The next topic of Chapter VI is the Cauchy problem for hyperbolic equa-
tions of order m ≥ 2, the domains of dependence of solutions to hyperbolic
equations, and H¨ ormander’s theory [H1] of propagation of singularities for
the equations of real principal type with applications to hyperbolic equa-
In Chapter VII the Fredholm property for elliptic boundary value prob-
lems and parametrices in smooth domains are studied following the approach
of the author’s book [E1]. The main application of the parametrix is the
study of heat trace asymptotics as t → 0. The parametrix construction
allows one to compute explicitly two leading terms of the heat trace asymp-
totics for the cases of Dirichlet and Neumann boundary conditions. Chapter
VII concludes with elements of the spectral theory of elliptic operators and
the proof of the index theorem for elliptic operators in
works of Atiyah-Singer [AtS1], [AtS2] and Seeley [Se3].
The last Chapter VIII is devoted to the theory of Fourier integral opera-
tors. Starting with the local theory of FIO, we proceed to the global theory.
We consider only a subclass of H¨ ormander’s FIOs (see [H1]), assuming that
the Lagrangian manifold of the FIO corresponds to the graph of a canoni-
cal transformation. In particular, having a global canonical transformation,
we construct a global FIO corresponding to this canonical transformation.
Next, following Maslov [M1], [M2], [MF], we construct a global geomet-
ric optic solution for a second order hyperbolic equation on arbitrary time
interval [0,T ].
Chapter VIII concludes with a section on the oblique derivative prob-
lem. The oblique derivative problem is a good example of nonelliptic bound-
ary value problem, and it attracted the attention of many mathematicians:
Egorov-Kondrat’ev [EgK], Malutin [Mal], Mazya-Paneah [MaP], Mazya
[Ma], and others. The section is based on the author’s paper [E3], and it
uses the FIOs to greatly simplify the problem. Similar results are obtained
independently by Sj¨ ostrand [Sj] and Duistermaat-Sj¨ ostrand [DSj].
At the end of each chapter there is a problem section. Some problems are
relatively simple exercises that help to study the material. Others are more
diﬃcult problems that cover additional topics not included in the book. In
those cases hints or references to the original sources are given.
I want to thank my friend and collaborator Jim Ralston for many fruitful
discussions and advice. I am very grateful to my former students Joe Ben-
nish, Brian Sako, Carol Shubin, Borislava Gutarz, Xiaosheng Li and others
who took notes during my classes. These notes were the starting point of
this book. I express my deep gratitude to the anonymous referees whose