##
**Analysis.**
*(English)*
Zbl 0873.26002

Graduate Studies in Mathematics. 14. Providence, RI: American Mathematical Society (AMS). 278 p. (1996).

This book is an unconventional introductory text on analysis starting with the most elementary facts on measure theory, on the Fourier transform, and on the commonly used function spaces, and ending with a short introduction to potential theory and to the calculus of variations. The essentials of modern analysis, needed to understand some modern developments, for instance, of quantum mechanics, are presented in a rigorous and pedagogical way. The readers (graduate students, mathematicians, physicists and other natural scientists) are guided to a level where they can read the current literature with understanding. After glancing or reading the material contained in the book, one can agree with the authors who say: “…relative beginners can get some flavour of research mathematics and the feeling that the subject is open-ended”. The treatment of the subject is as direct as possible, the most general and abstract formulation of a result and a theory is often avoided. Contrary to many texts, the authors does not make a big distinction between real analysis and functional analysis, as they state: “Analysis without functions does not go very far”. For instance, the material concerning the classical function spaces does not involve the Baire category theorems, or the Hahn-Banach theorems, and the notion of a Hilbert-space is hardly mentioned. Moreover, the pure existence theorems of standard mathematics courses are not overemphasized. In cases when the existence of a constant is stated, the constant is explicitely given, or at least an estimate is obtained. This helps students and even researchers to learn how to calculate.

The first chapter is a very brief introduction to the theory of Lebesgue integration. The most important notions and existence theorems are recalled (some of them without proofs). A special emphasis is devoted to the convergence theorems where also less-known results like “The missing term in Fatou’s lemma” are presented.

The \(L^p\) spaces are considered in the second chapter. In addition to the classical Hölder’s and Minkowski’s inequalities, Hanner’s inequality is proved, which leads to the uniform convexity property. The separating property of the linear functionals and the uniform boundedness principle are proved without using the machinery of functional analysis. The Banach-Alaoglu theorem is also obtained in a constructive way which does not require the use of the axiom of choice.

The third chapter contains rearrangement inequalities. This subject mixes geometry and integration theory. The authors emphasize that these inequalities are extremely useful analytic tools, since these inequalities yield that the minimizers of the Hardy-Littlewood-Sobolev and the Sobolev inequalities are spherically symmetric functions.

Two important integral inequalities, Young’s inequality and the Hardy-Littlewood-Sobolev inequality are studied in the fourth chapter. The authors present both the simple versions and the full versions (with the sharp constants) of these inequalities. The simple versions (when only the existence of some constants is stated) have much easier proofs, the full versions are more involved and give complete insight.

The fifth chapter is devoted to the Fourier transform. Among others, the Plancherel theorem, the inversion formula, and the sharp Hausdorff-Young inequality are discussed here.

The sixth chapter is a brief introduction to distribution theory. This theory is developed around that every \(L^1_{\text{loc}}\) function is differentiable. The results known from the classical calculus (e.g., Newton-Leibniz rule, chain rule) are generalized to the distributional setting. The Sobolev spaces \(W^{1,p}_{\text{loc}}\) and \(W^{1,p}\) are introduced also here. The last result of the chapter is the statement that a positive distribution can be identified with a positive Borel regular measure.

The Sobolev spaces \(H^1\) and \(H^{1/2}\) are treated in the seventh chapter. The space \(H^1= W^{1,2}\) is particularly important since its norm comes from an inner product. This space is used to study the Schrödinger’s partial differential equation. The space \(H^{1/2}\) is discussed for two reasons, first it provides a good exercise in fractional differentiation. The second reason is that it can be used to describe a version of Schrödinger’s equation that incorporates some features of Einstein’s special relativity theory.

The eighth chapter contains Sobolev inequalities. These inequalities in general mean an estimation of lower order derivatives of a function in terms of higher order derivatives. These inequalities are standard tools in the regularity theory of partial differential equations, in the calculus of variations, in the geometric measure theory, etc. This chapter discusses only the most basic and simplest inequalities.

Chapter 9 is devoted to potential theory and Coulomb energies. The chapter starts with harmonic, subharmonic and superharmonic functions. The strong maximum principle, Newton’s theorem, the positivity properties of the Coulomb energy, and lower bounds on Schrödinger’s wave function are obtained in the sequel.

The regularity of solutions of Poisson’s equation is the topic of Chapter 10. The results presented here summarize various regularity properties (such as continuity, Hölder continuity, first and higher order differentiability) of the solutions depending on the regularity properties of the ‘right hand side’ of the equation.

The last chapter offers an introduction to the calculus of variations. This chapter is of illustratory character. The authors demonstrate how to use the mathematics developed in the previous chapters to solve optimization problems. The first example is taken from quantum mechanics and is the problem of determining the lowest energy of an atom. The second example, the Thomas-Fermi problem, arises in chemistry. The third problem is the capacitor problem from electrostatics. The theory is used to show that in all cases a minimizer exists and hence it is a solution of a partial differential equation. Of course, the methods of the book are not limited to these examples, but these examples show a general strategy to attack optimization problems.

The chapters are always followed by exercises. Their solution may deepen the understanding of the results and the methods of the material. As the authors state, they selected from those topics that had been useful in their own research and were among those that practicing analysts need in their kit-bag. This latter statement coincides with the opinion of the reviewer, as well.

The first chapter is a very brief introduction to the theory of Lebesgue integration. The most important notions and existence theorems are recalled (some of them without proofs). A special emphasis is devoted to the convergence theorems where also less-known results like “The missing term in Fatou’s lemma” are presented.

The \(L^p\) spaces are considered in the second chapter. In addition to the classical Hölder’s and Minkowski’s inequalities, Hanner’s inequality is proved, which leads to the uniform convexity property. The separating property of the linear functionals and the uniform boundedness principle are proved without using the machinery of functional analysis. The Banach-Alaoglu theorem is also obtained in a constructive way which does not require the use of the axiom of choice.

The third chapter contains rearrangement inequalities. This subject mixes geometry and integration theory. The authors emphasize that these inequalities are extremely useful analytic tools, since these inequalities yield that the minimizers of the Hardy-Littlewood-Sobolev and the Sobolev inequalities are spherically symmetric functions.

Two important integral inequalities, Young’s inequality and the Hardy-Littlewood-Sobolev inequality are studied in the fourth chapter. The authors present both the simple versions and the full versions (with the sharp constants) of these inequalities. The simple versions (when only the existence of some constants is stated) have much easier proofs, the full versions are more involved and give complete insight.

The fifth chapter is devoted to the Fourier transform. Among others, the Plancherel theorem, the inversion formula, and the sharp Hausdorff-Young inequality are discussed here.

The sixth chapter is a brief introduction to distribution theory. This theory is developed around that every \(L^1_{\text{loc}}\) function is differentiable. The results known from the classical calculus (e.g., Newton-Leibniz rule, chain rule) are generalized to the distributional setting. The Sobolev spaces \(W^{1,p}_{\text{loc}}\) and \(W^{1,p}\) are introduced also here. The last result of the chapter is the statement that a positive distribution can be identified with a positive Borel regular measure.

The Sobolev spaces \(H^1\) and \(H^{1/2}\) are treated in the seventh chapter. The space \(H^1= W^{1,2}\) is particularly important since its norm comes from an inner product. This space is used to study the Schrödinger’s partial differential equation. The space \(H^{1/2}\) is discussed for two reasons, first it provides a good exercise in fractional differentiation. The second reason is that it can be used to describe a version of Schrödinger’s equation that incorporates some features of Einstein’s special relativity theory.

The eighth chapter contains Sobolev inequalities. These inequalities in general mean an estimation of lower order derivatives of a function in terms of higher order derivatives. These inequalities are standard tools in the regularity theory of partial differential equations, in the calculus of variations, in the geometric measure theory, etc. This chapter discusses only the most basic and simplest inequalities.

Chapter 9 is devoted to potential theory and Coulomb energies. The chapter starts with harmonic, subharmonic and superharmonic functions. The strong maximum principle, Newton’s theorem, the positivity properties of the Coulomb energy, and lower bounds on Schrödinger’s wave function are obtained in the sequel.

The regularity of solutions of Poisson’s equation is the topic of Chapter 10. The results presented here summarize various regularity properties (such as continuity, Hölder continuity, first and higher order differentiability) of the solutions depending on the regularity properties of the ‘right hand side’ of the equation.

The last chapter offers an introduction to the calculus of variations. This chapter is of illustratory character. The authors demonstrate how to use the mathematics developed in the previous chapters to solve optimization problems. The first example is taken from quantum mechanics and is the problem of determining the lowest energy of an atom. The second example, the Thomas-Fermi problem, arises in chemistry. The third problem is the capacitor problem from electrostatics. The theory is used to show that in all cases a minimizer exists and hence it is a solution of a partial differential equation. Of course, the methods of the book are not limited to these examples, but these examples show a general strategy to attack optimization problems.

The chapters are always followed by exercises. Their solution may deepen the understanding of the results and the methods of the material. As the authors state, they selected from those topics that had been useful in their own research and were among those that practicing analysts need in their kit-bag. This latter statement coincides with the opinion of the reviewer, as well.

Reviewer: Zs.Páles (Debrecen)

### 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 |