we recite the investigative current circumstance of Q_pspaces, recall some definitions about the Heisenberg group and some resultsabout Hardy space theory on the Heisenberg group. Some known conclusionsand the development of Q_p spaces which we interested in, are introduced. Thebackground and significance of our studies are also noted.In the second chapter, the tent spaces on the Siegel upper half space, whichare defined in terms of Choquet integrals with respect to Hausdorff capacityon the Heisenberg group, are introduced. And some characterizations of thefractional Carleson measures in the Siegel upper half space are discussed. Thenthe atomic decomposition and the duality of the tent spaces on the Siegel upperhalf space are obtained.In the third chapter, by the results of the tent spaces and the fractionalCarleson measures on the Siegel upper half space in chapter 2, we introduce theHardy-Hausdorff spaces on the Heisenberg group. And the atomic decomposi- tion of the Hardy-Hausdorff spaces is discussed. Then we obtained that the dualspaces of the Hardy-Hausdorff spaces are Q_p spaces on the Heisenberg group. Itis that the predual of Q_p spaces on the Heisenberg group is the Hardy-Hausdorffspaces.We continue to discuss Q_p spaces on the Heisenberg group in the lastchapter. And using the dyadic results and the atomic decomposition of theHardy-Hausdorff spaces and duality, the atomic decomposition of Q_p, spaces onthe Heisenberg group are obtained.This research is supported in part by grants from the National NaturalScience Foundation of China (No. 10671115) and the Natural Science Foundationof Guangdong Province (No.04011000).