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Preprints Archive: Abstract of IC2010043 (2010)

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Outer functions in analytic weighted Lipschitz algebras

by B. Bouya

Document info: Pages 16, Figures 0.

The analytic weighted Lipschitz algebra $\cL$ is the Banach algebra of all analytic functions on the unit disk $\D,$ that are continuous on $\overline{\D}$ and such that $\sup_{z,w\in\overline{\D}\atop z\neq w}\frac{|f(z)-f(w)|}{\omega(|z-w|)}< +\infty$, where $\omega$ is a modulus of continuity. We give a new characterization of outer functions in $\Lambda_{\omega},$ by their modulus in $\T$. As application, we obtain a refinement of Shirokov's construction of outer functions in $\cL$ vanishing on a given $\omega-$Carleson set. We obtain also an extension of Havin-Shamoyan-Carleson-Jacobs Theorem to an arbitrary modulus of continuity.

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