# Analytic capacity

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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

It was first introduced by Ahlfors in the 1940s while studying the removability of singularities of bounded analytic functions.

## Definition

Let KC be compact. Then its analytic capacity is defined to be

γ ( K ) = sup { | f ( ) | ;   f H ( C K ) ,   f 1 ,   f ( ) = 0 }

Here, H ( U ) denotes the set of bounded analytic functions UC, whenever U is an open subset of the complex plane. Further,

f ( ) := lim z z ( f ( z ) f ( ) ) f ( ) := lim z f ( z )

(note that usually f ( ) lim z f ( z ) )

## Ahlfors function

For each compact KC, there exists a unique extremal function, i.e. f H ( C K ) such that f 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 4n, 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 ) = 2 but γ(K) = 0.

## Vitushkin's conjecture

Suppose dimH(K) = 1 and H1(K) > 0. Vitushkin's conjecture states that

γ ( K ) = 0     K    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

Analytic capacity Wikipedia

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