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The main change in this edition is the inclusion of exercises with answers and hints. This is meant to emphasize that this volume has been written as a general course in modern analysis on a graduate student level and not only as the beginning of a specialized course in partial differen tial equations. In particular, it could also serve as an introduction to harmonic analysis. Exercises are given primarily to the sections of gen eral interest; there are none to the last two chapters. Most of the exercises are just routine problems meant to give some familiarity with standard use of the tools...

Singularities of solutions of differential equations forms the common theme of these papers taken from a seminar held at the Institute for Advanced Study in Princeton in 1977-1978. While some of the lectures were devoted to the analysis of singularities, others focused on applications in spectral theory. As an introduction to the subject, this volume treats current research in the field in such a way that it can be studied with profit by the non-specialist.

This volume presents the unpublished articles of Lars Hormander from 1951 to 2007.The organisation of these articles and the notes explaining their origins, are by Lars Hormander himself, as he fully intended these to be published in this form.

From the reviews:

"Volumes III and IV complete L. Hormander'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...

The term convexity used to describe these lectures given at the Univer- sity of Lund in 1991-92 should be understood in a wide sense. Only Chap- ters I and II are devoted to convex sets and functions in the traditional sense of convexity. The following chapters study other kinds of convexity which occur in analysis. Most prominent is the pseudo-convexity (plurisubh- monicity) in the theory of functions of several complex variables discussed in Chapter IV. It relies on the theory of subharmonic functions in R DEGREES, so Chapter III is devoted to subharmonic functions in R" DEGREES for any n....

On March 17-19 and May 19-21,1995, analysis seminars were organized jointly at the universities of Copenhagen and Lund, under the heading "Danish-Swedish Analysis Seminar." The main topic was partial differen- tial equations and related problems of mathematical physics. The lectures given are presented in this volume, some as short abstracts and some as quite complete expositions or survey papers. They span over a large vari- ety of topics. The most frequently occurring theme is the use of microlocal analysis which is now important also in the study of non-linear differential equations...

In this introductory textbook, a revised and extended version of well-known lectures by L. Hormander from 1986, four chapters are devoted to weak solutions of systems of conservation laws. Apart from that the book only studies classical solutions. Two chapters concern the existence of global solutions or estimates of the lifespan for solutions of nonlinear perturbations of the wave or Klein-Gordon equation with small initial data. Four chapters are devoted to microanalysis of the singularities of the solutions. This part assumes some familiarity with pseudodifferential operators which are...

This volume is an expanded version of Chapters III, IV, V and VII of my 1963 book "Linear partial differential operators." In addition there is an entirely new chapter on convolution equations, one on scattering theory, and one on methods from the theory of analytic functions of several complex variables. The latter is somewhat limited in scope though since it seems superfluous to duplicate the monographs by Eh renpreis and by Palamodov on this subject. The reader is assumed to be familiar with distribution theory as presented in Volume I. Most topics discussed here have in fact been...

On March 17-19 and May 19-21,1995, analysis seminars were organized jointly at the universities of Copenhagen and Lund, under the heading "Danish-Swedish Analysis Seminar." The main topic was partial differen tial equations and related problems of mathematical physics. The lectures given are presented in this volume, some as short abstracts and some as quite complete expositions or survey papers. They span over a large vari ety of topics. The most frequently occurring theme is the use of microlocal analysis which is now important also in the study of non-linear differential equations although...

From the reviews:

"Volumes III and IV complete L. Hormander'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...

The aim of this book is to give a systematic study of questions con- cerning existence, uniqueness and regularity of solutions of linear partial differential equations and boundary problems. Let us note explicitly that this program does not contain such topics as eigenfunction expan- sions, although we do give the main facts concerning differential operators which are required for their study. The restriction to linear equations also means that the trouble of achieving minimal assumptions concerning the smoothness of the coefficients of the differential equations studied would not be worth...

Marcel Riesz (1886-1969) was the younger of the famed pair of mathematicians and brothers. Although Hungarian he spent most of his professional life in Sweden. He worked on summability theory, analytic functions, the moment problem, harmonic and functional analysis, potential theory and the wave equation. The depth of his research and the clarity of his writing place his work on the same level as that of his brother Frederic Riesz. This edition of his Collected Papers contains most of Marcel Riesz's published papers with the exception of a few papers in Hungarian that were subsumed into later...