R. de la Llave, R. Obaya
Regularity of the composition operator 
  in spaces of H\"older functions.
(99K, Latex 2.09)

ABSTRACT.  We study the regularity of the
composition operator ($(f,g) \mapsto g\circ f) $
in spaces of H\"older differentiable functions.
Depending on the smooth norms used to
topologize $f,g$ and their composition, the
operator  has different differentiability properties.
We give complete and sharp results for
the classical H\"older spaces of
functions defined on geometrically well behaved
open sets in Banach spaces.  We also provide
examples that show that the regularity conclusions are
sharp and also that if the geometric conditions fail, 
even in finite dimensions, many elements of
the theory of functions (smoothing, interpolation,
extensions) can have somewhat unexpected properties.