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Vol. I of Lars H÷rmander's 4-volume treatise was an exposition of the theory of distributions and Fourier analysis preparing for the study of linear partial differential operators.

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Vol. 1 of Lars Hörmander's influential 4-volume treatise is a detailed exposition of the theory of distributions. From the reviews: 'In order to illustrate the richness of the book: in my review of the 1983 edition [...] I gave a list of 20 subjects which were new compared to Hörmander's book of 1963. Most of these subjects concern important, basic and highly nontrivial theorems in analysis. Hörmander's treatment of these is extremely clear and efficient and often highly original. [...] Most of the exercises are witty, with an interesting point. The phrasing of both the exercises and the answers and hints is very careful [...] In all, the book can be highly recommended, both as a textbook for advanced students, and as background and reference for introductory courses on distributions and Fourier analysis. J.J. Duistermaat in Mededelingen van het Wiskundig Genootschap. ... Read more

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the definitive guide to micro-local analysis
Hormander's four volume guide to linear partial differential operators is the definitive guide to smooth micro-local analysis from the man who invented a large fraction of it.

Micro-local analysis is an approach tothe analysis of singularities of variable coefficient linear partialdifferential equations based on the use of the Fourier transform and ofsymplectic geometry. Ideas from Hamiltonian physics and quantum theory alsofeature heavily. The fact that the subject is a mixture of analysis andgeometry makes it beautiful and powerful.

The four volume set is hardwork and in some ways is more useful as a reference work than as a learningbook but on the other hand, all the material is there and the books doreward the reader who is willing to spend the time to study themcarefully.

The first volume is the most accessible and the secondpaperback edition carries the added benefit of included exercises withpartial answers which are often highly informative. ... Read more

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The first two chapters of this book are devoted to convexity in the classical sense, for functions of one and several real variables respectively. This gives a background for the study in the following chapters of related notions which occur in the theory of linear partial differential equations and complex analysis such as (pluri-)subharmonic functions, pseudoconvex sets, and sets which are convex for supports or singular supports with respect to a differential operator. In addition, the convexity conditions which are relevant for local or global existence of holomorphic differential equations are discussed, leading up to Trépreau’s theorem on sufficiency of condition (capital Greek letter Psi) for microlocal solvability in the analytic category.

At the beginning of the book, no prerequisites are assumed beyond calculus and linear algebra. Later on, basic facts from distribution theory and functional analysis are needed. In a few places, a more extensive background in differential geometry or pseudodifferential calculus is required, but these sections can be bypassed with no loss of continuity. The major part of the book should therefore be accessible to graduate students so that it can serve as an introduction to complex analysis in one and several variables. The last sections, however, are written mainly for readers familiar with microlocal analysis.

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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.

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This volume presents lectures from the Danish-Swedish Analysis Seminar 1995, covering themes in mathematical physics, spanning a wide variety of topics. ... Read more

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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

Honours awarded to Lars Hörmander: Fields Medal 1962, Speaker at International Congress 1970, Wolf Prize 1988, AMS Steele Prize 2006

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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 Frédéric 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 books. It also includes a translation by J. Horváth of Riesz's thesis on summable trigonometric series and summable power series. They are thus a valuable reference work for libraries and for researchers. ... Read more

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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 Frédéric 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 books. It also includes a translation by J. Horváth of Riesz's thesis on summable trigonometric series and summable power series. They are thus a valuable reference work for libraries and for researchers. ... Read more

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From the reviews: "With the basic theory of distributions and Fourier analysis set down, volume II proceeds deeper into the study of linear PDE, particularly with constant coefficients. Detailed analysis is made of the regularity of certain fundamental solutions of general constant coefficient operators P(D), including a classification of when P(D) is hypoelliptic." American Mathematical Monthly#1 "The books are very careful written. They give a good introduction into the areas considered. The reviewer's opinion is that L. Hörmander`s books will be a basis for studying differential equations for a long time." ZAMM#2 ... Read more

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In this introductory textbook, a revised and extended version of well-known lectures by L. Hrmander 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 standard in the theory of linear differential operators, but the extension to the more exotic classes of opertors needed in the nonlinear theory is presented in complete detail. ... Read more

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In Fourier analysis, a multiplier operator is a type of linear operator, or transformation of functions. These operators act on a function by altering its Fourier transform. Specifically they multiply the Fourier transform of a function by a specified function known as the multiplier or symbol. Occasionally, the term "multiplier operator" itself is shortened simply to "multiplier". In simple terms, the multiplier reshapes the frequencies involved in any function. This class of operators turns out to be broad: general theory shows that a translation-invariant operator on a group which obeys some regularity conditions can be expressed as a multiplier operator, and conversely. Many familiar operators, such as translations and differentiation, are multiplier operators, although there are many more complicated examples such as the Hilbert transform. In signal processing, a multiplier operator is called a "filter", and the multiplier is the filter's frequency response. In the wider context, multiplier operators are special cases of spectral multiplier operators, which arise from the functional calculus of an operator. ... Read more

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Chapters: Lars Hörmander, Arvid Carlsson, Ernst Wigforss, Gustaf Retzius, Etzel Cardeña, Johannes Rydberg, Anders Jahan Retzius, Lauritz Weibull, Anders Retzius, Per Stjernquist, Albert Victor Bäcklund, Olle Schmidt, Per Georg Scheutz, Marcel Riesz, Lars Gårding, Martin Wiberg, Carl Hellmuth Hertz, Arne Meurman. Source: Wikipedia. Pages: 56. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: Lars Valter Hörmander (born January 24, 1931) is a Swedish mathematician who has been called "the foremost contributor to the modern theory of linear partial differential equations". He was awarded the Fields Medal in 1962, and the Wolf Prize in 1988. His Analysis of Linear Partial Differential Operators IIV is considered a standard work on the subject of linear partial differential operators. Hörmander completed his Ph.D. in 1955 at Lund University. Hörmander then worked at Stockholm University, at Stanford University, and at the Institute for Advanced Study in Princeton, New Jersey. He returned to Lund University as professor from 1968 until 1996, when he retired with the title of professor emeritus. Hörmander was born in Mjällby, a village in Blekinge in southern Sweden where his father was a teacher. Like his older brothers and sisters before him, he attended the local elementary school, realskola in a nearby town to which he commuted by train, and the gymnasium (high school) in Lund from which he graduated in 1948. At the time when he entered the gymnasium, the principal had instituted an experiment of reducing the period of the education from three to two years, and the daily activities to three hours. This freedom to work on his own, " than the universities offer in Sweden today", suited Hörmander "very well". He was also positively influenced by his enthusiastic mathematics teacher, a docent at Lu...More: http://booksllc.net/?id=500958 ... Read more

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Chapters: Sofia Kovalevskaya, Per Martin-Löf, Per Enflo, Lars Hörmander, Harald Cramér, Herman Wold, Henry Wallman, Anders Lindstedt, Olav Kallenberg, Ivar Otto Bendixson, Gösta Mittag-Leffler, Paco Lagerstrom, David Enskog, Waloddi Weibull, Björn Engquist, Lennart Carleson, Filip Lundberg, Germund Dahlquist, Arne Beurling, Albert Victor Bäcklund, Conny Palm, Anders Martin-Löf, Viggo Stoltenberg-Hansen, Anders Björner, Erik Ivar Fredholm, Helge Von Koch, Torkel Franzén, Johan Håstad, Marcel Riesz, Lars Gårding, Oskar Sandberg, Carl Johan Malmsten, Hans Riesel, Samuel Klingenstierna, Torsten Carleman, Thomas Hakon Grönwall, Åke Pleijel, Arne Meurman, Erland Samuel Bring, Fritz Carlson, Toby Gärnö Heikenen. Source: Wikipedia. Pages: 136. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: Per Erik Rutger Martin-Löf (born 1942) is a Swedish logician, philosopher, and mathematical statistician. He is internationally renowned for his work on the foundations of probability, statistics, mathematical logic, and computer science. Since the late 1970s, Martin-Löf's publications have been mainly in logic. In philosophical logic, Martin-Löf has wrestled with the philosophy of logical consequence and judgment, partly inspired by the work of Brentano, Frege, and Husserl. In mathematical logic, Martin-Löf has been active in developing intuitionistic type theory as a constructive foundation of mathematics; Martin-Löf's work on type theory has influenced computer science. Per Martin-Löf holds a joint chair for Mathematics and Philosophy at Stockholm University. His brother Anders Martin-Löf is now emeritus professor of mathematical statistics at Stockholm University; the two brothers have collaborated in research in probability and statistics. The research of Anders and Per Martin-Löf has influenced statistic...More: http://booksllc.net/?id=1925764 ... Read more