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Opérateurs intégraux singuliers sur certaines courbes du plan complexe. (French) Zbl 0537.42016
Let \(\Gamma\) be a rectifiable curve in the plane, and \(x\to z(x)\) its arc length parametrization. One can define the Cauchy integral operator by \(Tf(x)=\lim_{\epsilon \to 0}\int_{| x-y|<\epsilon}1/z(x)- z(y)f(y)dz(y).\) We prove that the class of rectifiable curves for which this operator extends to a bounded operator on \(L^ 2({\mathbb{R}})\) is the class of Ahlfors’ ”regular curves”. A curve \(\Gamma\) is regular if for some \(C>0\), the arc length of the intersection of \(\Gamma\) with any disc is always less than C times the radius of the disc. (For instance, ord- arc curves are regular.) The regular Jordan curves are therefore the rectifiable Jordan curves \(\Gamma\) for which \(L^ 2(\Gamma)\) is the direct sum of the two Hardy spaces of analytic functions.
The boundedness of T was proved by A. P. Calderon [Proc. Natl. Acad. Sci. USA 74, 1324-1327 (1977; Zbl 0373.44003)] when \(\Gamma\) is the graph of a Lipschitz function \(\phi\) with \(\| \phi '\|_{\infty}<\delta\) for some (unknown) constant \(\delta\). R. R. Coifman, A. McIntosh and Y. Meyer [L’intégrale de Cauchy difinit un opérateur borné sur \(L^ 2\) pour les courbes Lipschitziennes, Ann. Math., II. Ser. 116, 361-387 (1982; Zbl 0497.42012)] extended this result to any \(\delta>0.\)
We also give a ”geometric” proof of the Coifman-McIntosh-Meyer theorem. A slightly improved version of it can be found in the CMA Research Report 22, 1984 Australian National University (Canberra, Australia). More precise estimates are obtained by T. Murai [”Boundedness of singular integral operators of Calderon type (V)”, Prepr. Ser., Nagoya Univ. Coll. Gen. Educ. (to appear)].

MSC:
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
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References:
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