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Convolution singular integrals on Lipschitz surfaces. (English) Zbl 0763.42009
The authors analyse some classes of convolution singular integral operators on $\bbfR\sp{n+1}$, embedded in the Clifford algebra $\bbfR\sb{(n)}$ with identity $e\sb 0$. Let $\Sigma=\{g(x)e\sb 0+{\bold x}\in\bbfR\sp{n+1}$: ${\bold x}\in\bbfR\sp n\}$, where $g$ is a Lipschitz function which satisfies $\vert\nabla g\vert\sb \infty\leq\tan\omega$, $0\leq\omega<\pi/2$. Consider a right-monogenic (right-Clifford-regular) function $\varphi$ which satisfies $\vert\varphi(x)\vert\leq C\vert x\vert\sp{-n}$ on a sector $S\sb \mu=\{x=x\sb 0+{\bold x}\in\bbfR\sp{n+1}$: $\vert x\sb 0\vert<\vert{\bold x}\vert\tan\mu\}$ where $\omega<\mu<\pi/2$. Suppose there exists a bounded function $\Phi$ satisfying $\Phi(R)- \Phi(r)=\int\sb{x\in\bbfR\sp n, r<\vert x\vert<R} \varphi(x)dx$. Then the related convolution singular integral operator $$(Tu)(x)=\lim\sb{\varepsilon\to 0+}\bigl\{ \int\sb{y\in\Sigma, \vert x- y\vert\geq\varepsilon} \varphi(x-y)n(y)u(y)dS\sb y+\Phi(\varepsilon n(x))u(x)\bigr\}$$ is bounded on $L\sb p(\Sigma)$ for $1<p<\infty$. Here $dS\sb y$ 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 $L\sp p$-boundedness of these operators.
Reviewer: K.Yabuta (Nara)

42B20Singular and oscillatory integrals, several variables
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