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Rectifiability and analytic capacity in the complex plane

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Analytic capacity and removable sets In this chapter we shall discuss a classical problem in complex analysis and its relations to the rectifiability of sets in the complex plane C . The problem is the following: which compact sets E ⊃ C are removable for bounded analytic functions in the following sense? (19.1) If U is an open set in C containing E and f : U\E → C is a bounded analytic function, then f has an analytic extension to U . This problem has been studied for almost a century, but a geometric characterization of such removable sets is still lacking. We shall prove some partial results and discuss some other results and conjectures. For many different function classes a complete solution has been given in terms of Hausdorff measures or capacities. For example, if the boundedness is replaced by the Holder continuity with exponent α, 0 E is that H 1+α ( E ) = 0, see Exercise 4, Dolzenko [1] and Uy [2], and for the corresponding question for harmonic functions Carleson [1]. Kral [1] proved that for the analytic BMO functions the removable sets E are characterized by the condition H 1 ( E ) = 0. The problem (19.1) is more delicate, because the metric size is not the only thing that matters; the rectifiability structure also seems to be essential as we shall see. Ahlfors [1] introduced a set function γ, called analytic capacity , whose null-sets are exactly the removable sets of (19.1).