This is an announcement for the paper "New thoughts on the vector-valued Mihlin-H"ormander multiplier theorem" by Tuomas P. Hytonen.

Abstract: Let X be a UMD space with type t and cotype q, and let T be a Fourier multiplier operator with a scalar-valued symbol m. If the Mihlin multiplier estimate holds for all partial derivatives of m up to the order n/max(t,q')+1, then T is bounded on the X-valued Bochner spaces. 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 n/r+1, where r 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.

Archive classification: math.FA

Mathematics Subject Classification: 42B15; 46B09; 46B20

Remarks: 8 pages, submitted

The source file(s), cotype-multipliers.bbl: 2535 bytes

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http://front.math.ucdavis.edu/0909.3225

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http://arXiv.org/abs/0909.3225

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