Distributions and Operators: (252 Graduate Texts in Mathematics)

This textbook gives an introduction to distribution theory with emphasis on applications using functional analysis. In more advanced parts of the book, pseudodi?erential methods are introduced. Distributiontheoryhasbeen developedprimarilytodealwithpartial(and ordinary) di?erential equations in general situations. Functional analysis in, say, Hilbert spaces has powerful tools to establish operators with good m- ping properties and invertibility properties. A combination of the two allows showing solvability of suitable concrete partial di?erential equations (PDE). When partial di?erential operators are realized as operators in L (?) for 2 n anopensubset?ofR ,theycomeoutasunboundedoperators.Basiccourses infunctionalanalysisareoftenlimitedtothestudyofboundedoperators,but we here meet the necessityof treating suitable types ofunbounded operators; primarily those that are densely de?ned and closed. Moreover, the emphasis in functional analysis is often placed on selfadjoint or normal operators, for which beautiful results can be obtained by means of spectral theory, but the cases of interest in PDE include many nonselfadjoint operators, where diagonalizationbyspectraltheoryisnotveryuseful.Weincludeinthisbooka chapter on unbounded operatorsin Hilbert space (Chapter 12),where classes of convenient operators are set up, in particular the variational operators, including selfadjoint semibounded cases (e.g., the Friedrichs extension of a symmetric operator), but with a much wider scope. Whereas the functional analysis de?nition of the operators is relatively clean and simple, the interpretation to PDE is more messy and complicated.

Table of Contents:
Distributions and derivatives.- Motivation and overview.- Function spaces and approximation.- Distributions. Examples and rules of calculus.- Extensions and applications.- Realizations and Sobolev spaces.- Fourier transformation of distributions.- Applications to differential operators. The Sobolev theorem.- Pseudodifferential operators.- Pseudodifferential operators on open sets.- Pseudodifferential operators on manifolds, index of elliptic operators.- Boundary value problems.- Boundary value problems in a constant-coefficient case.- Pseudodifferential boundary operators.- Pseudodifferential methods for boundary value problems.- Topics on Hilbert space operators.- Unbounded linear operators.- Families of extensions.- Semigroups of operators.

Review :
From the reviews: "The book is directed at graduate students and 'researchers interested in its special topics' ... . Distribution and Operators is split into five parts ... . Well-written and scholarly, and equipped with many exercises of considerable pedagogical importance ... Grubb's book is a fine contribution to the literature and should soon occupy a solid place among graduate texts for aspiring hard analysts specializing, e.g., in PDE." (Michael Berg, MAA Online, December, 2008) "This textbook gives a very clear introduction to distribution theory with emphasis on applications using functional analysis. In more advanced parts of the book, pseudodifferential methods are introduced, specially tailored for the study of boundary value problems. ... The whole book is carefully and nicely written and each chapter ends with a number of exercises ... . perfect for a Ph.D. course ... . most of the material has been used frequently at the University of Copenhagen for several graduate courses." (Fabio Nicola, Mathematical Reviews, Issue 2010 b) "The book under review is a complete self-contained introduction to classical distribution theory with applications to the study of linear partial differential operators and, in particular, of (elliptic) boundary value problems, an area where the author has much experience ... . very clear and precise. Almost all the results given in the book are proved, and the proofs give plenty of details and are easy to follow. There are a lot of examples and exercises ... ." (David Jornet, Zentralblatt MATH, Vol. 1171, 2009)