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Une minoration de la norme de l’opérateur de Cauchy sur les graphes lipschtiziens. (A reduction of the norm of the Cauchy operator on the lipschitzian graphs). (French) Zbl 0639.42013
The author proves that the estimate of the norm of the operator \[ T_ Af(x)=\lim_{\epsilon \to +0}\int_{| x-y| >\epsilon}[x- y+i(A(x)-A(y))]^{-1} f(y)dy \] established by T. Murai is optimal by showing that \(\| TA\|_{L^ 2(R),L^ 2(R)}\geq M^{1/2}/10\) if \(\| A'\|_{\infty}\leq M\), where A is a Lipschitz function on R. The idea of the proof is to use the Cantor set of the complex plane known under the name “Garnette’s example”.
Reviewer: H.Tanabe

MSC:
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
30C55 General theory of univalent and multivalent functions of one complex variable
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