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Fourier multipliers on weighted $L^{p}$-spaces

Author: T. S. Quek
Journal: Proc. Amer. Math. Soc. 127 (1999), 2343-2351
MSC (1991): Primary 42A45
Published electronically: April 9, 1999
MathSciNet review: 1486747
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Abstract: In his 1986 paper in the Rev. Mat. Iberoamericana, A. Carbery proved that a singular integral operator is of weak type $(p,p)$ on $L^{p}(\mathbb{R}^{n})$ if its lacunary pieces satisfy a certain regularity condition. In this paper we prove that Carbery's result is sharp in a certain sense. We also obtain a weighted analogue of Carbery's result. Some applications of our results are also given.

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Additional Information

T. S. Quek
Affiliation: Department of Mathematics, National University of Singapore, Singapore 119260, Republic of Singapore

Keywords: Singular integral operators, Fourier multipliers, weighted $L^{p}$-spaces
Received by editor(s): August 20, 1996
Received by editor(s) in revised form: October 31, 1997
Published electronically: April 9, 1999
Dedicated: Dedicated to Professor Leonard Y. H. Yap on the ocassion of his sixtieth birthday
Communicated by: J. Marshall Ash
Article copyright: © Copyright 1999 American Mathematical Society

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