Ebook: Distributions and Operators

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.

This book gives an introduction to distribution theory, based on the work of Schwartz and of many other people. 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 ordinary and partial differential equations. In some of the latter chapters, the author illustrates how distribution theory is used to define pseudodifferential operators and how they are applied in the discussion of solvability of PDE, with or without boundary conditions. Each chapter has been enhanced with many exercises and examples, and a bibliography of relevant books and papers is collected at the end.

A few of the unique topics include: (1) Boundary value problems in a constant-coefficient case; (2) Pseudodifferential Boundary Operators; (3) Families of extensions.

Gerd Grubb is Professor of Mathematics at University of Copenhagen.