New Research On Q_p And Hyperbolic Q_p Spaces

The history of studying on Qp spaces is not very long. It has been actively investigated since they were introduced . There still exist many problems waiting for solution. Many elegant achievements and series of consequences gained during the past ten years have far-reaching influence on the development of function theory and other research fields concerned. Therefore, the development of Qp spaces is attracting more and more attention.Firstly, this paper introduces the background of Qp spaces and their development updated. Secondly, this paper roundly contains research results by author, many of which, though have been proved before, are reproved here in new and more convenient ways. Also, this paper answers a question in the literature [AuLa] and generalizes some well-known results. Finally, this paper defines a new space, the hyperbolic Qp space (0 < p < oo), denoted by Qp. Some profound research on this kind of space are made, such as the equivalent definitions, Mobius invariance, comparisons among several hyperbolic spaces, and so on.This research is supported in part by grants from the National Natural Science Foundation of China (No.10171058) and the Natural Science Foundation of Guangdong Province (No.010446).