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 < α < 1, then the necessary and sufficient condition for the removability of E is that H1+α(E) = 0, see Exercise 4, Dolzenko [1] and Uy [2], and for the corresponding question for harmonic functions Carleson [1]. Král [1] proved that for the analytic BMO functions the removable sets E are characterized by the condition H1(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).