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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 Ehrenpreis 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 encountered in Volume I in special cases, which should provide the necessary motivation and background for a more systematic and precise exposition. The main technical tool in this volume is the Fourier- Laplace transformation. More powerful methods for the study of operators with variable coefficients will be developed in Volume III. However, constant coefficient theory has given the guidance for all that work. Although the field is no longer very active - perhaps because of its advanced state of development - and although it is possible to pass directly from Volume I to Volume III, the material presented here should not be neglected by the serious student who wants to gain a balanced perspective of the theory of linear partial differential equations.
List of contents
Existence and Approximation of Solutions of Differential Equations.- Interior Regularity of Solutions of Differential Equations.- The Cauchy and Mixed Problems.- Differential Operators of Constant Strength.- Scattering Theory.- Analytic Function Theory and Differential Equations.- Convolution Equations.
About the author
Lars Hörmander, born 1931 in Sweden, did his secondary schooling as well as his undergraduate and doctoral studies in Lund. His principle teacher and adviser at the University of Lund was Marcel Riesz until he returned, then Lars Gårding. In 1956 he worked in the USA, at the universities of Chicago, Kansas, Minnesota and New York, before returning to a chair at the University of Stockholm. He remained a frequent visitor to the US, particularly to Stanford and was Professor at the IAS, Princeton from 1964 to 1968. In 1968 he accepted a chair at the University of Lund, Sweden, where, today, he is Emeritus Professor.§§Hörmander's lifetime work has been devoted to the study of partial differential equations and its applications in complex analysis. In 1962 he was awarded the Fields Medal for his contributions to the general theory of linear partial differential operators.
Summary
In this second volume (4 volume work), Lars Hörmander looks at operators with constant coefficients. An analysis of the existence and regularity of (fundamental) solutions in the first two chapters is followed by a thorough study of the Cauchy problem. He then moves on to cover spectral theory of short range perturbations of operators with constant coefficients, and Fourier-Laplace representations of solutions of homogeneous differential equations with constant coefficients. The last chapter presents a study of the closely related subject of convolution operators.
Additional text
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 "This volume focuses on linear partial differential operators with constant coefficients … . Each chapter ends with notes on the literature, and there is a large bibliography. … The binding of this softcover reprint seems quite good … . Overall, it is great to have this book back at an affordable price. It really does deserve to be described as a classic." (Fernando Q. Gouvêa, MathDL, January, 2005)
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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
"This volume focuses on linear partial differential operators with constant coefficients ... . Each chapter ends with notes on the literature, and there is a large bibliography. ... The binding of this softcover reprint seems quite good ... . Overall, it is great to have this book back at an affordable price. It really does deserve to be described as a classic." (Fernando Q. Gouvêa, MathDL, January, 2005)
