Analytic capacity
Encyclopedia
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
from
can become. Roughly speaking, 
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.
be compact
. Then its analytic capacity is defined to be
Here,
denotes the set of bounded
analytic functions
, whenever U is an open
subset of the complex plane
. Further,
(note that usually
)
, there exists a unique extremal function, i.e. 
such that 
, 
and 
. This function is called the Ahlfors function of K. Its existence can be proved by using a normal family argument involving Montel's theorem
.
denote Hausdorff dimension
and
denote 1-dimensional Hausdorff measure
. Then
implies 
while 
guarantees 
.
However, the case when
and 
is more difficult.
and its analytic capacity, it might be
conjectured that
. 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
be the unit square. Then, 
is the union of 4 squares of side length 
and these squares are located in the corners of 
. In general, 
is the union of 
squares (denoted by 
) of side length 
, each 
being in the corner of some 
. Put 
Then
but 
and 
. Vitushkin's conjecture states that
In this setting, K is (purely) unrectifiable if and only if
for all rectifiable curves (or equivalently, 
-curves or (rotated) Lipschitz graphs) 
.
Guy David published a proof in 1998 for the case when, in addition to the hypothesis above,
. Until now, very little is known about the case when 
is infinite (even sigma-finite).
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
are removable?"
It is easy to see that K is removable if and only if
. 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.
Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
, the analytic capacity of a compact subset K of the complex plane
Complex plane
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...
is a number that denotes "how big" a bounded
Bounded function
In mathematics, a function f defined on some set X with real or complex values is called bounded, if the set of its values is bounded. In other words, there exists a real number M...
analytic function
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, categories that are similar in some ways, but different in others...
from
can become. Roughly speaking, 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
Mathematical singularity
In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...
of bounded analytic functions.
Definition
Let be compactCompact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
. Then its analytic capacity is defined to be
Here,
denotes the set of boundedBounded function
In mathematics, a function f defined on some set X with real or complex values is called bounded, if the set of its values is bounded. In other words, there exists a real number M...
analytic functions
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
, whenever U is an openOpen set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
subset of the complex plane
Complex plane
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...
. Further,
(note that usually
)Ahlfors function
For each compact, there exists a unique extremal function, i.e. such that , and . This function is called the Ahlfors function of K. Its existence can be proved by using a normal family argument involving Montel's theoremMontel'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 Paul Montel, and give conditions under which a family of holomorphic functions is normal....
.
Analytic capacity in terms of Hausdorff dimension
Let denote Hausdorff dimensionHausdorff dimension
thumb|450px|Estimating the Hausdorff dimension of the coast of Great BritainIn mathematics, the Hausdorff dimension is an extended non-negative real number associated with any metric space. The Hausdorff dimension generalizes the notion of the dimension of a real vector space...
and
denote 1-dimensional Hausdorff measureHausdorff measure
In mathematics a Hausdorff measure is a type of outer measure, named for Felix Hausdorff, that assigns a number in [0,∞] to each set in Rn or, more generally, in any metric space. The zero dimensional Hausdorff measure is the number of points in the set or ∞ if the set is infinite...
. Then
implies while guarantees .However, the case when
and is more difficult.Positive length but zero analytic capacity
Given the partial correspondence between the 1-dimensional Hausdorff measure of a compact subset of and its analytic capacity, it might beconjectured that
. However, this conjecture is false.A counterexample was first given by A. G. Vitushkin
Anatoli Georgievich Vitushkin
Anatoli Georgievich Vitushkin was a Soviet mathematician noted for his work on mathematical analysis and analytic capacity.-Career:...
, 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
be the unit square. Then, is the union of 4 squares of side length and these squares are located in the corners of . In general, is the union of squares (denoted by ) of side length , each being in the corner of some . Put Then
but Vitushkin's Conjecture
Suppose and . Vitushkin's conjecture states thatIn this setting, K is (purely) unrectifiable if and only if
for all rectifiable curves (or equivalently, -curves or (rotated) Lipschitz graphs) .Guy David published a proof in 1998 for the case when, in addition to the hypothesis above,
. Until now, very little is known about the case when 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
are removable?"It is easy to see that K is removable if and only if
. 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.