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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 function|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 [[mathematical singularity|singularities]] of bounded analytic functions.


==Definition==
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Let ''K'' ⊂ '''C''' be [[compact space|compact]]. Then its analytic capacity is defined to be
 
:<math>\gamma(K) = \sup \{|f'(\infty)|;\ f\in\mathcal{H}^\infty(\mathbf{C}\setminus K),\ \|f\|_\infty\leq 1,\ f(\infty)=0\}</math>
 
Here, <math>\mathcal{H}^\infty (U) </math> denotes the set of [[bounded function|bounded]] analytic [[Function (mathematics)|functions]] ''U'' → '''C''', whenever ''U'' is an [[open set|open]] subset of the [[complex plane]]. Further,
 
:<math> f'(\infty):= \lim_{z\to\infty}z\left(f(z)-f(\infty)\right) </math>
:<math> f(\infty):= \lim_{z\to\infty}f(z) </math>
 
(note that usually <math> f'(\infty)\neq \lim_{z\to\infty} f'(z) </math>)
 
==Ahlfors function==
For each compact ''K'' ⊂ '''C''', there exists a unique extremal function, i.e. <math>f\in\mathcal{H}^\infty(\mathbf{C}\setminus K)</math> such that <math>\|f\|\leq 1</math>, ''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<sub>''H''</sub> denote [[Hausdorff dimension]] and ''H''<sup>1</sup> denote 1-dimensional [[Hausdorff measure]]. Then ''H''<sup>1</sup>(''K'') = 0 implies γ(''K'') = 0 while dim<sub>''H''</sub>(''K'') > 1 guarantees γ(''K'') > 0. However, the case when dim<sub>''H''</sub>(''K'') = 1 and ''H''<sup>1</sup>(''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 ''H''<sup>1</sup>(''K'') = 0. However, this conjecture is false. A counterexample was first given by [[Anatoli Georgievich Vitushkin|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 ''K''<sub>0</sub> := [0, 1] × [0, 1] be the unit square. Then, ''K''<sub>1</sub> is the union of 4 squares of side length 1/4 and these squares are located in the corners of ''K''<sub>0</sub>. In general, ''K<sub>n</sub>'' is the union of 4<sup>''n''</sup> squares (denoted by <math>Q_n^j</math>) of side length 4<sup>−''n''</sup>, each <math>Q_n^j</math> being in the corner of some <math>Q_{n-1}^k</math>. Take ''K'' to be the intersection of all ''K<sub>n</sub>'' then <math>H^1(K)=\sqrt{2}</math> but γ(''K'') = 0.
 
===Vitushkin's Conjecture===
Suppose dim<sub>''H''</sub>(''K'') = 1 and ''H''<sup>1</sup>(''K'') > 0. Vitushkin's conjecture states that
 
:<math> \gamma(K)=0\ \Leftrightarrow\ K \ \text{ is purely unrectifiable} </math>
 
In this setting, ''K'' is (purely) [[unrectifiable]] if and only if ''H''<sup>1</sup>(''K'' ∩ Γ) = 0 for all [[rectifiable curve]]s (or equivalently, ''C''<sup>1</sup>-curves or (rotated) Lipschitz graphs) Γ.
 
Guy David published a proof in 1998 for the case when, in addition to the hypothesis above, ''H''<sup>1</sup>(''K'') < ∞. Until now, very little is known about the case when ''H''<sup>1</sup>(''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 [[Removable singularity#Riemann's theorem|Riemann's theorem for removable singularities]], every [[singleton (mathematics)|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==
* {{cite book |last=Mattila |first=Pertti |title=Geometry of sets and measures in Euclidean spaces |year=1995 |publisher=Cambridge University Press |isbn=0-521-65595-1}}
* {{cite book |last=Pajot |first=Hervé |title=Analytic Capacity, Rectifiability, Menger Curvature and the Cauchy Integral |year=2002 |series=Lecture Notes in Mathematics |publisher=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
* {{cite book |last=Dudziak |first=James J. |title=Vitushkin's Conjecture for Removable Sets |year=2010 |series=Universitext |publisher=Springer-Verlag |isbn=978-14419-6708-4}}
[[Category:Analytic functions|*]]

Latest revision as of 07:48, 27 August 2014


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