Editing Analytic capacity

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 function|bounded]] [[analytic function]] on '''C'''&nbsp;\&nbsp;''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 [[Lars Ahlfors]] in the 1940s while studying the removability of [[mathematical singularity|singularities]] of bounded analytic functions.

==Definition==
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 <math>f'(\infty) = g'(0)</math>, where <math>g(z) = f(1/z)</math>. However, usually <math> f'(\infty)\neq \lim_{z\to\infty} f'(z)</math>.

Equivalently, the analytic capacity may be defined as<ref>{{eom| title = Capacity| last = Solomentsev| first = E. D.}}</ref>
:<math>\gamma(K)=\sup \left|\frac1{2\pi} \int_C f(z)dz\right|</math>
where ''C'' is a contour enclosing ''K'' and the supremum is taken  over ''f'' satisfying the same conditions as above: ''f'' is bounded analytic outside ''K'', the bound is one, and <math>f(\infty)=0.</math>

If ''A'' ⊂ '''C''' is an arbitrary set, then we define

:<math>\gamma(A) = \sup \{ \gamma(K) : K \subset A, \, K \text{ compact} \}.</math>

==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&nbsp;\&nbsp;''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.

==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 [[John B. 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===
Let ''K'' ⊂ '''C''' be a compact set. Vitushkin's conjecture states that

:<math> \gamma(K)=0\ \iff \ \int_0^\pi \mathcal H^1(\operatorname{proj}_\theta(K)) \, d\theta = 0 </math>

where <math>\operatorname{proj}_\theta(x,y) := x \cos \theta + y\sin\theta</math> denotes the orthogonal projection in direction θ. By the results described above, Vitushkin's conjecture is true when dim<sub>''H''</sub>''K'' ≠ 1.

[[Guy David (mathematician)|Guy David]] published a proof in 1998 of Vitushkin's conjecture for the case dim<sub>''H''</sub>''K'' = 1 and ''H''<sup>1</sup>(''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 <math>K = \bigcup_{i=1}^\infty K_i</math>, where each ''K''<sub>''i''</sub> is a Borel set, then <math>\gamma(K) \leq C \sum_{i=1}^\infty\gamma(K_i)</math>.

David's and Tolsa's theorems together imply that Vitushkin's conjecture is true when ''K'' is ''H''<sup>1</sup>-[[sigma-finite]].

In the non ''H''<sup>1</sup>-sigma-finite case, Pertti Mattila proved in 1986<ref>{{Cite journal |last=Mattila |first=Pertti |date=1986 |title=Smooth Maps, Null-Sets for Integralgeometric Measure and Analytic Capacity |url=https://www.jstor.org/stable/1971273 |journal=Annals of Mathematics |volume=123 |issue=2 |pages=303–309 |doi=10.2307/1971273 |jstor=1971273 |issn=0003-486X|url-access=subscription }}</ref> that the conjecture is false, but his proof did not specify which implication of the conjecture fails. Subsequent work by Jones and Muray<ref>{{Cite journal |last1=Jones |first1=Peter W. |last2=Murai |first2=Takafumi |title=Positive analytic capacity but zero Buffon needle probability. |url=https://msp.org/pjm/1988/133-1/pjm-v133-n1-p06-s.pdf |journal=Pacific Journal of Mathematics |date=1988 |volume=133 |issue=1 |pages=99–114|doi=10.2140/pjm.1988.133.99 }}</ref> produced an example of a set with zero Favard length and positive analytic capacity, explicitly disproving one of the directions of the conjecture. As of 2023 it is not known whether the other implication holds but some progress has been made towards a positive answer by Chang and Tolsa.<ref>{{Cite journal |last1=Chang |first1=Alan |last2=Tolsa |first2=Xavier |date=2020-10-05 |title=Analytic capacity and projections |url=https://ems.press/doi/10.4171/jems/1004 |journal=Journal of the European Mathematical Society |language=en |volume=22 |issue=12 |pages=4121–4159 |doi=10.4171/JEMS/1004 |issn=1435-9855|arxiv=1712.00594 }}</ref>

==See also==
* {{annotated link|Capacity of a set}}
* {{annotated link|Conformal radius}}

==References==
{{Reflist}}
* {{cite book |last=Mattila |first=Pertti|author-link = Pertti Mattila |title=Geometry of sets and measures in Euclidean spaces |url=https://archive.org/details/geometryofsetsme0000matt |url-access=registration |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}}
* {{cite book |last=Tolsa |first=Xavier |title=Analytic Capacity, the Cauchy Transform, and Non-homogeneous Calderón–Zygmund Theory |year=2014 |series=Progress in Mathematics |publisher=Birkhäuser Basel |isbn=978-3-319-00595-9}}

[[Category:Analytic functions|*]]