Opérateurs d’intégrale singulière les surfaces régulières. (Singular integral operators on regular surfaces). (French) Zbl 0655.42013

The aim of the paper is to find a rather large class of k-dimensional surfaces in \({\mathbb{R}}^ n \)on which the natural singular integral operators are bounded on \({\mathbb{L}}^ 2.\) A surface belonging to this class is called regular and is characterized by a Lipschitz function z(x) having the property \(| \{x\in {\mathbb{R}}^ k;\)z(x)\(\in B\}| \leq Cr({\mathbb{B}})\) k for any ball \({\mathbb{B}}\) of radius r(\({\mathbb{B}})\) in \({\mathbb{R}}^ n.\) It is shown that if z(x) is a regular surface and if \({\mathbb{K}}:{\mathbb{R}}\) \(n\setminus \{0\}\to {\mathbb{C}}\) is an odd, \(C^{\infty}\), homogeneous of degree -k function then the kernel \({\mathbb{K}}(z(x)-z(y))\) defines a bounded operator on \({\mathbb{L}}\) 2(\({\mathbb{R}}\) k). Thus, if \(k=n-1\), the double-layer potential defines a bounded operator on \({\mathbb{L}}\) 2(\({\mathbb{S}})\) when \({\mathbb{S}}\) is a regular surface of dimension n-1.
Reviewer: L.Goras


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