Ideals in big Lipschitz algebras of analytic functions

Thomas Vils Pedersen¹

¹ Department of Mathematics and Physics The Royal Veterinary and Agricultural University Thorvaldsensvej 40 DK-1871 Frederiksberg C, Denmark

Studia Mathematica, Tome 161 (2004) no. 1, pp. 33-59

Résumé

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 $Q$.

Détail

DOI : 10.4064/sm161-1-3

Keywords: gamma mit lambda gamma lipschitz algebra functions analytic unit disc mathbb which satisfy lipschitz condition order gamma overline mathbb closed set unit circle mathbb inner function gamma closed ideal mit lambda gamma consisting those functions mit lambda gamma which hbox f z w gamma rightarrow iii mit lambda gamma closed ideal mit lambda gamma mathbb hbox every greatest common divisor inner parts non zero functions main conjecture about ideal structure mit lambda gamma gamma i subseteq every closed ideal mit lambda gamma confirm conjecture closed ideals mit lambda gamma which countable obtain partial results where moreover every wks closed ideal mit lambda gamma form mit lambda gamma mit lambda gamma closed set subseteq mathbb inner function