Analysis (2nd ed.)

✍ Scribed by Lieb, Elliott H. & Loss, Michael

• Publisher: American Mathematical Society
• Year: 2011
• Tongue: English
• Leaves: 366
• Series: Graduate Studies in Mathematics
• Category: Library

✦ Synopsis

Teschl cites this as a good introduction to Lebesgue measure theory.

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From the preface: "Since the publication of the first edition [MR1415616] four years ago, we have received many helpful comments from colleagues and students. Not only were typographical errors pointed out, but interesting suggestions were also made for improvements and clarification. We, too, wanted to add more topics which, in the spirit of the book, are hopefully of use to students and practitioners.
"This led to a second edition, which contains all the 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'. Egorov'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.''

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1st ed. review:
In their preface the authors state that they wrote this unconventional book on analysis for students (and teachers) who "prefer to do something with the material, as it is learned, rather than wait for a full-fledged development of all basic principles''.
The first chapter, on measure and integration, leaves many technical (i.e. boring) details to the reader as exercises or to be looked up in one of the many existing books. Chapter 2 treats Lp-spaces without mentioning Banach spaces (Hilbert spaces are introduced very briefly). For instance, the authors say that the fact that the dual of Lp separates elements of Lp is "normally proved with the Hahn-Banach theorem'' but they avoid this by showing that if f≠0, then g=df|f|p−2f¯¯¯∈Lp′ and ∫fg≠0. Similarly, they prove Mazur's theorem, according to which if fj→f in Lp (1<p<∞) then a convex combination of the functions fj converges strongly to f, without the Hahn-Banach theorem.
Corresponding to the research interests of the authors there is a strong emphasis on inequalities. They try not to say "there exists a constant such that …'' but give it explicitly or at least an estimate of it. In Chapter 2, besides the Jensen and the Hölder inequalities, they present the Minkowski integral inequality and the two inequalities of Olof Hanner: it is left as an exercise to decide in what sense they express the uniform convexity and the uniform smoothness of the unit sphere. Chapters 3, 4 and 8 have the titles "Rearrangement inequalities'', "Integral inequalities'', and "Sobolev inequalities'', respectively.
Denoting the symmetric decreasing rearrangement of a function f defined on Rn by f∗, the authors first prove the "simplest inequality'': ⟨f,g⟩≤⟨f∗,g∗⟩ if f,g≥0. Next they prove as a lemma the Frederick Riesz inequality ⟨f,g∗h⟩≤⟨f∗,g∗∗h∗⟩ for n=1, and then generalize it to Rn for any n. They define the Steiner symmetrization and the Schwarz symmetrization of functions introduced by H. J. Brascamp, Lieb and J. M. Luttinger [J. Functional Analysis 17 (1974), 227–237; MR0346109], and then give two proofs of the theorem: a "compactness proof'' related to the proof in the above-cited paper, and a "symmetry proof'' using some ideas about competing symmetries due to E. A. Carlen and Loss [J. Funct. Anal. 88 (1990), no. 2, 437–456; MR1038450]. The general rearrangement inequality of Brascamp-Lieb-Luttinger is stated without proof.
Chapter 4 starts with the Young inequality (∗) |⟨f,g∗h⟩|≤Cp,q,r;n∥f∥p∥g∥q∥h∥r, where 1/p+1/q+1/r=2; the sharp constant Cp,q,r;n=(bpbqbr)n/2 with bp=p1/p/s1/s (1/p+1/s=1) was found by W. Beckner [Ann. of Math. (2) 102 (1975), no. 1, 159–182; MR0385456] and Brascamp and Lieb [Advances in Math. 20 (1976), no. 2, 151–173; MR0412366]. Without the sharp constant, Young's inequality follows from Hölder's, but the authors also present the full version giving the sharp constant and the functions for which equality holds (for the proof that these functions are unique they refer to the original paper). The "ultimate generalization of Young's inequality'' due to Lieb [Invent. Math. 102 (1990), no. 1, 179–208; MR1069246] is also stated without proof.
Next the authors consider the Hardy-Littlewood-Sobolev inequality, where g in (∗) is replaced by |x|−λ and q by n/λ (0<λ<n). Here the sharp constant is known for p=r=2n/(2n−λ) and was found by Lieb [Ann. of Math. (2) 118 (1983), no. 2, 349–374; MR0717827]; for p≠r only an upper estimate is known and it is given. Again the authors present two proofs: a simple one without the sharp constant, and the long and difficult one for p=r which yields the sharp constant and involves fundamentally the competing symmetries of Carlen and Loss. The Sobolev inequality ∥∇f∥2≥C∥f∥q, where q=2n/(n−2), with the sharp constant C, is deduced in Chapter 8 from the Hardy-Littlewood-Sobolev inequality. The authors also consider −Δ−−−√ instead of ∇ on the left-hand side, and the Sobolev inequality for a general Wm,p is quoted without proof.
Chapter 5 is an 11-page introduction to Fourier transforms on Lp (1≤p≤2). The sharp Hausdorff-Young inequality ∥Ff∥p′≤b1/np∥f∥p due to Beckner is stated without proof; a reference is given to Lieb's 1990 paper where the shortest proof, and also the functions for which equality is achieved, can be found.
Chapters 6 and 7 briefly present the distributions of Laurent Schwartz, the Sobolev spaces Wm,p, in particular H1=W1,2, and H1/2. The last three chapters apply the theories developed thus far. Chapter 9 introduces harmonic and subharmonic functions, potentials and energies, and proves the theorem of Frederick Riesz according to which a subharmonic function is the potential of a positive measure. The short Chapter 10 proves the hypoellipticity of the Laplace operator. Chapter 11 has the title "Introduction to the calculus of variations'' and considers three problems: (1) the eigenvalues and eigenfunctions of the Schrödinger equation −ΔΨ(x)+V(x)Ψ(x)=EΨ(x); (2) the Thomas-Fermi problem of finding a minimizer for the energy functional E(ρ)=
35∫R3ρ(x)5/3dx−∫R3z|x|ρi(x)dx+12∫R3∫R3ρi(x)ρ(y)|x−y|dxdy
under the conditions ρ≥0, ρ∈L1∩L5/3 and ∫R3ρ=N or ∫R3ρ≤N; (3) to find f∈L2(Rn) such that the Newtonian capacity of the bounded set A equals ∫Rnf2 and ϕ=|x|1−n∗f is ≥1 quasi-everywhere on A.
Reviewed by J. Horváth