In complex analysis, the analytic capacity of a compact subset K of the complex plane is a number that denotes "how big" a bounded analytic function on C K can become. Roughly speaking, γ(K) measures the size of the unit ball of the space of bounded analytic functions outside K.
Contents
- Definition
- Ahlfors function
- Analytic capacity in terms of Hausdorff dimension
- Positive length but zero analytic capacity
- Vitushkins conjecture
- Removable sets and Painlevs problem
- References
It was first introduced by Ahlfors in the 1940s while studying the removability of singularities of bounded analytic functions.
Definition
Let K ⊂ C be compact. Then its analytic capacity is defined to be
Here,
(note that usually
Ahlfors function
For each compact K ⊂ C, there exists a unique extremal function, i.e.
Analytic capacity in terms of Hausdorff dimension
Let dimH denote Hausdorff dimension and H1 denote 1-dimensional Hausdorff measure. Then H1(K) = 0 implies γ(K) = 0 while dimH(K) > 1 guarantees γ(K) > 0. However, the case when dimH(K) = 1 and H1(K) ∈ (0, ∞] is more difficult.
Positive length but zero analytic capacity
Given the partial correspondence between the 1-dimensional Hausdorff measure of a compact subset of C and its analytic capacity, it might be conjectured that γ(K) = 0 implies H1(K) = 0. However, this conjecture is false. A counterexample was first given by A. G. Vitushkin, and a much simpler one by J. Garnett in his 1970 paper. This latter example is the linear four corners Cantor set, constructed as follows:
Let K0 := [0, 1] × [0, 1] be the unit square. Then, K1 is the union of 4 squares of side length 1/4 and these squares are located in the corners of K0. In general, Kn is the union of 4n squares (denoted by
Vitushkin's conjecture
Suppose dimH(K) = 1 and H1(K) > 0. Vitushkin's conjecture states that
In this setting, K is (purely) unrectifiable if and only if H1(K ∩ Γ) = 0 for all rectifiable curves (or equivalently, C1-curves or (rotated) Lipschitz graphs) Γ.
Guy David published a proof in 1998 for the case when, in addition to the hypothesis above, H1(K) < ∞. Until now, very little is known about the case when H1(K) is infinite (even sigma-finite).
Removable sets and Painlevé's problem
The compact set K is called removable if, whenever Ω is an open set containing K, every function which is bounded and holomorphic on the set Ω K has an analytic extension to all of Ω. By Riemann's theorem for removable singularities, every singleton is removable. This motivated Painlevé to pose a more general question in 1880: "Which subsets of C are removable?"
It is easy to see that K is removable if and only if γ(K) = 0. However, analytic capacity is a purely complex-analytic concept, and much more work needs to be done in order to obtain a more geometric characterization.