On the preservation of Lipschitz classes under the Faber transformation
Let [omega] be a nonempty open, simply connected subset of the complex plane and let [delta] = \z ǁ ǁ z ǁ < 1. Then there is a bijective analytic mapping g: \doubc - [macron][omega] → \doubc - [macron][delta]. We note that g has a continuous extension to the boundary [gamma] of [omega]. Let A([delta]) be the Banach algebra of functions continuous on [macron][delta] and analytic on [delta];We consider the tranformation T mapping f in A([delta]) to (Tf)(w) = 1 [over] 2[pi] i ϵt[limits][subscript][gamma] f(g(t)) [over] t-w dtin the Banach algebra of functions continuous on [macron][omega] and analytic on [omega]. The mapping T is called the Faber transformation and is an injective continuous linear transformation;It is of interest to determine the properties of f which are preserved under the Faber transformation T. In particular, we assume that f satisifies a Lipschitz condition on [macron] [delta] and consider the Lipschitz class of Tf on [macron][omega]. We show that the Lipschitz class of Tf is affected by the Lipschitz class of f and the smoothness of [gamma], and, under suitable conditions on [gamma], we obtain qualitative results when g satisfies a Lipschitz condition.