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

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

2009-09-17

arxiv.org/abs/0909.3225v1

Mathematics

Functional Analysis

8 pages, submitted

Scientific paper

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.

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