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)


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