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The Analysis of Linear Partial Differential Operators II: Differential Operators with Constant Coefficients
Vol. I of Lars Hvrmander's 4-volume treatise was an exposition of the theory of distributions and Fourier analysis preparing for the study of linear partial differential operators. The present Vol. II is mainly devoted to 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. One chapter is devoted to the spectral theory of short range perturbations of operators with constant coefficients, and another presents Fourier-Laplace representations of solutions of homogeneous differential equations with constant coefficients. The last chapter is a study of the closely related subject of convolution operators.
- GenresMathematics
Kindle Edition
First published June 1, 1983
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About the author
Lars Hörmander
29 books2 followersLars Valter Hörmander (born 24 January 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, the Wolf Prize in 1988, and the Leroy P. Steele Prize in 2006. His Analysis of Linear Partial Differential Operators I–IV 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.
http://en.wikipedia.org/wiki/Lars_Hör...
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.
http://en.wikipedia.org/wiki/Lars_Hör...
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January 8, 2024Chapter X. Existence and Approximation of solutions of differential equations.
Definition 10.1.1. A positive function k in R is a temperate weight function if k(xi+eta)<= (1+C|xi|)^N k(eta))
This set is denoted as K.
Definition 10.1.6. If k in K, B_{p,k} is the set of all tempered distributions u such that Fourier(u) is a function and
|u|_{p,k}:=[int |k(xi) Fourier(u)(xi)|^pdxi]^(1/p) < infinity.
If p=infinity,
|u|_{infinity, k}= essential supremum of k(xi)fourier(u)(xi) (standard)
Theorem 10.1.7. Schwartz space subset B_{p,k} subset tempered distributions (in topological sense)
Chapter XIII. Differential operators of constant strength.
Differential operators defined on the space B_{k,p} are studied.
Definition 10.1.1. A positive function k in R is a temperate weight function if k(xi+eta)<= (1+C|xi|)^N k(eta))
This set is denoted as K.
Definition 10.1.6. If k in K, B_{p,k} is the set of all tempered distributions u such that Fourier(u) is a function and
|u|_{p,k}:=[int |k(xi) Fourier(u)(xi)|^pdxi]^(1/p) < infinity.
If p=infinity,
|u|_{infinity, k}= essential supremum of k(xi)fourier(u)(xi) (standard)
Theorem 10.1.7. Schwartz space subset B_{p,k} subset tempered distributions (in topological sense)
Chapter XIII. Differential operators of constant strength.
Differential operators defined on the space B_{k,p} are studied.
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