The smallest positive eigenvalue of a quasisymmetric automorphism of the unit circle

Abstract

Abstract. This paper provides sufficient conditions on a quasisymmetric automorphism γ of the unit circle which guarantee the existence of the smallest positive eigenvalue of γ. They are expressed by means of a regular quasiconformal Teichmüller self-mapping ϕ of the unit disc ∆. In particular, the norm of the generalized harmonic conjugation operator Aγ : H → H is determined by the maximal dilatation of ϕ. A characterization of all eigenvalues of a quasisymmetric automorphism γ in terms of the smallest positive eigenvalue of some other quasisymmetric automorphism σ is given.

Harmonic extensions of quasisymmetric mappings

“…If y E QOT(M) then both E , and F , are quasiconformal in A and their maximal dilatations are equal. In [PI,Th. 1.21 several distortion results concerning the harmonic map Pl,20, were given for a E A and y E QOT(M), in terms of M and la], which enabled the author to estimate the maximal dilatation of F , and E , in terms of M .…”

Generalized Fourier coefficients of a quasisymmetric homeomorphism and the Fredholm eigenvalue

“…In a basic paper [Pa2], Partyka introduced the notion of generalized Fourier coefficients and discussed these coefficients for a quasisymmetric homeomorphism. Recently, Kühnau ([Ku1,Ku2,Ku3,Ku4]) gave some new characterizations of the Fredholm eigenvalue for a quasicircle by means of these generalized Fourier coefficients.…”

“…In this note, we give a fast approach to Kühnau's results and also answer some questions posed by Kühnau. Let = {z : |z| < 1} denote the unit disk in the extended complex planeĈ; * =Ĉ − is the exterior of , and S 1 = ∂ = ∂ * is the unit circle. According to Partyka [Pa2] (see also [NS], [TT]), the (m, n)-th generalized Fourier coefficient for a function h on the unit circle is defined as Note that h(m, n) is the m-th Fourier coefficient for h n in the usual sense. Partyka [Pa2] discussed these generalized Fourier coefficients for a quasisymmetric homeomorphism.…”

“…According to Partyka [Pa2] (see also [NS], [TT]), the (m, n)-th generalized Fourier coefficient for a function h on the unit circle is defined as Note that h(m, n) is the m-th Fourier coefficient for h n in the usual sense. Partyka [Pa2] discussed these generalized Fourier coefficients for a quasisymmetric homeomorphism. Let Hom + (S 1 ) denote the set of all sense-preserving homeomorphisms of S 1 onto itself.…”

On a modification of the Poisson integral operator

“…e.g. [14], [24], [21], [22], [23] and [25]. Therefore the author is much indebted to professor Jan Krzyż for introducing him to the theory of Fredholm eigenvalues of a Jordan curve.…”