The Analysis of Linear Partial Differential Operators II: Differential Operators with Constant Coefficients

by Lars Hörmander

Author received the 1962 Fields Medal

Author received the 1988 Wolf Prize (honoring achievemnets of a lifetime)

Author is leading expert in partial differential equations

Genres: Mathematics

406 pages, Kindle Edition

First published June 1, 1983

About the author

Lars 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...

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[HikBook]

Chapter 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.