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From the reviews: "Volumes III and IV complete L. Hörmander's treatise on linear partial differential equations. They constitute the most complete and up-to-date account of this subject, by the author who has dominated it and made the most significant contributions in the last decades.....It is a superb book, which must be present in every mathematical library, and an indispensable tool for all - young and old - interested in the theory of partial differential operators." L. Boutet de Monvel in Bulletin of the American Mathematical Society, 1987 "This treatise is outstanding in every respect and must be counted among the great books in mathematics. It is certainly no easy reading (...) but a careful study is extremely rewarding for its wealth of ideas and techniques and the beauty of presentation." J. Brüning in Zentralblatt MATH, 1987
Vol. II of Lars Hörmander's 4-volume treatise is mainly devoted to operators with constant coefficients. From the reviews: "...these volumes are excellently written and make for greatly profitable reading. For years to come they will surely be a main reference for anyone wishing to study partial differential operators." Mathematical Reviews
I had mixed feelings when I thought how I should prepare the book for the second edition. It was clear to me that I had to correct all mistakes and misprints that were found in the book during the life of the first edition. This was easy to do because the mistakes were mostly minor and easy to correct, and the misprints were not many. It was more difficult to decide whether I should update the book (or at least its bibliography) somehow. I decided that it did not need much of an updating. The main value of any good mathematical book is that it teaches its reader some language and some skills. It can not exhaust any substantial topic no matter how hard the author tried. Pseudodifferential operators became a language and a tool of analysis of partial differential equations long ago. Therefore it is meaningless to try to exhaust this topic. Here is an easy proof. As of July 3, 2000, MathSciNet (the database of the American Mathematical Society) in a few seconds found 3695 sources, among them 363 books, during its search for "pseudodifferential operator". (The search also led to finding 963 sources for "pseudo-differential operator" but I was unable to check how much the results ofthese two searches intersected). This means that the corresponding words appear either in the title or in the review published in Mathematical Reviews.
Let X be an abstract not necessarily compact orientable CR manifold of dimension 2n−1, n⩾2, and let Lk be the k-th tensor power of a CR complex line bundle L over X. Given q∈{0,1,…,n−1}, let □(q)b,k be the Gaffney extension of Kohn Laplacian for (0,q) forms with values in Lk. For λ≥0, let Π(q)k,≤λ:=E((−∞,λ]), where E denotes the spectral measure of □(q)b,k. In this work, the author proves that Π(q)k,≤k−N0F∗k, FkΠ(q)k,≤k−N0F∗k, N0≥1, admit asymptotic expansions with respect to k on the non-degenerate part of the characteristic manifold of □(q)b,k, where Fk is some kind of microlocal cut-off function. Moreover, we show that FkΠ(q)k,≤0F∗k admits a full asymptotic expansion with respect to k if □(q)b,k has small spectral gap property with respect to Fk and Π(q)k,≤0 is k-negligible away the diagonal with respect to Fk. By using these asymptotics, the authors establish almost Kodaira embedding theorems on CR manifolds and Kodaira embedding theorems on CR manifolds with transversal CR S1 action.
This book is an excellent, comprehensive introduction to semiclassical analysis. I believe it will become a standard reference for the subject. --Alejandro Uribe, University of Michigan Semiclassical analysis provides PDE techniques based on the classical-quantum (particle-wave) correspondence. These techniques include such well-known tools as geometric optics and the Wentzel-Kramers-Brillouin approximation. Examples of problems studied in this subject are high energy eigenvalue asymptotics and effective dynamics for solutions of evolution equations. From the mathematical point of view, semiclassical analysis is a branch of microlocal analysis which, broadly speaking, applies harmonic analysis and symplectic geometry to the study of linear and nonlinear PDE. The book is intended to be a graduate level text introducing readers to semiclassical and microlocal methods in PDE. It is augmented in later chapters with many specialized advanced topics which provide a link to current research literature.

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