Some Studies On Doubling Measures,δ-monotonicity And Quasisymmetric Maps

In this paper, we discuss the definitions and properties of doubling measures, δ-monotonicity and quasisymmetric maps. We prove:(1) Let μ be a doubling mea-sure on Rn that satisfies the decay condition∫|z|＞1|z|-1dμ(z)＜∞. Then the mapping fμ defined by (2.4) is η-quasisymmetric mapping. Furthermore,fμ is δ-monotonicity.(2) Let f:Rnâ†'Rn be a nonconstant δ-monotone mapping, n≥2. Then the weight‖Df‖is isotropic doubling.It contains five parts. In part one, we summarize the works done by former re-searchers. Then we draw forth the problems we will discuss. In part two, we show the main results of this paper. In parts three and four, we give some knowledge regarding doubling measures, monotone mappings and isotropic doubling measures and prove our main results. In part five, we give two questions which can be discussed further.