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''analytic measure, Ahlfors analytic measure''
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{{MSC|30C85|31A15}}
  
A set function in the plane, introduced by L. Ahlfors [[#References|[1]]], which is an analogue of the logarithmic [[Capacity|capacity]], and which is suited for the characterization of sets of removable singularities of bounded analytic functions. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012180/a0121801.png" /> be a bounded closed set in the plane, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012180/a0121802.png" /> be the set of functions which are analytic outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012180/a0121803.png" />, vanish at infinity and are bounded everywhere outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012180/a0121804.png" /> by the constant 1. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012180/a0121805.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012180/a0121806.png" />. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012180/a0121807.png" />, is called the analytic capacity of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012180/a0121808.png" />. The analytic capacity of an arbitrary set is usually defined as the supremum of the analytic capacities of its bounded closed subsets.
 
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012180/a0121809.png" /> is a bounded closed set, then a necessary and sufficient condition in order that each function which is analytic and bounded outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012180/a01218010.png" /> be extendable to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012180/a01218011.png" />, is that the analytic capacity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012180/a01218012.png" /> be zero (the Ahlfors theorem).
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[[Category:Analysis]]
  
There also exist several concepts related to analytic capacity which are suited to the metrics of various other spaces of analytic functions (see, for example, [[#References|[2]]] and [[#References|[3]]]).
 
  
The concept of analytic capacity proved to be well suited to certain problems in approximation theory, where the solution of a number of fundamental problems is formulated in terms of analytic capacity. Thus, [[#References|[4]]], any continuous function on a bounded closed set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012180/a01218013.png" /> in the plane can be uniformly approximated by rational functions to any desired degree of accuracy, if and only if the equality
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012180/a01218014.png" /></td> </tr></table>
 
  
holds for any disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012180/a01218015.png" /> of radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012180/a01218016.png" />.
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===Definitions===
  
====References====
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Analytic capacity was introduced by L.V. Ahlfors in {{Cite|A}} in 1947 for the characterization of removable singularities of bounded analytic functions. Let $K$ be a compact set in the complex plane $\mathbb C$. The analytic capacity of $K$ is defined by
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> L.V. Ahlfors, "Bounded analytic functions" ''Duke Math. J.'' , '''14''' (1947) pp. 1–11 {{MR|0021108}} {{ZBL|0030.03001}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> P.R. Garabedian, "The classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012180/a01218017.png" /> and conformal mapping" ''Trans. Amer. Math. Soc.'' , '''69''' (1950) pp. 392–415 {{MR|0039072}} {{ZBL|0040.33001}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> S.O. Sinanyan, "Approximation by analytic functions and polynomials in a real mean" ''Dokl. Akad. Nauk ArmSSR'' , '''35''' : 3 (1962) pp. 107–112 (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> A.G. Vitushkin, "Analytic capacity of sets in problems of approximation theory" ''Russian Math. Surveys'' , '''22''' : 6 (1967) pp. 139–200 ''Uspekhi Mat. Nauk'' , '''22''' : 6 (1967) pp. 141–199 {{MR|}} {{ZBL|0164.37701}} </TD></TR></table>
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$$\gamma(K)=\sup\{\lim_{|z|\to\infty}|zf(z)|: f\in A(K)\}$$
 +
where $A(K)$ is the set of functions which are analytic outside $K$, vanish at infinity and for which $|f(z)|\leq1$ for
 +
$z\in\mathbb C\setminus K$.  
  
 +
A related concept, which is more useful in rational approximation, is continuous analytic capacity $\alpha(K)$. It is defined as $\gamma(K)$ but the test functions $f$ are additionally required to be defined and continuous in the whole complex plane.
  
  
====Comments====
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===Removable sets===
A good reference to the general area is [[#References|[a1]]].
 
  
====References====
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Ahlfors proved in {{Cite|A}} that a compact set $K$ is removable for bounded analytic functions if and only if  $\gamma(K)=0$. The removability means that whenever $U$ is an open set in $\mathbb C$ containing $K$ and $f$ is bounded and analytic in $U\setminus K$, then $f$ has an analytic extension to $U$. It is fairly easy to show with the help of the Cauchy integral formula and Liouville's theorem that $K$ is removable for bounded analytic functions if and only if every bounded analytic function in $\mathbb C\setminus K$ is constant. Similarly, $\alpha(K)=0$ if and only if every bounded continuous function in $\mathbb C$ which is analytic in $\mathbb C\setminus K$ is constant.
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> T.W. Gamelin, "Uniform algebras" , Prentice-Hall (1969) {{MR|0410387}} {{ZBL|0213.40401}} </TD></TR></table>
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 +
An outstanding problem is to find a characterization in geometric terms for the null-sets of the analytic capacity. This is called Painlevè's  problem since P. Painlevè studied it already in 1888 and proved a sufficient condition, property (i) below.
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 +
 
 +
 
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===Properties===
 +
 
 +
Some basic properties of the analytic capacity are the following three:
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 +
(i) If $K$ has length (that is, one-dimensional [[Hausdorff measure]]) zero, then $\gamma(K)=0$.
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 +
(ii) If $K$ has [[Hausdorff dimension]] greater than 1 (in particular, if $K$ has interior points), then $\gamma(K)>0$.
 +
 
 +
(iii) If $K$ is a subset of a rectifiable curve, then $\gamma(K)=0$ if and only if $K$ has length zero.
 +
 
 +
The first two are fairly easy, but the third one is deep, see, e.g., {{Cite|P}}.
 +
 
 +
 
 +
===Rational approximation and semiadditivity===
 +
Solutions of many fundamental problems in rational approximation can be formulated in terms of analytic capacity. Thus any continuous function on a compact set $K$ in the plane can be uniformly approximated by rational functions with poles off $K$ if and only if
 +
$$\alpha(D\setminus K)=\alpha(D\setminus \text{interior}(K))\ \text{for any disc}\ D.$$
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The work of the Moscow school (A.G. Vitushkin, M.S. Melnikov, and others) in the 1960's was particularly important in this development. Vitushkin also formulated the semiadditivity problem:
 +
 
 +
does there exist a constant $C$ such that for all compact subsets $K_1$ and $K_2$ of the plane,
 +
$$\gamma(K_1\cup K_2)\leq C(\gamma(K_1)+\gamma(K_2))?$$
 +
 
 +
===Tolsa's solutions===
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 +
 
 +
In {{Cite|T}} X. Tolsa solved both  Painlevè's problem and the semiadditivity problem. The solutions depend on the so-called Menger curvature $c(z_1,z_2,z_3)$ for triples of points in  $\mathbb C$ and a formula of M.S. Melnikov relating it to the Cauchy kernel $1/z$. By definition the Menger curvature is the reciprocal of the radius of the circle passing through the points $z_1,z_2,z_3$; it is equal to zero if and only if the three points lie on one line. Tolsa proved the following two results:
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 +
'''Theorem 1'''
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For a compact set $K\subset \mathbb C$, $\gamma(K)>0$ if and only there is a positive non-trivial Borel measure $\mu$ on $\mathbb C$ such that
 +
$\mu(D)\leq {\rm diam}\, (D)$ for all discs $D$ in $\mathbb C$ and $\int\int\int c(z_1,z_2,z_3)^2d\mu z_1d\mu z_2d\mu z_3<\infty$.
 +
 
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'''Theorem 2'''
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The analytic capacity is semiadditive: there exists a constant $C$ such that for all compact subsets $K_1,K_2,\dots$, of the plane,
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$$\gamma(\cup_jK_j)\leq C\sum_j\gamma(K_j).$$
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 +
Good general references are {{Cite|G}}, {{Cite|P}}, {{Cite|V}} and {{Cite|Z}}.
 +
 
 +
===References===
 +
 
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{|
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 +
|-
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 +
|valign="top"|{{Ref|A}}|| L.V. Ahlfors, "Bounded analytic functions" Duke Math. J. , 14 (1947) pp. 1–11.  {{MR|0021108}}{{ZBL|0030.03001}}
 +
 
 +
|-
 +
 
 +
|valign="top"|{{Ref|G}}|| J.B. Garnett, "Analytic Capacity and Measure" Spinger-Verlag Lecture Notes 297, 1972. {{MR|0454006}} {{ZBL|0253.30014}}
 +
 
 +
|-
 +
 
 +
|valign="top"|{{Ref|P}}|| H. Pajot, "Analytic Capacity, Rectifiability, Menger Curvature and the Cauchy Integral" , Spinger-Verlag Lecture Notes 1799, 2002. {{MR|1952175}} {{ZBL|1043.28002}}
 +
 
 +
|-
 +
 
 +
|valign="top"|{{Ref|T}}|| X. Tolsa, "Painlevè's problem and the semiadditivity of analytic capacity" , Acta Mathematica 190 (2003), 105-149. {{MR|1982794}} {{ZBL|1060.30031}}
 +
 
 +
|-
 +
 
 +
|valign="top"|{{Ref|V}}|| A.G. Vitushkin, "Analytic capacity of sets in problems of approximation theory"" Russian Math. Surveys , 22 : 6 (1967) pp. 139–200 Uspekhi Mat. Nauk , 22 : 6 (1967) pp. 141–199. {{MR|1857292}}{{ZBL|0164.37701}}
 +
 
 +
|-
 +
 
 +
|valign="top"|{{Ref|Z}}|| L. Zalcman, "Analytic capacity rational approximation" Spinger-Verlag Lecture Notes 50, 1968.

Latest revision as of 07:23, 3 October 2012

2020 Mathematics Subject Classification: Primary: 30C85 Secondary: 31A15 [MSN][ZBL]


Definitions

Analytic capacity was introduced by L.V. Ahlfors in [A] in 1947 for the characterization of removable singularities of bounded analytic functions. Let $K$ be a compact set in the complex plane $\mathbb C$. The analytic capacity of $K$ is defined by $$\gamma(K)=\sup\{\lim_{|z|\to\infty}|zf(z)|: f\in A(K)\}$$ where $A(K)$ is the set of functions which are analytic outside $K$, vanish at infinity and for which $|f(z)|\leq1$ for $z\in\mathbb C\setminus K$.

A related concept, which is more useful in rational approximation, is continuous analytic capacity $\alpha(K)$. It is defined as $\gamma(K)$ but the test functions $f$ are additionally required to be defined and continuous in the whole complex plane.


Removable sets

Ahlfors proved in [A] that a compact set $K$ is removable for bounded analytic functions if and only if $\gamma(K)=0$. The removability means that whenever $U$ is an open set in $\mathbb C$ containing $K$ and $f$ is bounded and analytic in $U\setminus K$, then $f$ has an analytic extension to $U$. It is fairly easy to show with the help of the Cauchy integral formula and Liouville's theorem that $K$ is removable for bounded analytic functions if and only if every bounded analytic function in $\mathbb C\setminus K$ is constant. Similarly, $\alpha(K)=0$ if and only if every bounded continuous function in $\mathbb C$ which is analytic in $\mathbb C\setminus K$ is constant.

An outstanding problem is to find a characterization in geometric terms for the null-sets of the analytic capacity. This is called Painlevè's problem since P. Painlevè studied it already in 1888 and proved a sufficient condition, property (i) below.


Properties

Some basic properties of the analytic capacity are the following three:

(i) If $K$ has length (that is, one-dimensional Hausdorff measure) zero, then $\gamma(K)=0$.

(ii) If $K$ has Hausdorff dimension greater than 1 (in particular, if $K$ has interior points), then $\gamma(K)>0$.

(iii) If $K$ is a subset of a rectifiable curve, then $\gamma(K)=0$ if and only if $K$ has length zero.

The first two are fairly easy, but the third one is deep, see, e.g., [P].


Rational approximation and semiadditivity

Solutions of many fundamental problems in rational approximation can be formulated in terms of analytic capacity. Thus any continuous function on a compact set $K$ in the plane can be uniformly approximated by rational functions with poles off $K$ if and only if $$\alpha(D\setminus K)=\alpha(D\setminus \text{interior}(K))\ \text{for any disc}\ D.$$ The work of the Moscow school (A.G. Vitushkin, M.S. Melnikov, and others) in the 1960's was particularly important in this development. Vitushkin also formulated the semiadditivity problem:

does there exist a constant $C$ such that for all compact subsets $K_1$ and $K_2$ of the plane, $$\gamma(K_1\cup K_2)\leq C(\gamma(K_1)+\gamma(K_2))?$$

Tolsa's solutions

In [T] X. Tolsa solved both Painlevè's problem and the semiadditivity problem. The solutions depend on the so-called Menger curvature $c(z_1,z_2,z_3)$ for triples of points in $\mathbb C$ and a formula of M.S. Melnikov relating it to the Cauchy kernel $1/z$. By definition the Menger curvature is the reciprocal of the radius of the circle passing through the points $z_1,z_2,z_3$; it is equal to zero if and only if the three points lie on one line. Tolsa proved the following two results:

Theorem 1 For a compact set $K\subset \mathbb C$, $\gamma(K)>0$ if and only there is a positive non-trivial Borel measure $\mu$ on $\mathbb C$ such that $\mu(D)\leq {\rm diam}\, (D)$ for all discs $D$ in $\mathbb C$ and $\int\int\int c(z_1,z_2,z_3)^2d\mu z_1d\mu z_2d\mu z_3<\infty$.

Theorem 2 The analytic capacity is semiadditive: there exists a constant $C$ such that for all compact subsets $K_1,K_2,\dots$, of the plane, $$\gamma(\cup_jK_j)\leq C\sum_j\gamma(K_j).$$

Good general references are [G], [P], [V] and [Z].

References

[A] L.V. Ahlfors, "Bounded analytic functions" Duke Math. J. , 14 (1947) pp. 1–11. MR0021108Zbl 0030.03001
[G] J.B. Garnett, "Analytic Capacity and Measure" Spinger-Verlag Lecture Notes 297, 1972. MR0454006 Zbl 0253.30014
[P] H. Pajot, "Analytic Capacity, Rectifiability, Menger Curvature and the Cauchy Integral" , Spinger-Verlag Lecture Notes 1799, 2002. MR1952175 Zbl 1043.28002
[T] X. Tolsa, "Painlevè's problem and the semiadditivity of analytic capacity" , Acta Mathematica 190 (2003), 105-149. MR1982794 Zbl 1060.30031
[V] A.G. Vitushkin, "Analytic capacity of sets in problems of approximation theory"" Russian Math. Surveys , 22 : 6 (1967) pp. 139–200 Uspekhi Mat. Nauk , 22 : 6 (1967) pp. 141–199. MR1857292Zbl 0164.37701
[Z] L. Zalcman, "Analytic capacity rational approximation" Spinger-Verlag Lecture Notes 50, 1968.
How to Cite This Entry:
Analytic capacity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Analytic_capacity&oldid=24364
This article was adapted from an original article by A.G. Vitushkin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article