I have tried in this book to describe those aspects of
pseudodifferential and Fourier integral operator theory whose usefulness
seems proven and which, from the viewpoint of organization and
"presentability," appear to have stabilized. Since, in my opinion, the
main justification for studying these operators is pragmatic, much
attention has been paid to explaining their handling and to giving
examples of their use. Thus the theoretical chapters usually begin with
a section in which the construction of special solutions of linear
partial differential equations is carried out, constructions from which
the subsequent theory has emerged and which continue to motivate it:
parametrices of elliptic equations in Chapter I (introducing
pseudodifferen- tial operators of type 1, 0, which here are called
standard), of hypoelliptic equations in Chapter IV (devoted to
pseudodifferential operators of type p, 8), fundamental solutions of
strongly hyperbolic Cauchy problems in Chap- ter VI (which introduces,
from a "naive" standpoint, Fourier integral operators), and of certain
nonhyperbolic forward Cauchy problems in Chapter X (Fourier integral
operators with complex phase). Several chapters-II, III, IX, XI, and
XII-are devoted entirely to applications. Chapter II provides all the
facts about pseudodifferential operators needed in the proof of the
Atiyah-Singer index theorem, then goes on to present part of the results
of A. Calderon on uniqueness in the Cauchy problem, and ends with a new
proof (due to J. J. Kohn) of the celebrated sum-of-squares theorem of L.
Hormander, a proof that beautifully demon- strates the advantages of
using pseudodifferential operators.
