Prépublication d'Orsay numéro 90 (24/12/1998)

ANALYTIC CAPACITY, CALDERON-ZYGMUND OPERATORS, AND RECTIFIABILITY
DAVID, Guy - Arithmétique et Géométrie Algébrique, Université Paris-Sud, Bât. 425, 91405 Orsay cedex

Keywords : analytic capacity; Cauchy kernel; Calderón-Zygmund operators

Classification MSC : 30C85; 30E20

Abstract :

The aim of this text is to give an account of developments in Calderón-Zygmund theory and geometric measure theory (rectifiability) that led to a proof of Vitushkin's conjecture on analytic capacity. These developments will be presented with a definite bias; a big part of the agenda for this text is to convince the reader that this problem about analytic capacity needed a fair amount of technology to be solved, and thus it was natural that it waited essentially until now to be solved, and that it was likely that this would happen about now.

We shall insist here a little more on the aspects related to singular integral operators, and leave the geometric measure theory more in the shadow. This is definitely not by lack of interest, but rather to make the exposition easier.

For a more impartial account, the reader may consult [Ma5] and its references.

The author wishes to dedicate this work to the memory of A. P. Calderón. This is a little presomptuous, but certainly not out of context, since the main result in the theory of analytic capacity is probably Calderón's contribution to the proof of the Denjoy conjecture. (See the beginning of Section 4).

It is a pleasure to thank Joan Verdera for a careful reading of this text and various improvements

Article : Fichier Postscript
Contact : Guy.DAVID@math.u-psud.fr