## A Real Variable Method for the Cauchy Transform, and Analytic CapacityThis research monograph studies the Cauchy transform on curves with the object of formulating a precise estimate of analytic capacity. The note is divided into three chapters. The first chapter is a review of the Calderón commutator. In the second chapter, a real variable method for the Cauchy transform is given using only the rising sun lemma. The final and principal chapter uses the method of the second chapter to compare analytic capacity with integral-geometric quantities. The prerequisites for reading this book are basic knowledge of singular integrals and function theory. It addresses specialists and graduate students in function theory and in fluid dynamics. |

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### Contents

The Calder6n commutator 8 proofs of its boundedness | 1 |

A real variable method for the Cauchy transform on graphs | 31 |

Analytic capacities of cranks | 71 |

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### Common terms and phrases

0-jumps 0-simple 1-previsible according to theorem Algebraic Analysis analytic analytic capacity Ao,xo apply assume assumption Banach Spaces bounded bounded variation branch codimension compact compensated completes the proof Const continuous cyclic group define definition Dellacherie denote disjoint Edited eigenvalues equation equivariant exists finite flip doubling Floquet multipliers frozen wave function geometric given global bifurcation Hence homotopy invariance Hopf bifurcation implies inequality integrable increasing process interval isotropy jumps Lemma linear martingale minimal period O-sequence obtain one-parameter orbit index orthogonal p-integrable pairwise parameter periodic solutions Poincare Proceedings proof of Theorem prove Q++-continuous quadratic variation random variables regular martingale representation resp rotating waves secondary bifurcations Seminar sequence simple sets square integrable square integrable martingales stationary solutions stochastic stopping lines submartingale theorem 2.9 Theory tingale Topology two-parameter virtual period virtual symmetry