Painlevé's problem and analytic capacity
Painlevé's problem and analytic capacity
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Painlevé 的问题和分析能力
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发表时间：
2006
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通讯作者：
X. Tolsa
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作者：
X. Tolsa
本文综述了有关Painleve问题和解析容量γ的半可加性的一些最新结果。特别地，我们给出了容量γ+的半可加性的详细证明，并且我们几乎完全给出了证明γ和γ+之间的可比性的所有论据。本文源于2004年6月在埃尔埃斯科里亚（马德里）举行的第7届调和分析和偏微分方程国际会议上的一系列三个讲座。在这篇文章中，我们将讨论和评论最近的一些结果与所谓的Painleve的问题和半可加性的分析能力，以及其他相关的问题。本书的某些部分，特别是那些纯粹是暂时性的部分，非常接近我们以前（但最近）的调查论文[To 10]。然而，本文包含了更详细的信息。例如，与[To 10]不同的是，它包括了证明γ+的半可加性以及γ和γ+之间的可比性的详细论据。本文的计划如下。在第一部分，这是介绍性的和纯粹的解释，分析能力和Painleve的问题的概念进行了解释。给出了解析容量的一些基本性质，并给出了解决Vitushkin猜想的Guy大卫定理.本节的最后一部分涉及柯西变换和容量γ+。在第二节中，我们引入了测度的曲率的概念，并且一方面说明了它与柯西变换的L范数的密切关系，另一方面说明了它与可求正性的密切关系。这一部分也主要是临时性的。在第三节中，我们综述了非加倍测度下Calderon-Zygmund理论的几个结果。这一理论在分析能力的研究中起着关键作用。本文讨论了与解析容量有关的一些结果，如Calderon-Zygmund算子的弱（1，1）有界性，Cotlar不等式，T（1）和T（B）定理.第四节讨论了容量γ+的半可加性。我们给出了这个结果的详细证明。此外，我们给出了一个新的（据我们所知）证明，它避免了T（1）定理的使用（尽管我们也解释了基于T（1）定理使用的替代论点）。在第五节中，我们陈述了关于γ与γ+之间的可比性定理，它特别地暗示了解析容量的半可加性。描述了相似性定理证明的主要思想和难点。我们给赠款部分支持MTM 2004 -00519（西班牙）和2001-SGR-00431（Generalitat de卡塔卢尼亚）.
In this paper we survey some recent results in connection with the so called Painleve’s problem and the semiadditivity of analytic capacity γ. In particular, we give the detailed proof of the semiadditivity of the capacity γ+, and we show almost completely all the arguments for the proof of the comparability between γ and γ+. This paper arose from a series of three lectures given at the 7 International Conference on Harmonic Analysis and Partial Differential Equations, at El Escorial (Madrid), in June 2004. In this article we will discuss and review some recent results in connection with the so called Painleve’s problem and the semiadditivity of analytic capacity, as well as other related questions. Some parts of this work, specially those sections which are purely expository, follow quite closely our previous (but very recent) survey paper [To10]. However, the present article contains much more detailed information. For instance, unlike [To10], it includes the detailed arguments for the proofs of the semiadditivity of γ+ and of the comparability between γ and γ+. The plan of the paper is the following. In the first section, which is introductory and purely expository, the notions of analytic capacity and Painleve’s problem are explained. Also, some basic properties of analytic capacity are shown and the theorem of Guy David which solves Vitushkin’s conjecture is stated. The final part of this section deals with the Cauchy transform and the capacity γ+. In the second section we introduce the notion of curvature of a measure, and we show its close relationship with the L norm of the Cauchy transform on the one hand, and with rectifiability on the other hand. This section is also mainly expository. In Section 3 we survey several results on Calderon-Zygmund theory with non doubling measures. This theory plays a key role in the study of analytic capacity. We discuss some of the results more useful in connection with analytic capacity, such as the weak (1, 1) boundedness of Calderon-Zygmund operators, Cotlar’s inequality, and the T (1) and T (b) theorems. Section 4 deals with the semiadditivity of the capacity γ+. We give the detailed proof of this result. Further, we show a new (as far as we know) proof which avoids the use of the T (1) theorem (although we also explain the alternative arguments based on the use of the T (1) theorem). In Section 5 we state the theorem about the comparability between γ and γ+, which in particular implies the semiadditivity of analytic capacity. We describe the main ideas and difficulties in the proof of the comparability theorem. We give the Partially supported by grants MTM2004-00519 (Spain) and 2001-SGR-00431 (Generalitat de Catalunya).