 
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.
Introduction
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
