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 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.

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 dim_H denote Hausdorff dimension and H¹ denote 1-dimensional Hausdorff measure. Then H¹(K) = 0 implies γ(K) = 0 while dim_H(K) > 1 guarantees γ(K) > 0. However, the case when dim_H(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, K_n 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 K_n then $H^{1}(K)={\sqrt {2}}$ but γ(K) = 0.

Vitushkin's conjecture

Suppose dim_H(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.

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