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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Convolution singular integrals on Lipschitz surfaces
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by Chun Li, Alan McIntosh and Stephen Semmes PDF
J. Amer. Math. Soc. 5 (1992), 455-481 Request permission

Abstract:

We prove the ${L_p}$-boundedness of convolution singular integral operators on a Lipschitz surface \[ \Sigma = \{ g({\mathbf {x}}){e_0} + {\mathbf {x}} \in {\mathbb {R}^{n + 1}}:{\mathbf {x}} \in {\mathbb {R}^n}\} \] where $g$ is a Lipschitz function which satisfies ${\left \| {\nabla g} \right \|_\infty } \leq {\text {tan}}\omega < \infty$. 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 $\left | {\phi (x)} \right | \leq C{\left | x \right |^{ - n}}$ on a sector \[ S_\mu ^o = \{ x = {x_0} + {\mathbf {x}} \in {\mathbb {R}^{n + 1}}:\left | {{x_0}} \right | < \left | {\mathbf {x}} \right |{\text {tan}} \mu \} \] where $\mu > \omega$. Provided there exists an ${L_\infty }$ function $\underline \phi$ satisfying \[ \underline \phi (R) - \underline \phi (r) = \int _{\substack {r < |x| < R\\x \in {\mathbb {R}^{n}}}} \phi (x)\;dx \], then the related convolution singular integral operator \[ ({T_{(\phi ,\underline {\phi )} }}u)(x) = \lim _{\varepsilon \to 0+}\left \{\int _{\substack {y \in \Sigma \\|x - y| \geq \varepsilon }} \phi (x - y)n(y)u(y)\;d{S_y} + \underline \phi (\varepsilon n(x))u(x) \right \} \] is bounded on ${L_p}(\Sigma )$ for $1 < p < \infty$.
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Additional Information
  • © Copyright 1992 American Mathematical Society
  • 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