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Analytic capacity
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 from \( C \ K \) can become. Roughly speaking, γ(K) measures the size of the unit ball of the space of bounded analytic functions outside K.
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
\( \gamma(K) = \sup \{|f'(\infty)|;\ f\in\mathcal{H}^\infty(\mathbf{C}\setminus K),\ \|f\|_\infty\leq 1,\ f(\infty)=0\} \)
Here, \(\mathcal{H}^\infty (U) \) denotes the set of bounded analytic functions U → C, whenever U is an open subset of the complex plane. Further,
\( f'(\infty):= \lim_{z\to\infty}z\left(f(z)-f(\infty)\right) \)
\( f(\infty):= \lim_{z\to\infty}f(z) \)
(note that usually \(f'(\infty)\neq \lim_{z\to\infty} f'(z) ) \)
Ahlfors function
For each compact K ⊂ C, there exists a unique extremal function, i.e. \( f\in\mathcal{H}^\infty(\mathbf{C}\setminus K) \) such that \( \|f\|\leq 1, f(∞) = 0 \) 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 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 \( Q_n^j) \) of side length 4−n, each \(Q_n^j \)being in the corner of some Q_{n-1}^k. Take K to be the intersection of all Kn then \( H^1(K)=\sqrt{2} \) but γ(K) = 0.
Vitushkin's conjecture
Suppose dimH(K) = 1 and H1(K) > 0. Vitushkin's conjecture states that
\( \gamma(K)=0\ \Leftrightarrow\ K \ \text{ is purely unrectifiable} \)
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.
References
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.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
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