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Convolution singular integrals on Lipschitz surfaces


Authors: Chun Li, Alan McIntosh and Stephen Semmes
Journal: J. Amer. Math. Soc. 5 (1992), 455-481
MSC: Primary 42B20; Secondary 30G35, 47B35, 47G10
DOI: https://doi.org/10.1090/S0894-0347-1992-1157291-5
MathSciNet review: 1157291
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Abstract: We prove the $ {L_p}$-boundedness of convolution singular integral operators on a Lipschitz surface

$\displaystyle \Sigma = \{ g({\mathbf{x}}){e_0} + {\mathbf{x}} \in {\mathbb{R}^{n + 1}}:{\mathbf{x}} \in {\mathbb{R}^n}\} $

where $ g$ is a Lipschitz function which satisfies $ {\left\Vert {\nabla g} \right\Vert _\infty } \leq {\text{tan}}\omega < \infty $. Here we have embedded $ {\mathbb{R}^{n + 1}}$ in the Clifford algebra $ {\mathbb{R}_{(n)}}$ with identity $ {e_0}$, and are considering convolution with right-monogenic functions $ \phi $ which satisfy $ \left\vert {\phi (x)} \right\vert \leq C{\left\vert x \right\vert^{ - n}}$ on a sector

$\displaystyle S_\mu ^o = \{ x = {x_0} + {\mathbf{x}} \in {\mathbb{R}^{n + 1}}:\... ...t {{x_0}} \right\vert < \left\vert {\mathbf{x}} \right\vert{\text{tan}} \mu \} $

where $ \mu > \omega $. Provided there exists an $ {L_\infty }$ function $ \underline \phi $ satisfying

$\displaystyle \underline \phi (R) - \underline \phi (r) = \int_{\substack{\mathop r < \vert x\vert < R\\ x \in {\mathbb{R}^{n}}}} \phi (x)\;dx $

, then the related convolution singular integral operator

$\displaystyle ({T_{(\phi ,\underline {\phi )} }}u)(x) = \lim_{\varepsilon \to 0... ...\phi (x - y)n(y)u(y)\;d{S_y} + \underline \phi (\varepsilon n(x))u(x) \right\} $

is bounded on $ {L_p}(\Sigma )$ for $ 1 < p < \infty $.

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Additional Information

DOI: https://doi.org/10.1090/S0894-0347-1992-1157291-5
Article copyright: © Copyright 1992 American Mathematical Society

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