Analysis. 2nd ed. (English) Zbl 0966.26002

Lieb, Elliott H.; Loss, Michael

Graduate Studies in Mathematics. 14. Providence, RI: American Mathematical Society (AMS). xxii, 346 p. (2001).

See the review of the first edition (1996) in Zbl 0873.26002.
From the Preface to the second edition:
“This second edition contains corrections and some fresh items. Chief among these is Chapter 12 in which we explain several topics concerning eigenvalues of the Laplacian and the Schrödinger operator, such as the min-max principle, coherent states, semiclassical approximation and how to use these to get bounds on eigenvalues and sums of eigenvalues. But there are other additions, too, such as more on Sobolev spaces (Chapter 8) including a compactness criterion, and Poincaré, Nash and logarithmic Sobolev inequalities. The latter two are applied to obtain smoothing properties of semigroups.
Chapter 1 (Measure and integration) has been supplemented with a discussion of the more usual approach to integration theory using simple functions, and how to make this even simpler by using ‘really simple functions’. Egoroff’s theorem has also been added. Several additions were made to Chapter 6 (Distributions) including one about the Yukawa potential.
There are, of course, many more exercises as well”.

MSC:

26-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to real functions
28-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to measure and integration
42-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to harmonic analysis on Euclidean spaces
46-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functional analysis
49-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to calculus of variations and optimal control
26D10	Inequalities involving derivatives and differential and integral operators
26D15	Inequalities for sums, series and integrals
31B05	Harmonic, subharmonic, superharmonic functions in higher dimensions
31B15	Potentials and capacities, extremal length and related notions in higher dimensions
46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46F05	Topological linear spaces of test functions, distributions and ultradistributions
46F10	Operations with distributions and generalized functions
49R50	Variational methods for eigenvalues of operators (MSC2000)
81Q05	Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics

Keywords:

measure theory; Fourier transform; function spaces; potential theory; calculus of variations; Banach-Alaoglu theorem; rearrangement inequalities; integral inequalities; Young’s inequality; Hardy-Littlewood-Sobolev inequality; distribution theory; Coulomb energies; regularity; eigenvalues; Laplacian; Schrödinger operator; min-max principle; Sobolev spaces; compactness criterion; Egoroff’s theorem; Yukawa potential

Citations:

Zbl 0873.26002