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Math. Nachr. (1996), OperatorValued Fourier Multipliers,

Summary: Math. Nachr. (1996),
Operator­Valued Fourier Multipliers,
Vector­Valued Besov Spaces, and Applications
By Herbert Amann of Z¨urich
(Received August 16, 1996)
Abstract. It is shown that translation­invariant operators with operator­valued symbols act con­
tinuously on Besov spaces of Banach­space­valued distributions. This result is then used to extend
and complement the known theory of vector­valued Besov spaces. In addition, its power is demon­
strated by giving applications to a variety of problems from elliptic and parabolic differential and
integrodifferential equations.
Fourier multiplier theorems provide one of the most important tools in the study
of partial differential and pseudodifferential equations. Among them Mikhlin's theo­
rem, guaranteeing the continuity of pseudodifferential operators on L p ­spaces, plays a
predominant r“ole.
The simplest case of a pseudodifferential operator is provided by a translation­
invariant operator which can always be written in the form
a(D) := F \Gamma1 aF
where a, the symbol of a(D), is a sufficiently smooth function on IR n , and F denotes the
Fourier transform. We are particularly interested in the case where a takes its value in


Source: Amann, Herbert - Institut für Mathematik, Universität Zürich


Collections: Mathematics