6533b7d0fe1ef96bd125ab78
RESEARCH PRODUCT
Rectifiability and analytic capacity in the complex plane
Pertti Mattilasubject
Pure mathematicsBounded functionMathematical analysisComplex measureAnalytic capacityOpen setHausdorff spaceFunction (mathematics)Complex planeMathematicsAnalytic functiondescription
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).
year | journal | country | edition | language |
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1995-04-28 |