Difference between revisions of "Painlevé problem"
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| − | === | + | ===The problem and first results=== |
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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 | ||
| + | $\mu(D)\leq diam(D)$ 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. |
Painlevé problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Painlev%C3%A9_problem&oldid=16072