Abstract
A quasisymmetric graph is a curve whose projection onto a line is a quasisymmetric map. We show that this class of curves is related to solutions of the reduced Beltrami equation and to a generalization of the Zygmund class Λ*. This relation makes it possible to use the tools of harmonic analysis to construct nontrivial examples of quasisymmetric graphs and of quasiconformal maps. Kovalev was supported by the NSF grant DMS-0968756.
Highlights
Let X and Y be subsets of a Euclidean space Rn
Y is quasisymmetric if there is a homeomorphism η : [0, ∞) → [0, ∞) such that for any triple of distinct points a, b, x ∈ X
We call a set Γ ⊂ C a quasisymmetric graph if the orthogonal projection of Γ onto R is a quasisymmetric homeomorphism between Γ and R
Summary
Y is quasisymmetric if there is a homeomorphism η : [0, ∞) → [0, ∞) such that for any triple of distinct points a, b, x ∈ X (1.1). There exists a constant s0 > 0 such that any s-quasisymmetric graph Γ ⊂ C with s < s0 is the image of R under a reduced quasiconformal mapping f : C → C. There exists a reduced quasiconformal mapping f : C → C whose restriction to the line segment [0, 1] has infinite Φq-variation for every 0 < q < 1. This result was previously known only for q < 1/2 [25, Remark 4.1]. The images of R under reduced quasiconformal maps C → C are precisely quasisymmetric graphs and vertical lines. It would be interesting to investigate, e.g., 2-dimensional quasisymmetric graphs in R4, but we do not pursue this direction here
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