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A note on operator-valued Fourier multipliers on Besov spaces

✍ Scribed by Shangquan Bu; Jin-Myong Kim

Publisher
John Wiley and Sons
Year
2005
Tongue
English
Weight
134 KB
Volume
278
Category
Article
ISSN
0025-584X

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✦ Synopsis

Abstract

Let X be a Banach space. We show that each m : ℝ \ {0} → L (X ) satisfying the Mikhlin condition sup~x ≠0~(‖m (x )‖ + ‖xm ′(x )‖) < ∞ defines a Fourier multiplier on B ^s^ ~p,q~ (ℝ; X ) if and only if 1 < p < ∞ and X is isomorphic to a Hilbert space; each bounded measurable function m : ℝ → L (X ) having a uniformly bounded variation on dyadic intervals defines a Fourier multiplier on B ^s^ ~p,q~ (ℝ; X ) if and only if 1 < p < ∞ and X is a UMD space. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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