Extensions of the Laurent Decomposition and the spaces $A^p(Ω)$

Abstract

We generalize the classical Laurent decomposition in the setting of domains Ω ⊆ C bounded by Jordan curves. This leads us to study the Fréchet spaces A p (Ω), and their relation to the spaces C p (∂Ω). In the final section, we examine the case of a non Jordan domain Ω. CONTENTS 1. Introduction 1 2. Preliminaries 2 3. The case of the circle 3 4. Jordan Domain 7 5. Internally Tangent Circles 13 References 14

One sided conformal collars and the reflection principle

“…We also can extend the above fact for every finite number p + 1 of derivatives, provided that the one-sided conformal collar is of class A p . ( [3], [5]) Furthermore, instead of considering the unit circle we can replace it by one-sided free boundary arcs ( [1])which are analytic arcs and obtain similar results (see Th.2.2 below) We also mention that our result relates to the considerations of ( [2]) Finally, a second application of the method of the present paper can be found in [7], see proof of Th. 4.2 where it is proven the following.…”