  We consider the category of perverse sheaves on a complex vector space smooth
with respect to a stratification given by an arrangement of hyperplanes with
real equations. As shown in an earlier wotk of two of the authors, this
category can be described in terms of certain diagrams of vector spaces
labelled by all the faces of the real arrangement (we call such diagrams
hyperbolic sheaves). In this paper we calculate, in these terms, several
fundamental operations of sheaf theory such as forming the space of vanishing
cycles, specialization and the Fourier-Sato transform.
