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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Fourier multipliers on Lipschitz curves


Authors: Alan McIntosh and Tao Qian
Journal: Trans. Amer. Math. Soc. 333 (1992), 157-176
MSC: Primary 42B15; Secondary 47B35, 47G99
MathSciNet review: 1062194
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Abstract: We develop the theory of Fourier multipliers acting on $ {L_p}(\gamma )$ where $ \gamma $ is a Lipschitz curve of the form $ \gamma = \{ x + ig(x)\} $ with $ \left\Vert g\right\Vert _\infty < \infty $ and $ \left\Vert g\prime\right\Vert _\infty < \infty $ . The aim is to better understand convolution singular integrals $ B$ defined naturally on such curves by

$\displaystyle Bu(z) = {\text{p.v.}}\int_\gamma {\varphi (z - \zeta )u(\zeta )d\zeta } $

for almost all $ z \in \gamma $ .

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1992-1062194-7
PII: S 0002-9947(1992)1062194-7
Article copyright: © Copyright 1992 American Mathematical Society