*The Analysis of Linear Partial Differential Operators*is in four volumes but the first two appeared together as did the second two. Volumes 1 and 2 have been reviewed together as have Volumes 3 and 4. We have therefore treated Volumes 1 and 2 as a single work and Volumes 3 and 4 as a single work.

**1. Linear Partial Differential Operators (1963), by Lars Hörmander.**

**1.1. Review by: Stephen Hoffman.**

*Amer. Math. Monthly*

**73**(3) (1966), 323.

This book is divided into three parts. Part 1 contains a chapter summarising the theory of distributions and a chapter on the special spaces of distributions to be considered. Part 2, on differential operators with constant coefficients, has chapters on the existence of solutions, the interior regularity of solutions and the Cauchy problem. Part 3 considers differential operators with variable coefficients. A brief chapter on differential equations with no solutions is followed by chapters on operators of constant strength, operators with simple characteristics, the Cauchy problem, and a concluding chapter on elliptic boundary value problems. The treatment is thorough with relatively abbreviated proofs. In the preface the author states that his aim is to give a systematic study of questions of existence, uniqueness, and regularity of solutions of partial differential equations and boundary problems. This claim is met.

**1.2. Review by: M Schechter.**

*Mathematical Reviews* MR0161012 **(28 #4221)**.

This book contains some of the recent developments in the theory of linear partial differential equations, in particular, those topics close to the interests of the author. There are three parts. The first deals with background material in functional analysis and develops the spaces in which the equations are embedded. The second deals with constant coefficient operators considering existence of solutions and interior regularity. The last and largest part discusses operators with variable coefficients. ... The book will be invaluable to many. The researcher in the field will find it a ready reference. For others it will serve as a readable account of current research. It certainly does not cover all major trends in partial differential equations, but comes as close as any book can. On the other hand, it can hardly be recommended for the novice. The introduction to each topic is good, but little is done in the way of motivation. The exposition is essentially of the form: definition, lemma, theorem. References for the methods employed in the book are very good. However, it is difficult to trace the origin of a theorem. ... In content and exposition the author has done an excellent job. The book cannot be praised too highly.

**1.3. Review by: L Ehrenpreis.**

*Science, New Series* **143** (3603) (1964), 234.

In the late 1940's Laurant Schwartz devised the theory of distributions, a formulation of some of the important classical concepts of mathematical analysis in the modern language of topological vector spaces. At the time many analysts were sceptical of the interest this work attracted, a typical comment being that Schwartz was acting like a linguist and not like a mathematician. Yet, during the 15-year interval, the theory has had a profound effect on partial differential equations and on some branches of physics and has, to a great extent, revolutionized some important trends in these fields. Linear Partial Differential Operators, by Hörmander, represents the progress that has been made in partial differential equations as a result of this new view- point. The author himself is a major contributor to that progress. ... Linear Partial Differential Operators contains very deep results. It is lucid and extremely readable, and it should become one of the classics in the field.

**2. An Introduction to Complex Analysis in Several Variables (1966), by Lars Hörmander.**

**2.1. From the Preface.**

Two recent developments in the theory of partial differential equations have caused this book to be written. One is the theory of overdetermined systems of differential equations with constant coefficients, which depends very heavily on the theory of functions of several complex variables. The other is the solution of the so-called partial-Neumann problem, which has made possible a new approach to complex analysis through methods from the theory of partial differential equations. Solving the Cousin problems with such methods gives automatically certain bounds for the solution, which are not easily obtained with the classical methods, and results of this type are important for the applications to overdetermined systems of differential equations.

**2.2. Review by: M Schechter.**

*Mathematical Reviews* MR0203075 **(34 #2933)**.

This book develops the theory of analytic functions of several complex variables from the viewpoint of the theory of partial differential equations. ... The treatment is not intended to achieve completeness in any direction, but rather to provide an introduction to the theory for those whose main interests are in analysis. The reader is expected to have basic knowledge in integration and distribution theory, functional analysis, and the calculus of differential forms. ... The book is well written and very clear. ... For anyone wishing to obtain a grasp of the basic theory, this book is an excellent introduction.

**2.3. Review by: J E Smoller.**

*SIAM Review* **33** (2) (1991), 338.

The book gives a fairly complete treatment of those aspects of distribution theory and Fourier analysis which are needed as background for studying linear partial differential operators. In this reviewer's opinion, the high point of the book is Chapter VIII, 'Spectral Analysis of Singularities', where the author presents and excellent discussion of the wavefront set, a notion he introduced some years ago to study singularities of solution of linear partial differential equations, and which, in the context of microlocal analysis, is being applied nowadays to study singularities of solutions of certain nonlinear equations.

**3. Integration theory (Swedish) (1970), by Tomas Claesson and Lars Hörmander.**

**3.1. Review by: Edwin Hewitt.**

*Mathematical Reviews*MR0267061

**(42 #1963).**

This small book, which evolved from lectures by one of the authors at Stockholm University in 1959, is intended to furnish students in Swedish universities with the information about the theory of integration they will need for various stages in the graduate program in mathematics in Swedish universities. In particular, the entire book is intended to suffice for courses for the doctor's examination. ... Within its brief compass, this book presents an astonishing amount of material. Every proof is uncomplicated and elegant. Many fine points of the theory have been sacrificed, and no applications whatever are mentioned. Nevertheless, within the compass of the authors' intentions, it is a striking success.

**4. (a) The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis (1983), by Lars Hörmander.**

**(b) The Analysis of Linear Partial Differential Operators II: Differential Operators with Constant Coefficients (1983), by Lars Hörmander.**

**4.1. Review by: F Treves.**

*Bull. Amer. Math. Soc.*

**10**(2) (1984), 337-340.

The first seven chapters of Volume I of the books under review here are devoted to a detailed exposition of distribution theory. In my opinion it is the best now available in print. As in the earlier books of the same author, the economy of the argument is unequalled. All the theorems are there (among them the Schwartz kernel theorem), and they all have short, sometimes very short, yet always complete, proofs. The reasoning often follows the most clever shortcuts. The accent is on precision, reliability and brevity. This can make for arduous reading on the part of the inexperienced student, but it provides an ideal text to the mathematician who would like to teach a course or base a seminar on some of the topics discussed in the book. ... Volume II is devoted to PDE proper, and essentially to differential operators with constant coefficients. ... To some extent it is an updated version of certain chapters of the book [Hörmander, *Linear partial differential operators,* 1963] covering fundamental solutions, inhomogeneous equations, hypoellipticity and the Cauchy problem. But it contains material that could not be found in the earlier book ...

**4.2. Review by: Michael E Taylor.**

*Amer. Math. Monthly* **92** (1985), 745-749.

Calculus was perfected by Newton to formulate and solve differential equations arising in mechanics. Work on continua, modelling things such as moving fluids and vibrating solids, quickly led to the formulation of partial differential equations, and the subject of PDE has enjoyed a long and productive history. A number of important developments in analysis have been motivated by, and particularly effective in, the development of linear PDE, the subject of the volumes by Professor Hörmander. ... These volumes are to some degree a greatly extended rewrite of Hörmander's 'Linear Partial Differential Operators', published by *Springer-Verlag* in 1964. They cover an enormous amount of analysis. ... Some of these topics will be more completely appreciated when they are used in subsequent volumes. The two volumes which are out, and their companions which will follow, will not likely serve as the texts for one's first brush with PDE, but the serious analyst will find here an elegant presentation of a vast amount of material on linear PDE, by a consummate master of the subject.

**4.3. Review by: L Cattabriga.**

*Mathematical Reviews* MR0717035 **(85g:35002a)**.

*Mathematical Reviews* MR0705278 **(85g:35002b)**.

These books are the first two volumes of a comprehensive work on the theory of linear partial differential operators intended to give a much more expanded and up-to-date treatment of this theory than the author's well-known book ['Linear partial differential operators' (1963)], which first appeared in 1963. The enormous progress in this field that occurred since the early 1960s, and to which the author contributed greatly, has caused a profound change both in the perspectives and in the techniques proper of this theory. This has been mainly due to the strong development of the study of the singularities of solutions of partial differential equations and to the introduction of pseudo-differential and Fourier integral operators. The first of these volumes provides a thorough treatment of a great deal of these new tools and techniques in the setting of Schwartz distribution theory. Its content can also be considered as a broad development of this theory with particular emphasis on Fourier analysis. ... Volume II is mainly devoted to partial differential operators with constant coefficients, with the only exception of Chapter XIII which treats differential operators of constant strength, and part of it obviously concerns problems already considered in the above-mentioned earlier work of the author ... Needless to say, 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.

**4.4. Review by: Calvin H Wilcox.**

*SIAM Review* **27** (2) (1985), 311-313.

The theory of partial differential equations is one of the oldest fields of mathematics. Its development can be said to begin with papers by d'Alembert (1746) and Euler (1748) on the vibration of strings. Problems arising in the study of partial differential equations have motivated many of the principal developments in classical and modern analysis. Examples include harmonic analysis (Fourier), complex analysis (Cauchy, Riemann), the theory of integral equations (Fredholm, Hilbert), Hilbert and Banach space theory, fixed point theorems (Schauder), the theory of distributions (L Schwartz) and many others. The theory of partial differential equations is one of the most active fields of modern mathematics. Thus in 1983 Mathematical Reviews published reviews of some 1582 papers and books in the field. Some of the problems analysed in these publications are derived from applications. However, a casual reading of Mathematical Reviews reveals that most of the papers on partial differential equations are not motivated by specific applications. Instead, the subject has become an autonomous field of pure mathematics with its own problems and methods. Professor Hörmander has been a major contributor to this field for nearly 30 years. A survey of recent publications in the field shows that the problems, concepts and methods that he has introduced have been taken up by many researchers and have led to several new areas of research. The books under review present a systematic exposition of much of professor Hörmander's work.

**5. (a) The Analysis of Linear Partial Differential Operators III: Pseudo-Differential Operators (1985), by Lars Hörmander.**

**(b) The Analysis of Linear Partial Differential Operators IV: Fourier Integral Operator (1985), by Lars Hörmander.**

**5.1. From the Preface.**

The first (*Distribution theory and Fourier analysis*) and second (*Differential operators with constant coefficients*) volumes of this monograph [1983] can be regarded as an expansion and updating of my book *Linear partial differential operators* in the Grundlehren series in 1963. However, Volumes III and IV are almost entirely new.

**5.2. Review by: Louis Boutet de Monvel.**

*Bull. Amer. Math. Soc.* **16** (1) (1987), 161-167.

These two volumes 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. The subject of the book is the study, in general, of linear partial differential equations (the book does not deal with nonlinear problems, except some which are used as tools, in particular in symplectic geometry, e.g., for the construction of normal forms). The first two volumes of the treatise mainly dealt with L Schwartz's distribution theory, Fourier transformation, and differential operators with constant coefficients or perturbations of these. The author himself describes them as a - rather thorough and far-reaching - updating of his book of 1963. Let us note, among the many new topics described in these books, the Malgrange preparation theorem, the method of the stationary phase and oscillatory integrals, and the definition of the wavefront set, which are important tools for the last two volumes. The two last volumes are more specifically a treatise on microlocal analysis and its applications. They arrive at a time when microlocal analysis, after twenty years, can be considered as mature and has given many fruits, and yet is still perfectly alive and promising.

**5.3. Review by: Min You Qi.**

*Mathematical Reviews* MR0781536** (87d:35002a)**.

*Mathematical Reviews* MR0781537** (87d:35002b)**.

The 1960s and 1970s saw great developments in the theory of linear PDE, owing mainly to the appearance of the pseudodifferential operators (PsDO) and the Fourier integral operators (FIO). ... Although the theory of PsDO has as its precursor the theory of singular integral operators, which has a longer history and is interesting in its own right, it began to play an ever more significant role in the theory of PDE since the appearance of Calderón's uniqueness theorem and the Atiyah-Singer-Bott index theorem in the early 1960s. Both of them should be noted as very important breakthroughs for the theory of linear PDE. Since then, PsDO have been a very powerful tool in this area. The theory of FIO, in its turn, originates from the connection between geometrical and physical optics, which has an even longer history, as it can be dated back to the conflict of Newton's and Huygens' theories on the nature of light and also the very interesting and profound story of the relation between classical and quantum mechanics. The theory of FIO also calls to the stage of linear PDE those very old and time-honoured ideas and methods of classical mechanics, of course, in its most up-to-date formulation. By this is meant the theory of symplectic geometry, which is indispensable in the theory of linear PDE. Thus, PsDO and FIO mark a new stage in the development of the theory of linear PDE, and the monograph under review is just a survey of all these developments and has PsDO and FIO as its main topics. Since the author is one of the founders of these theories and has made important contributions to almost all aspects of the theories, he is highly qualified to give that survey, which is so masterfully accomplished in this monograph. Actually, the author has almost rewritten many chapters of the theory, always simplified, and with many new insights, new methods and new problems. His style is always very accurate and precise (maybe a little too succinct). All these make reading the monograph very rewarding and also very demanding for the reader.

**5.4. Review by: Calvin H Wilcox.**

*SIAM Review* **28** (2) (1986), 285-287.

Volumes I and II of this work were reviewed in this Review [see above]. The purpose of this supplementary review is to describe some of the topics treated in Volumes III and IV that are of potential interest to applied mathematicians. These include asymptotic properties of eigenvalues (Chapters XVII and XXIX), pseudo-differential operators (Chapter XVIII), and the propagation of singularities (Chapters XXIV and XXVI).

**6. Notions of Convexity (1994), by Lars Hörmander.**

**6.1. Review by: T Hawai.**

*Bull. Amer. Math. Soc.*

**32**(4) (1995), 429-431.

This is an excellent exposition on the notions of "convexity in a wide sense", with a strong emphasis on their application to the theory of linear partial differential equations and complex analysis. "Convexity in a wide sense" includes subharmonic functions (Chapter III), plurisubharmonic functions, Siamese twins of pseudo-convex sets (Chapter IV), convexity with respect to differential operators (the so-called P-convexity) (Chapter VI), etc. ... Subjects that are relevant to concrete problems in complex analysis and the theory of linear partial differential operators seem to have been given top priority, and I think this selection of topics is reasonable and appropriate. ... I would like to end this review by expressing my personal wish: I hope students in complex analysis read this book to find that the theory of partial differential equations is closely related to the theory of analytic functions of several complex variables, and still more, I wish they would try to attack this adjacent field. 'An introduction to complex analysis in several variables' is an excellent book also for this purpose, but this new book is probably better suited for this purpose, as it contains much more material which is peculiar to the theory of partial differential equations. As one of the old students of 'An introduction to complex analysis in several variables', I believe the new generation will benefit much from this new book by Hörmander.

**6.2. Review by: Steven George Krantz.**

*Mathematical Reviews* MR1301332 **(95k:00002)**.

The notion of convexity is an old one. It was used by Archimedes in his axiomatic treatment of arc length. Later, it was used by Fermat, Cauchy, Minkowski, and others as a tool ancillary to other studies. Not until the 1930s did Bonneson and Fenchel formalize the notion of convexity and do a systematic study in a monograph. Nowadays, convexity is an essential tool in analysis and geometry. Nonetheless, most treatments of the subject are specialized. They are usually directed to functional analysis considerations, minimax problems, or to classical geometry. There are few, if any, systematic treatments of the hard analysis aspects of convexity. Thus we are indebted to Hörmander for giving us such a study in the book under review. The book is written in Hörmander's customary careful and concise style. ... The book has few prerequisites, and graduate students of analysis should be encouraged to read it. Not only does the book acquaint the reader with the "right way'' to think about convexity, but it is an entrée to a great deal of important mathematical culture: Fourier analysis, the partial problem, isoperimetric inequalities, Lelong numbers, microlocal analysis, and symplectic geometry.

**7. Lectures on nonlinear hyperbolic differential equations (1997), by Lars Hörmander.**

**7.1. Review by: Jean-Marc Delort.**

*Mathematical Reviews*MR1466700

**(98e:35103)**.

The material of this volume is a revised version of lectures given by the author more than ten years ago. It introduces the reader to many recent aspects in the study of nonlinear hyperbolic differential equations, many of them being presented for the first time in book form. ...