Some Problems On Isotropic Doubling Measure,δ-monotone And Quasiconformal Mapping

In this paper, we discuss the definitions and properties of doubling measures δ-monotone mappings and quasisymmetric mappings. We prove:(1)Suppose that Ω(?)Rn,n≥2and f:Ωâ†'Rn is a non-constant δ-monotone mapping.If B is a closed ball such that B(x,2r)(?)Ω then f is η-quasisymmetric on B with η depending only on δ.(2) Let be f:Ωâ†'Rn a non-constant δ-monotone mapping, then the weight||Df||is isotropic doubling.(3) For every n≥2there exists an isotropic doubling measure μ on R" that is purely singular with respect to L".(4) For every n≥2there exists an isotropic doubling measure μ on R" and a bi-Lipschitz mapping f:Rnâ†'Rn such that the push-forward measure f#μ is not isotropic doubling.It contains four parts.(1) We summarize the works done by former researchers. Then we draw four problems we will discuss and we show the main result of this paper.(2) We respectively introduces the knowledge of doubling measures,δ-monotone mappings, quasisymmetric mappings and quasiconformal mappings.(3) We prove our main results.(4) We give three questions which can be discuss further.