Distributions and Operators

Gerd Grubb (auth.)0387848940, 9780387848945

This book gives an introduction to distribution theory, in the spirit of Laurent Schwartz. Additionally, the aim is to show how the theory is combined with the study of operators in Hilbert space by methods of functional analysis, with applications to partial and ordinary differential equations. Here, the author provides an introduction to unbounded operators in Hilbert space, including a complete theory of extensions of operators, and applications using contraction semigroups.

In more advanced parts of the book, the author shows how distribution theory is used to define pseudodifferential operators on manifolds, and gives a detailed introduction to the pseudodifferential boundary operator calculus initiated by Boutet de Monvel, which allows a modern treatment of elliptic boundary value problems.

This book is aimed at graduate students, as well as researchers interested in its special topics, and as such, the author provides careful explanations along with complete proofs, and a bibliography of relevant books and papers. Each chapter has been enhanced with many exercises and examples.

Unique topics include:

* the interplay between distribution theory and concrete operators;

* families of extensions of nonselfadjoint operators;

* an illustration of the solution maps between distribution spaces by a fully worked out constant-coefficient case;

* the pseudodifferential boundary operator calculus;

* the Calderón projector and its applications.

Gerd Grubb is Professor of Mathematics at University of Copenhagen.

Table of contents :
Front Matter….Pages I-XII
Front Matter….Pages 1-1
Motivation and overview….Pages 3-7
Function spaces and approximation….Pages 9-25
Distributions. Examples and rules of calculus….Pages 27-54
Front Matter….Pages 55-55
Realizations and Sobolev spaces….Pages 57-94
Fourier transformation of distributions….Pages 95-122
Applications to differential operators. The Sobolev theorem….Pages 123-160
Front Matter….Pages 161-161
Pseudodifferential operators on open sets….Pages 163-194
Pseudodifferential operators on manifolds, index of elliptic operators….Pages 195-215
Front Matter….Pages 216-216
Boundary value problems in a constant-coefficient case….Pages 219-249
Pseudodifferential boundary operators….Pages 251-304
Pseudodifferential methods for boundary value problems….Pages 305-333
Front Matter….Pages 335-335
Unbounded linear operators….Pages 337-372
Families of extensions….Pages 373-404
Semigroups of operators….Pages 405-422
Back Matter….Pages 423-464