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The problem of characterizing removable sets (cf. [[Removable set|Removable set]]) for a class of bounded single-valued analytic functions of the complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071090/p0710901.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071090/p0710902.png" /> be a compact set in the complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071090/p0710903.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071090/p0710904.png" /> is a domain. One has to determine the minimal conditions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071090/p0710905.png" /> under which any bounded single-valued analytic function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071090/p0710906.png" /> can be continued analytically to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071090/p0710907.png" /> and thus is a constant. P. Painlevé [[#References|[1]]] stated a sufficient condition: The linear [[Hausdorff measure|Hausdorff measure]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071090/p0710908.png" /> should vanish (such sets are sometimes called Painlevé sets); however, his arguments contain some errors (see [[#References|[2]]], [[#References|[3]]]). A necessary and sufficient condition on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071090/p0710909.png" /> is that the [[Analytic capacity|analytic capacity]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071090/p07109010.png" /> vanishes (Ahlfors' theorem). An example has been constructed of a set with zero analytic capacity but having positive linear measure [[#References|[5]]].
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{{MSC|53A04|53A35}}
  
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P. Painlevé,  "Leçons sur la théorie analytique des équations différentielles, professées à Stockholm (1895)" , Paris  (1897)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L. Zoretti,  "Sur les fonctions analytiques uniformes qui possèdent un ensemble parfait discontinu de points singuliers"  ''J. Math. Pure Appl.'' , '''1'''  (1905)  pp. 1–51</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L. Zoretti,  "Leçons sur la prolongement analytique" , Gauthier-Villars  (1911)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  L. Ahlfors,  "Bounded analytic functions"  ''Duke Math. J.'' , '''14'''  (1947)  pp. 1–11</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A.G. Vitushkin,  "Example of a set of positive length but of zero analytic capacity"  ''Dokl. Akad. Nauk SSSR'' , '''127''' :  2  (1959)  pp. 246–249  (In Russian)</TD></TR></table>
 
  
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[[Category:Analysis]]
  
  
====Comments====
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{{TEX|done}}
  
  
====References====
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===The problem and first results===
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.B. Garnett,   "Analytic capacity and measure" , ''Lect. notes in math.'' , '''297''' , Springer (1972)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Carleson,   "Selected problems on exceptional sets" , v. Nostrand  (1957)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W.K. Hayman,  P.B. Kennedy,   "Subharmonic functions" , '''1''' , Acad. Press  (1976) pp. 229ff</TD></TR></table>
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The Painlevé problem is to find a characterization in geometric terms for the removable singularities of bounded analytic functions, or equivalently, for the null-sets of the [[analytic capacity]]. P. Painlevé studied this problem already in 1888 and proved a sufficient condition:  if a compact plane set $K$ has length (that is, one-dimensional [[Hausdorff measure]]) zero, then it is removable for bounded analytic functions. The latter 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 $U$. In the other direction; if $K$ has [[Hausdorff dimension]] greater than 1 (in particular, if $K$ has interior points), then $K$ is not removable. A deep result due to A.P. Calder\'on from 1977 says that if $K$ is a subset of a [[rectifiable curve]], then $K$ is removable if and only if it has length zero.
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===Tolsa's solution===
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In {{Cite|T}} X. Tolsa solved  Painlevé's problem. The solution depends 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's solution is:
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A compact set $K\subset \mathbb C$ is not removable for bounded analytic functions if and only there is a positive non-trivial Borel measure $\mu$ on $\mathbb C$ such that
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$\mu(D)\leq diam(D)$ 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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Good general reference is  {{Cite|P}}.
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===References===
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{|
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|valign="top"|{{Ref|P}}|| H. Pajot, "Analytic Capacity, Rectifiability, Menger Curvature and the Cauchy Integral" , Spinger-Verlag Lecture Notes 1799, 2002. {{MR|}} {{ZBL|}}
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|valign="top"|{{Ref|T}}|| X. Tolsa, "Painlevè's problem and the semiadditivity of analytic capacity" , Acta Mathematica, 190 (2003), 105-149. {{MR|}} {{ZBL|}}

Latest revision as of 17:55, 24 September 2012

2020 Mathematics Subject Classification: Primary: 53A04 Secondary: 53A35 [MSN][ZBL]


The problem and first results

The Painlevé problem is to find a characterization in geometric terms for the removable singularities of bounded analytic functions, or equivalently, for the null-sets of the analytic capacity. P. Painlevé studied this problem already in 1888 and proved a sufficient condition: if a compact plane set $K$ has length (that is, one-dimensional Hausdorff measure) zero, then it is removable for bounded analytic functions. The latter 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 $U$. In the other direction; if $K$ has Hausdorff dimension greater than 1 (in particular, if $K$ has interior points), then $K$ is not removable. A deep result due to A.P. Calder\'on from 1977 says that if $K$ is a subset of a rectifiable curve, then $K$ is removable if and only if it has length zero.

Tolsa's solution

In [T] X. Tolsa solved Painlevé's problem. The solution depends 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's solution is:

A compact set $K\subset \mathbb C$ is not removable for bounded analytic functions if and only there is a positive non-trivial Borel measure $\mu$ on $\mathbb C$ such that $\mu(D)\leq diam(D)$ for all discs $D$ in $\mathbb C$ and $\int\int\int c(z_1,z_2,z_3)^2d\mu z_1d\mu z_2d\mu z_3<\infty$.


Good general reference is [P].

References

[P] H. Pajot, "Analytic Capacity, Rectifiability, Menger Curvature and the Cauchy Integral" , Spinger-Verlag Lecture Notes 1799, 2002.
[T] X. Tolsa, "Painlevè's problem and the semiadditivity of analytic capacity" , Acta Mathematica, 190 (2003), 105-149.
How to Cite This Entry:
Painlevé problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Painlev%C3%A9_problem&oldid=22875
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article