A note on operator‐valued Fourier multipliers on Besov spaces | Zendy

Bu Shangquan | Zendy; Kim JinMyong | Zendy

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A note on operator‐valued Fourier multipliers on Besov spaces

Author(s) -

Bu Shangquan,

Kim JinMyong

Publication year - 2005

Publication title -

mathematische nachrichten

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.913

H-Index - 50

eISSN - 1522-2616

pISSN - 0025-584X

DOI - 10.1002/mana.200310330

Subject(s) - mathematics , multiplier (economics) , bounded function , fourier transform , hilbert space , banach space , pure mathematics , bounded operator , space (punctuation) , function space , mathematical analysis , discrete mathematics , linguistics , philosophy , economics , macroeconomics

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 sp,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 sp,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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