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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Some inequalities for singular convolution operators in $ L\sp p$-spaces


Author: Andreas Seeger
Journal: Trans. Amer. Math. Soc. 308 (1988), 259-272
MSC: Primary 42B15; Secondary 46E35, 47B38
DOI: https://doi.org/10.1090/S0002-9947-1988-0955772-3
MathSciNet review: 955772
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Abstract: Suppose that a bounded function $ m$ satisfies a localized multiplier condition $ {\sup _{t > 0}}\vert\vert\phi m({t^P} \cdot )\vert{\vert _{{M_p}}} < \infty $, for some bump function $ \phi $. We show that under mild smoothness assumptions $ m$ is a Fourier multiplier in $ {L^p}$. The approach uses the sharp maximal operator and Littlewood-Paley-theory. The method gives new results for lacunary maximal functions and for multipliers in Triebel-Lizorkin-spaces.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1988-0955772-3
Keywords: Fourier multipliers, sharp maximal operator, Littlewood-Paley-theory
Article copyright: © Copyright 1988 American Mathematical Society