## 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

### MSC:

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

### Keywords:

singular integral operators; regular surface
Full Text:

### References:

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