## Convolution singular integrals on Lipschitz surfaces.(English)Zbl 0763.42009

The authors analyse some classes of convolution singular integral operators on $$\mathbb{R}^{n+1}$$, embedded in the Clifford algebra $$\mathbb{R}_{(n)}$$ with identity $$e_ 0$$. Let $$\Sigma=\{g(x)e_ 0+{\mathbf x}\in\mathbb{R}^{n+1}$$: $${\mathbf x}\in\mathbb{R}^ n\}$$, where $$g$$ is a Lipschitz function which satisfies $$|\nabla g|_ \infty\leq\tan\omega$$, $$0\leq\omega<\pi/2$$. Consider a right-monogenic (right-Clifford-regular) function $$\varphi$$ which satisfies $$|\varphi(x)|\leq C| x|^{-n}$$ on a sector $$S_ \mu=\{x=x_ 0+{\mathbf x}\in\mathbb{R}^{n+1}$$: $$| x_ 0|<|{\mathbf x}|\tan\mu\}$$ where $$\omega<\mu<\pi/2$$. Suppose there exists a bounded function $$\Phi$$ satisfying $$\Phi(R)- \Phi(r)=\int_{x\in\mathbb{R}^ n, r<| x|<R} \varphi(x)dx$$. Then the related convolution singular integral operator $(Tu)(x)=\lim_{\varepsilon\to 0+}\bigl\{ \int_{y\in\Sigma, | x- y|\geq\varepsilon} \varphi(x-y)n(y)u(y)dS_ y+\Phi(\varepsilon n(x))u(x)\bigr\}$ is bounded on $$L_ p(\Sigma)$$ for $$1<p<\infty$$. Here $$dS_ y$$ is the surface measure of $$\Sigma$$ and $$n(y)$$ is the outer unit normal defined for almost all $$y\in\Sigma$$. Their result is an extension of the known one-dimensional case, and at the same time gives other proofs of the known results on the singular Cauchy integral operators on $$\Sigma$$. They give some comments on bounds on the $$L^ p$$-boundedness of these operators.
Reviewer: K.Yabuta (Nara)

### MSC:

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

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