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New thoughts on the vector-valued Mihlin-Hörmander multiplier theorem

Author: Tuomas P. Hytönen
Journal: Proc. Amer. Math. Soc. 138 (2010), 2553-2560
MSC (2010): Primary 42B15; Secondary 46B09, 46B20
Published electronically: March 11, 2010
MathSciNet review: 2607885
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Abstract: Let $ X$ be a UMD space with type $ t$ and cotype $ q$, and let $ T_m$ be a Fourier multiplier operator with a scalar-valued symbol $ m$. If $ \vert\partial^{\alpha}m(\xi)\vert\lesssim\vert{\xi}\vert^{-\vert\alpha\vert}$ for all $ \vert\alpha\vert\leq\lfloor{n/\max(t,q')\rfloor}+1$, then $ T_m$ is bounded on $ L^p(\mathbb{R}^n;X)$ for all $ p\in(1,\infty)$. For scalar-valued multipliers, this improves the theorem of Girardi and Weis (J. Funct. Anal., 2003), who required similar assumptions for derivatives up to the order $ \lfloor{n/r}\rfloor+1$, where $ r\leq\min(t,q')$ is a Fourier-type of $ X$. However, the present method does not apply to operator-valued multipliers, which are also covered by the Girardi-Weis theorem.

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Additional Information

Tuomas P. Hytönen
Affiliation: Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68, FI-00014 Helsinki, Finland

Keywords: Fourier multiplier, type and cotype of Banach spaces
Received by editor(s): September 17, 2009
Received by editor(s) in revised form: November 23, 2009
Published electronically: March 11, 2010
Communicated by: Nigel J. Kalton
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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