In the mathematical discipline of 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.
Let K ⊂ C be compact. Then its analytic capacity is defined to be
Note that , where . However, usually .
If A ⊂ C is an arbitrary set, then we define
Removable sets and Painlevé's problemEdit
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.
For each compact K ⊂ C, there exists a unique extremal function, i.e. such that , f(∞) = 0 and f′(∞) = γ(K). This function is called the Ahlfors function of K. Its existence can be proved by using a normal family argument involving Montel's theorem.
Analytic capacity in terms of Hausdorff dimensionEdit
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 capacityEdit
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 John B. 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 ) of side length 4−n, each being in the corner of some . Take K to be the intersection of all Kn then but γ(K) = 0.
Let K ⊂ C be a compact set. Vitushkin's conjecture states that
where denotes the orthogonal projection in direction θ. By the results described above, Vitushkin's conjecture is true when dimHK ≠ 1.
Guy David published a proof in 1998 of Vitushkin's conjecture for the case dimHK = 1 and H1(K) < ∞. In 2002, Xavier Tolsa proved that analytic capacity is countably semiadditive. That is, there exists an absolute constant C > 0 such that if K ⊂ C is a compact set and , where each Ki is a Borel set, then .
David's and Tolsa's theorems together imply that Vitushkin's conjecture is true when K is H1-sigma-finite. However, the conjecture is still open for K which are 1-dimensional and not H1-sigma-finite.
- Mattila, Pertti (1995). Geometry of sets and measures in Euclidean spaces. Cambridge University Press. ISBN 0-521-65595-1.
- Pajot, Hervé (2002). Analytic Capacity, Rectifiability, Menger Curvature and the Cauchy Integral. Lecture Notes in Mathematics. Springer-Verlag.
- J. Garnett, Positive length but zero analytic capacity, Proc. Amer. Math. Soc. 21 (1970), 696–699
- G. David, Unrectifiable 1-sets have vanishing analytic capacity, Rev. Math. Iberoam. 14 (1998) 269–479
- Dudziak, James J. (2010). Vitushkin's Conjecture for Removable Sets. Universitext. Springer-Verlag. ISBN 978-14419-6708-4.
- Tolsa, Xavier (2014). Analytic Capacity, the Cauchy Transform, and Non-homogeneous Calderón–Zygmund Theory. Progress in Mathematics. Birkhäuser Basel. ISBN 978-3-319-00595-9.