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Painlevé’s problem and the semiadditivity of analytic capacity. (English) Zbl 1060.30031

A compact set $E$ in the complex plane $ℂ$ is said to be removable (for bounded holomorphic functions) if for every open set ${\Omega }$ containing $E$, every bounded function holomorphic in ${\Omega }\setminus E$ has a holomorphic extension in ${\Omega }$. L. V. Ahlfors in 1947 [Duke Math. J. 14, 1–11 (1947; Zbl 0030.03001)] introduced the notion of analytic capacity of $E$; this is defined by

$\gamma \left(E\right)=sup\left\{|{f}^{\text{'}}\left(\infty \right)|:f:C\setminus E\to C\phantom{\rule{4.pt}{0ex}}\text{holomorphic,}\phantom{\rule{4.pt}{0ex}}|f|\le 1\right\},$

where ${f}^{\text{'}}\left(\infty \right):={lim}_{z\to \infty }z\left(f\left(z\right)-f\left(\infty \right)\right)·$ Ahlfors showed that $E$ is removable if and only if $\gamma \left(E\right)=0$ and posed the problem of the geometric characterization of the sets $E$ with $\gamma \left(E\right)=0$. This is a fundamental problem in Complex Analysis; it is called Painlevé’s problem.

One of the main results in the present paper is that analytic capacity is semi-additive:

$\gamma \left(E\cup F\right)\le const·\left(\gamma \left(E\right)+\gamma \left(F\right)\right)·$

This fact was conjectured by A. G. Vitushkin [Russ. Math. Surv. 22, 139–200 (1967; Zbl 0164.37701)]. The semi-additivity of $\gamma$ has the following consequences on Painlevé’s problem:

If $E$ has $\sigma$-finite length, then $\gamma \left(E\right)=0$ if and only if ${H}^{1}\left(E\cap {\Gamma }\right)=0$ for all rectifiable curves ${\Gamma }$ (${H}^{1}$ is the one-dimensional Hausdorff measure).

For compact sets of finite length this had been proved by G. David [Rev. Mat. Iberoam. 14, 369–479 (1998; Zbl 0913.30012)]; David’s theorem was previously known as Vitushkin’s conjecture. For sets $E$ which are not $\sigma$-finite the above characterization of removable sets does not hold; this was proved by P. Mattila [Ann. Math. 123, 303–309 (1986; Zbl 0589.28006)]. Tolsa obtained the semi-additivity of $\gamma$ by relating the analytic capacity with another quantity, the analytic capacity ${\gamma }_{+}\left(E\right)$ defined in terms of Cauchy transforms of measures supported on $E$:

${\gamma }_{+}\left(E\right)=sup\left\{\mu \left(E\right):\mu \phantom{\rule{4.pt}{0ex}}\text{Radon}\phantom{\rule{4.pt}{0ex}}\text{measure}\phantom{\rule{4.pt}{0ex}}\text{on}\phantom{\rule{4.pt}{0ex}}E\phantom{\rule{4.pt}{0ex}}\text{with}\phantom{\rule{4.pt}{0ex}}\parallel \frac{1}{z}*\mu {\parallel }_{\infty }\le 1\right\}·$

Tolsa proved that

$\gamma \left(E\right)\le const·\phantom{\rule{4pt}{0ex}}{\gamma }_{+}\left(E\right)·$

This inequality, together with its converse (which is relatively easy) and the semi-additivity of ${\gamma }_{+}$ (proved by the author in [Indiana Math. J. 51, 317–343 (2002; Zbl 1041.31002)]) give the semi-additivity of $\gamma$.

In [Indiana Math. J. 51, 317–343 (2002; Zbl 1041.31002)] Tolsa has also shown that ${\gamma }_{+}$ has a rather precise description in terms of curvature of measures. The curvature $c\left(\mu \right)$ of a measure $\mu$ is defined as follows:

${c}^{2}\left(\mu \right)=\phantom{\rule{0.277778em}{0ex}}\int \int \int \phantom{\rule{0.277778em}{0ex}}\frac{1}{R{\left(x,y,z\right)}^{2}}\phantom{\rule{0.166667em}{0ex}}d\mu \left(x\right)\phantom{\rule{0.166667em}{0ex}}d\mu \left(y\right)\phantom{\rule{0.166667em}{0ex}}d\mu \left(z\right),$

where $R\left(x,y,z\right)$ is the radius of the circle passing through $x,y,z$. This quantity, introduced by M. S. Melnikov [Sb. Math. 186, No. 6, 827–846 (1995; Zbl 0840.30008)] is one of the main tools in the study of removable sets. A consequence of the equivalence of $\gamma$ and ${\gamma }_{+}$ is a characterization of removable sets previously conjectured by Melnikov:

$E$ is non-removable if and only if it supports a Radon measure with linear growth and finite curvature.

Further information and references on the connection of Painlevé’s problem with Cauchy transforms, T(b)-theorems, the continuous analytic capacity, and the theory of rational approximation, as well as more details on the main steps of the proofs of the important results in the paper under review can be found in two recent survey papers by the author: 1. Painlevé’s problem, analytic capacity and curvature of measures, Proceedings of the Fourth European Congress, 2004. 2. Painlevé’s problem and analytic capacity, Lecture notes of a minicourse given at El Escorial, 2004. Also, in the paper [“Bilipschitz maps, analytic capacity, and the Cauchy integral”, to appear in Ann. Math.] the author continues the research of the present paper.

##### MSC:
 30C85 Capacity and harmonic measure in the complex plane 31A15 Potentials and capacity, harmonic measure, extremal length (two-dimensional) 42B20 Singular and oscillatory integrals, several variables 42B35 Function spaces arising in harmonic analysis
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