Ideals in big Lipschitz algebras of analytic functions

Volume 161 / 2004

Thomas Vils Pedersen Studia Mathematica 161 (2004), 33-59 MSC: 46J20, 46J15, 30H05. DOI: 10.4064/sm161-1-3

Abstract

For $0<\gamma\le1$, let ${\mit\Lambda}_{\gamma}^+$ be the big Lipschitz algebra of functions analytic on the open unit disc ${\mathbb D}$ which satisfy a Lipschitz condition of order $\gamma$ on $\overline{\mathbb D}$. For a closed set $E$ on the unit circle ${\mathbb T}$ and an inner function $Q$, let $J_{\gamma}(E,Q)$ be the closed ideal in ${\mit\Lambda}_{\gamma}^+$ consisting of those functions $f\in{\mit\Lambda}_{\gamma}^+$ for which

(i) $f=0 \hbox{ on }E$,

(ii) $|f(z)-f(w)|=o(|z-w|^{\gamma})$ as $d(z,E),d(w,E)\rightarrow0$,

(iii) $f/Q\in{\mit\Lambda}_{\gamma}^+$.

Also, for a closed ideal $I$ in ${\mit\Lambda}_{\gamma}^+$, let $E_I=\{z\in{\mathbb T}:f(z)=0\hbox{ for every }f\in I\}$ and let $Q_I$ be the greatest common divisor of the inner parts of non-zero functions in $I$. Our main conjecture about the ideal structure in ${\mit\Lambda}_{\gamma}^+$ is that $J_{\gamma}(E_I,Q_I)\subseteq I$ for every closed ideal $I$ in ${\mit\Lambda}_{\gamma}^+$. We confirm the conjecture for closed ideals $I$ in ${\mit\Lambda}_{\gamma}^+$ for which $E_I$ is countable and obtain partial results in the case where $Q_I=1$. Moreover, we show that every \wks closed ideal in ${\mit\Lambda}_{\gamma}^+$ is of the form $\{f\in{\mit\Lambda}_{\gamma}^+:f=0$ on $E$ and $f/Q\in{\mit\Lambda}_{\gamma}^+\}$ for some closed set $E\subseteq{\mathbb T}$ and some inner function $Q$.

Authors

  • Thomas Vils PedersenDepartment of Mathematics and Physics
    The Royal Veterinary and Agricultural University
    Thorvaldsensvej 40
    DK-1871 Frederiksberg C, Denmark
    e-mail

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