Quasi-conformal Mapping. On The Heisenberg Group Royden Algebra Characterize Conformal Invariant

In chapter 2 , we establish the characterization of quasiconformal mappings on the Heisenberg group by using the Banach space M_P(Ω), where Ω is a bounded domain in the Heisenberg group. Namely, if Ω, Ω' are bounded domains in HP, and if f : Ω â†'Ω' is a homeomorphism, then f is a quasiconformal mapping if and only if the mapping φf : M_2n+2(Ω/) —> M_2n+2(Ω), u —> u o f, is a Banach algebra isomorphism.In chapter 3, we recall the definitions and some properties of quasiregular mapping on the Heisenberg group, and define the quantities μ_G, λ_g by using modulus of families of horizontal curves. Then we prove the following two results :(1) If f : G —> G' = fG is a K—quasiconformal mapping, where G, G' be domains in Hⁿ.Thenwhere x,y ∈ G and x ≠ y.(2)μ_G(x, y), λ_g(x, y) are conformal invariants.