The Douady-Earle extension of quasihomographies

• Autorzy: Ken-Ichi Sakan , Józef Zając
• Języki publikacji: EN

Abstrakty

EN

Quasihomography is a useful notion to represent a sense-preserving automorphism of the unit circle T which admits a quasiconformal extension to the unit disc. For K ≥ 1 let $A_T(K)$ denote the family of all K-quasihomographies of T. With any $f ∈ A_T(K)$ we associate the Douady-Earle extension $E_f$ and give an explicit and asymptotically sharp estimate of the $L_∞$ norm of the complex dilatation of $E_f$.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Banach Center Publications
• Rocznik: 1996
• Tom: 37
• Numer: 1
• Strony: 35-44

Daty

wydano

1996

Twórcy

autor

Ken-Ichi Sakan

• Dept. of Mathematics, Osaka City University, Sugimoto, Sumiyoshi-ku, Osaka, Japan

autor

Józef Zając

• Institute of Mathematics, Polish Academy of Sciences, Narutowicza 56, PL-90-136 Łódź, Poland

Bibliografia

• [DE] A. Douady and C.I. Earle, Conformally natural extension of homeomorphisms of the circle, Acta Math. 157 (1986), 23-48.
• [K] J.G. Krzyż, Quasicircles and harmonic measure, Ann. Acad. Sci. Fenn. 12 (1987), 19-24.
• [LP] A. Lecko and D. Partyka, An alternative proof of a result due to Douady and Earle, Ann. Univ. Mariae Curie-Skłodowska Sectio A 42 (1988), 59-68.
• [P1] D. Partyka, The maximal dilatation of Douady and Earle extension of a quasisymmetric automorphism of the unit circle, Ann. Univ. Mariae Curie-Skłodowska Sectio A 44 (1990), 45-57.
• [P2] D. Partyka, A distortion theorem for quasiconformal automorphisms of the unit disc, Ann. Polon. Math. 55 (1991), 277-281.
• [P3] D. Partyka, The maximal value of the function $[0;1] ∋ r ↦ Φ_{K}^{2}(√r) - r$, Bull. Soc. Sci. Lettres Łódź 45 Sér. Rech. Déform. 20 (1995), 49-55.
• [Z1] J. Zając, The distortion function $Φ_K$ and quasihomographies, Current Topics of Analytic Function Theory, (1992), 403-428.
• [Z2] J. Zając, Quasihomographies, Monograph, Preprint.