In the mathematical discipline of
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
, the analytic capacity of a
compact subset ''K'' of the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
is a number that denotes "how big" a
bounded analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
on C \ ''K'' can become. Roughly speaking, ''γ''(''K'') measures the size of the
unit ball
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* ''Unit'' (a ...
of the space of bounded analytic functions outside ''K''.
It was first introduced by
Lars Ahlfors
Lars Valerian Ahlfors (18 April 1907 – 11 October 1996) was a Finnish mathematician, remembered for his work in the field of Riemann surfaces and his text on complex analysis.
Background
Ahlfors was born in Helsinki, Finland. His mother, Si ...
in the 1940s while studying the removability of
singularities of bounded analytic functions.
Definition
Let ''K'' ⊂ C be
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in Britis ...
. Then its analytic capacity is defined to be
:
Here,
denotes the set of
bounded analytic
functions ''U'' → C, whenever ''U'' is an
open
Open or OPEN may refer to:
Music
* Open (band), Australian pop/rock band
* The Open (band), English indie rock band
* Open (Blues Image album), ''Open'' (Blues Image album), 1969
* Open (Gotthard album), ''Open'' (Gotthard album), 1999
* Open (C ...
subset of the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
. Further,
:
:
Note that
, where
. However, usually
.
If ''A'' ⊂ C is an arbitrary set, then we define
:
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.
Ahlfors function
For each compact ''K'' ⊂ C, there exists a unique extremal function, i.e.
such that
, ''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
In complex analysis, an area of mathematics, Montel's theorem refers to one of two theorems about families of holomorphic functions. These are named after French mathematician Paul Montel, and give conditions under which a family of holomorphi ...
.
Analytic capacity in terms of Hausdorff dimension
Let dim
''H'' denote
Hausdorff dimension
In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of ...
and ''H''
1 denote 1-dimensional
Hausdorff measure
In mathematics, Hausdorff measure is a generalization of the traditional notions of area and volume to non-integer dimensions, specifically fractals and their Hausdorff dimensions. It is a type of outer measure, named for Felix Hausdorff, that ...
. Then ''H''
1(''K'') = 0 implies ''γ''(''K'') = 0 while dim
''H''(''K'') > 1 guarantees ''γ''(''K'') > 0. However, the case when dim
''H''(''K'') = 1 and ''H''
1(''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''
1(''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''
0 :=
, 1×
, 1be the unit square. Then, ''K''
1 is the union of 4 squares of side length 1/4 and these squares are located in the corners of ''K''
0. In general, ''K
n'' is the union of 4
''n'' squares (denoted by
) of side length 4
−''n'', each
being in the corner of some
. Take ''K'' to be the intersection of all ''K''
''n'' then
but ''γ''(''K'') = 0.
Vitushkin's conjecture
Let ''K'' ⊂ C be a compact set. Vitushkin's conjecture states that
:
where
denotes the orthogonal projection in direction θ. By the results described above, Vitushkin's conjecture is true when dim
''H''''K'' ≠ 1.
Guy David published a proof in 1998 of Vitushkin's conjecture for the case dim
''H''''K'' = 1 and ''H''
1(''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
, where each ''K''
''i'' is a Borel set, then
.
David's and Tolsa's theorems together imply that Vitushkin's conjecture is true when ''K'' is ''H''
1-
sigma-finite. However, the conjecture is still open for ''K'' which are 1-dimensional and not ''H''
1-sigma-finite.
References
*
*
* 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=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
*