The analysis of Euclidean space is well-developed. The classical Lie
groups that act naturally on Euclidean space-the rotations, dilations,
and trans- lations-have both shaped and guided this development. In
particular, the Fourier transform and the theory of translation
invariant operators (convolution transforms) have played a central role
in this analysis. Much modern work in analysis takes place on a domain
in space. In this context the tools, perforce, must be different. No
longer can we expect there to be symmetries. Correspondingly, there is
no longer any natural way to apply the Fourier transform.
Pseudodifferential operators and Fourier integral operators can playa
role in solving some of the problems, but other problems require new,
more geometric, ideas. At a more basic level, the analysis of a smoothly
bounded domain in space requires a great deal of preliminary spadework.
Tubular neighbor- hoods, the second fundamental form, the notion of
"positive reach", and the implicit function theorem are just some of the
tools that need to be invoked regularly to set up this analysis. The
normal and tangent bundles become part of the language of classical
analysis when that analysis is done on a domain. Many of the ideas in
partial differential equations-such as Egorov's canonical transformation
theorem-become rather natural when viewed in geometric language. Many of
the questions that are natural to an analyst-such as extension theorems
for various classes of functions-are most naturally formulated using
ideas from geometry.