L BOUNDEDNESS OF THE CAUCHY INTEGRAL AND MENGER CURVATURE
L BOUNDEDNESS OF THE CAUCHY INTEGRAL AND MENGER CURVATURE
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柯西积分和门格曲率的 L 有界
DOI：
10.1090/conm/277/04543
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发表时间：
2002
影响因子：
1.3
通讯作者：
J. Verdera
中科院分区：
文献类型：
--
作者：
J. Verdera
在本文中，我们解释了相关的Menger曲率理解的柯西积分算子的L2有界性。在引入Menger曲率并描述其基本性质后，我们进一步证明了Lipschitz图上Cauchy积分的Coifman-McIntosh-Meyer定理。从这个思想循环中，我们得到了一个新的简单方法来证明第一个Calderón交换子的L2有界性。本文指出，Lipschitz图上柯西积分的L2有界性可以很容易地归结为第一交换子的有界性。在最后一节中，我们描述了解决Vitushkin猜想的解析能力的各个步骤，特别注意门格尔曲率所起的作用。
In this paper we explain the relevance of Menger curvature in understanding the L2 boundedness properties of the Cauchy Integral Operator. After introducing Menger curvature and describing its basic properties we proceed to prove the Coifman-McIntosh-Meyer Theorem on the Cauchy Integral on a Lipschitz graph. From this circle of ideas comes a new simple approach to the L2 boundedness of the first Calderón commutator. We point out that the L2 boundedness of the Cauchy Integral on a Lipschitz graph can be easily reduced to the boundedness of the first commutator. In the last section we describe the various steps in the solution of Vitushkin’s conjecture on analytic capacity paying special attention to the role played by Menger curvature.