Gerd Grubb’s Distributions and Operators is serious business. Weighing in at some 450 pages, it is a thorough and high level introduction to what is still a rather advanced subject, despite its ubiquity in (and outside of) analysis: Laurent Schwartz’s by now highly evolved distribution theory, here set in the context of operator theory.
The book is directed at graduate students and “researchers interested in its special topics,” and it is appropriate to add that the reader had better be serious about what he is up to. A thorough grounding in analysis is called for, beyond the opening interval of graduate school, with functional analysis and even some PDE already under one’s belt, as well as a strong orientation toward hard analysis. Sobolev spaces appear already on p. 57, while pseudo-differential operators show up on p. 163: again, serious business, not for the timid.
Distributions and Operators is split into five parts, as follows. Part I, “Distributions and derivatives,” is foundational, as is Part II, “ Extensions and Applications.” Caveat: this section contains a serious treatment of Fourier transforms vis à vis distributions and Sobolev’s Theorem (or inequality, I guess). Then Parts III and IV are, respectively, “Pseudodifferential operators” and “Boundary value problems,” with Fredholm theory, index theory, the Dirichlet problem, Neumann problem, and even some material due to A. P. Calderón making appearances: Grubb doesn’t waste any time getting to the more modern stuff. Finally, Part V is “Topics on Hilbert space operators,” starting off with the unbounded case (!).
Well-written and scholarly, and equipped with many exercises of considerable pedagogical importance (the serious student should clearly hit these as hard as possible), 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. A particularly nice feature of the book is the inclusion of a list of “miscellaneous exercises” (p. 147 ff.), being “problems used for examinations at Copenhagen University in ‘Modern Analysis’ since the 1980s.” It appears half-way through the book, suggesting a transition to the more specialized material of Part III and beyond.
Preface.- Motivation and overview.- Function spaces and approximation.- Distributions: Examples and rules of calculus.- Realizations and Sobolev spaces.- Fourier transformation of distributions.- Applications to differential operators: The Sobolev theorem.- Pseudodifferential operators on open sets.- Pseudodifferential operators on manifolds, index of elliptic operators.- Boundary value problems in a constant-coefficient case.- Pseudodifferential boundary operators.- Pseudodifferential methods for boundary value problems.- Unbounded linear operators.- Families of extensions.- Semigroups of operators.- Some notation and prerequisites.- Topological vector spaces.- Function spaces on sets with smooth boundary.- Bibliography.- Index.