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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A note on the cone multiplier


Author: Gerd Mockenhaupt
Journal: Proc. Amer. Math. Soc. 117 (1993), 145-152
MSC: Primary 42B15; Secondary 42B10, 42B25, 47B38, 47G10
MathSciNet review: 1098404
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Abstract: In this paper we study the convolution operator given on the Fourier transform side by multiplication by

$\displaystyle {m_\alpha }(x,z) = \phi (z)(1 - \vert x\vert/z)_ + ^\alpha ,\qquad (x,z) \in {{\mathbf{R}}^2} \times {\mathbf{R}},\;\alpha > 0,$

where $ \phi \in C_0^\infty (1,2)$. We will prove that $ {m_\alpha }$ defines a bounded operator on $ {L^4}({{\mathbf{R}}^3})$ if $ \alpha > \tfrac{1} {8}$. Furthermore, as a generalization of a result of C. Fefferman (Acta Math. 124 (1970), 9-36), we will show that an $ ({L^2},{L^p})$ restriction theorem for compact $ {C^\infty }$ submanifolds $ M \subset {{\mathbf{R}}^n}$ of arbitrary codimension imply results for multipliers having a singularity of the form $ \operatorname{dist} {(x,M)^\alpha }$ near $ M$.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1993-1098404-6
PII: S 0002-9939(1993)1098404-6
Article copyright: © Copyright 1993 American Mathematical Society