##
**Harmonic analysis in phase space.**
*(English)*
Zbl 0682.43001

Annals of Mathematics Studies, 122. Princeton, NJ: Princeton University Press. ix, 277 p. (1989).

“In the beginning was the Heisenberg group.... And the Heisenberg group begat quantum mechanics, the orbit method of representation theory, the symplectic group, theta functions, the oscillator semigroup, the Hermite operator, the \({\bar \partial}_ b\)-problem, pseudoconvex domains, pseudodifferential operators, wavelets, and others.” Actually, most of these descendents were alive and well before the Heisenberg group’s rôle as a common ancestor was brought to light over the last twenty years. The book under review is a valiant attempt to present an account of (most of) these areas of mathematics, with an emphasis on the analysis - quantum mechanics and pseudodifferential operators.

The first chapter (of five) is dedicated to the problem of representing the Heisenberg commutation relations, and some of its ramifications. The author shows us the Stone-von Neumann theorem, the Schrödinger, Fock- Bargmann, and nilmanifold (Zak) representations of the Heisenberg group and the intertwinings between them, and describes the parallels between Hamiltonian mechanics (Poisson brackets,...) and quantum mechanics (the uncertainty principle,...). Various aspects of analysis on the Heisenberg group are mentioned: the Weyl transform, twisted convolution, the Wigner transform and radar ambiguity functions should give the flavour.

The second chapter is concerned with pseudodifferential operators, though the starting point is in quantum mechanics: to a symbol \(\sigma\): \(\mathbb{R}^n\times \mathbb{R}^n\to \mathbb{C}\), the Weyl correspondence associates the operator \(\sigma(D,X)\), given (at least formally) by the rule \[ \sigma (D,X)f(x)=\int_{{\mathbb{R}}^ n}\int_{{\mathbb{R}}^ n}\sigma (\xi,(x+y)e^{-2\pi i(x-y)\xi}f(y)\,dy\,d\xi. \] The calculus of these pseudodifferential operators is developed, and the operators corresponding to the symbol classes \(S^m_{\rho,\delta}\) – \[ S^m_{\rho,\delta}=\{\sigma \in C^{\infty}(\mathbb{R}^n\times \mathbb{R}^n): | D^{\alpha}_{\xi}D_x^{\beta}\sigma (\xi,x)| \leq C_{\alpha \beta}(1+| \xi |)^{m-\rho | \alpha | +\delta | \beta |}\} \] – (where \(0\leq \delta \leq \rho \leq 1\) and \(\delta <1)\) are studied. To the relief of the reader who is used to the Kohn-Nirenberg correspondence between symbols and operators, defined by the formula \[ \sigma (D,X)_{KN}f(x)=\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\sigma (\xi,x)e^{2\pi ip\xi}f(x-p)\,dp\,d\xi \] (modulo the order of \(D\) and \(X\), or of \(\xi\) and \(x\), this is the usual definition). Folland shows that \(\sigma (D,X)_{KN}={\tilde \sigma}(D,X)\), where \({\tilde \sigma}\) belongs to \(S^m_{\rho,\delta}\) if \(\sigma\) does. He then proves the basic facts about the operators \(\sigma(D,X)\): product formulae, description of the kernels associated to the operators, and so on. Some of the meat of the theory is then presented: the Calderón-Vaillancourt theorem (with both the rather mysterious argument of R. E. Howe [J. Funct. Anal. 38, 188–254 (1980; Zbl 0449.35002)] and the “standard” proof) and Hörmander’s sharp Gårding inequality (with some nontrivial improvements in the vector-valued case). The chapter concludes by returning to quantum mechanics, but this time in the quantum field theoretic version, with a discussion of the Wick (and anti-Wick) correspondences between symbols and operators.

The next chapter, “Wave Packets and Wave Fronts”, develops the ideas of Chapter 2 in the directions indicated in the title, the main input being due to A. Córdoba and C. Fefferman [Commun. Partial Differ. Equations 3, 979–1005 (1978; Zbl 0389.35046)], and touches on the links with \(\vartheta\)-functions in discussing the limitations of Gabor wavelets (signals of the form \(Ce^{iax}e^{-b(x-c)^2})\) in signal analysis and synthesis.

The metaplectic representation is dissected in Chapter 4. The symplectic group \(\mathrm{Sp}(n,\mathbb{R})\) is constructed as a subgroup of \(\mathrm{GL}(2n,\mathbb{R})\), then its action on the Heisenberg group by automorphisms is used to find the metaplectic (or oscillator or harmonic or Segal-Shale-Weil) representation, first on \(L^2(\mathbb{R})\), and then on Fock space. The author returns to the quantum mechanical theme with the Groenewold-van Hove results on the impossibility of a first quantization procedure with all the properties that one would like, and then investigates the Siegel half space \(\Sigma_n\) and the Siegel disc \(\Delta_n\). The group structure of \(\mathrm{Sp}(n, \mathbb{R})\) is also used to give an explanation of the privileged rôle of the Gaussian in harmonic analysis on phase space.

The final chapter deals with the oscillator semigroup, i.e., the semigroup of integral operators on \(L^2(\mathbb{R}^n)\) whose kernels are Gaussians centred at 0. Most of the results are (quite similar to results) in R. E. Howe’s recent paper [Proc. Symp. Pure Math. 48, 61–132 (1988; Zbl 0687.47034)] but the author has added his own development of a parallel theory of operators on Fock space.

After this outline, the reader may well ask what is not in the book. The connections with \(\vartheta\)-functions are barely touched on, and the links with pseudoconvex domains are mentioned only in a “Notes and Remarks” section. It seems a pity that no discussion of the unit disc in \({\mathbb{C}}^ n\) and the corresponding Siegel domain was included, especially since (to this reviewer’s way of thinking) this symmetric space is easier to understand than the Siegel half space and disc. Another notable omission is any discussion of subelliptic operators, in whose understanding the author himself played a key rôle. But perhaps we should await Volume II.

There are a few references which might perhaps be added to those given: in particular, I would suggest that A. A. Kirillov’s paper [Russ. Math. Surv. 17, No. 4, 53–104 (1962); translation from Usp. Mat. Nauk 17, No. 4(106), 57–110 (1962; Zbl 0106.25001)] is a useful introduction to the representation theory of nilpotent Lie groups and the orbit method for a reader who has just mastered the Heisenberg group, and I would add the recent paper of A. Grossmann, J. Morlet and T. Paul [J. Math. Phys. 26, 2473–2479 (1985; Zbl 0571.22021)] to the references on “wavelets” at the end of Chapter 4.

The presentation is very good – the author has taken great pains to express himself clearly and the text has been carefully typeset and proofread – an unusual occurrence in this day and age. The notation is consistent throughout, which is quite an achievement considered that the various areas represented in the book tend to talk different languages. (Perhaps this is why the interaction with complex analysis was omitted – it must have been a major task to translate much of the material on pseudodifferential operators from the Kohn-Nirenberg language into the Weyl language, and it would have required further translation in order to make the complex parametrizations consistent with the rest of the book; this may have been too formidable a prospect even for Folland to contemplate).

All in all, the author should be congratulated on a very valuable addition to the library of harmonic analysis.

The first chapter (of five) is dedicated to the problem of representing the Heisenberg commutation relations, and some of its ramifications. The author shows us the Stone-von Neumann theorem, the Schrödinger, Fock- Bargmann, and nilmanifold (Zak) representations of the Heisenberg group and the intertwinings between them, and describes the parallels between Hamiltonian mechanics (Poisson brackets,...) and quantum mechanics (the uncertainty principle,...). Various aspects of analysis on the Heisenberg group are mentioned: the Weyl transform, twisted convolution, the Wigner transform and radar ambiguity functions should give the flavour.

The second chapter is concerned with pseudodifferential operators, though the starting point is in quantum mechanics: to a symbol \(\sigma\): \(\mathbb{R}^n\times \mathbb{R}^n\to \mathbb{C}\), the Weyl correspondence associates the operator \(\sigma(D,X)\), given (at least formally) by the rule \[ \sigma (D,X)f(x)=\int_{{\mathbb{R}}^ n}\int_{{\mathbb{R}}^ n}\sigma (\xi,(x+y)e^{-2\pi i(x-y)\xi}f(y)\,dy\,d\xi. \] The calculus of these pseudodifferential operators is developed, and the operators corresponding to the symbol classes \(S^m_{\rho,\delta}\) – \[ S^m_{\rho,\delta}=\{\sigma \in C^{\infty}(\mathbb{R}^n\times \mathbb{R}^n): | D^{\alpha}_{\xi}D_x^{\beta}\sigma (\xi,x)| \leq C_{\alpha \beta}(1+| \xi |)^{m-\rho | \alpha | +\delta | \beta |}\} \] – (where \(0\leq \delta \leq \rho \leq 1\) and \(\delta <1)\) are studied. To the relief of the reader who is used to the Kohn-Nirenberg correspondence between symbols and operators, defined by the formula \[ \sigma (D,X)_{KN}f(x)=\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\sigma (\xi,x)e^{2\pi ip\xi}f(x-p)\,dp\,d\xi \] (modulo the order of \(D\) and \(X\), or of \(\xi\) and \(x\), this is the usual definition). Folland shows that \(\sigma (D,X)_{KN}={\tilde \sigma}(D,X)\), where \({\tilde \sigma}\) belongs to \(S^m_{\rho,\delta}\) if \(\sigma\) does. He then proves the basic facts about the operators \(\sigma(D,X)\): product formulae, description of the kernels associated to the operators, and so on. Some of the meat of the theory is then presented: the Calderón-Vaillancourt theorem (with both the rather mysterious argument of R. E. Howe [J. Funct. Anal. 38, 188–254 (1980; Zbl 0449.35002)] and the “standard” proof) and Hörmander’s sharp Gårding inequality (with some nontrivial improvements in the vector-valued case). The chapter concludes by returning to quantum mechanics, but this time in the quantum field theoretic version, with a discussion of the Wick (and anti-Wick) correspondences between symbols and operators.

The next chapter, “Wave Packets and Wave Fronts”, develops the ideas of Chapter 2 in the directions indicated in the title, the main input being due to A. Córdoba and C. Fefferman [Commun. Partial Differ. Equations 3, 979–1005 (1978; Zbl 0389.35046)], and touches on the links with \(\vartheta\)-functions in discussing the limitations of Gabor wavelets (signals of the form \(Ce^{iax}e^{-b(x-c)^2})\) in signal analysis and synthesis.

The metaplectic representation is dissected in Chapter 4. The symplectic group \(\mathrm{Sp}(n,\mathbb{R})\) is constructed as a subgroup of \(\mathrm{GL}(2n,\mathbb{R})\), then its action on the Heisenberg group by automorphisms is used to find the metaplectic (or oscillator or harmonic or Segal-Shale-Weil) representation, first on \(L^2(\mathbb{R})\), and then on Fock space. The author returns to the quantum mechanical theme with the Groenewold-van Hove results on the impossibility of a first quantization procedure with all the properties that one would like, and then investigates the Siegel half space \(\Sigma_n\) and the Siegel disc \(\Delta_n\). The group structure of \(\mathrm{Sp}(n, \mathbb{R})\) is also used to give an explanation of the privileged rôle of the Gaussian in harmonic analysis on phase space.

The final chapter deals with the oscillator semigroup, i.e., the semigroup of integral operators on \(L^2(\mathbb{R}^n)\) whose kernels are Gaussians centred at 0. Most of the results are (quite similar to results) in R. E. Howe’s recent paper [Proc. Symp. Pure Math. 48, 61–132 (1988; Zbl 0687.47034)] but the author has added his own development of a parallel theory of operators on Fock space.

After this outline, the reader may well ask what is not in the book. The connections with \(\vartheta\)-functions are barely touched on, and the links with pseudoconvex domains are mentioned only in a “Notes and Remarks” section. It seems a pity that no discussion of the unit disc in \({\mathbb{C}}^ n\) and the corresponding Siegel domain was included, especially since (to this reviewer’s way of thinking) this symmetric space is easier to understand than the Siegel half space and disc. Another notable omission is any discussion of subelliptic operators, in whose understanding the author himself played a key rôle. But perhaps we should await Volume II.

There are a few references which might perhaps be added to those given: in particular, I would suggest that A. A. Kirillov’s paper [Russ. Math. Surv. 17, No. 4, 53–104 (1962); translation from Usp. Mat. Nauk 17, No. 4(106), 57–110 (1962; Zbl 0106.25001)] is a useful introduction to the representation theory of nilpotent Lie groups and the orbit method for a reader who has just mastered the Heisenberg group, and I would add the recent paper of A. Grossmann, J. Morlet and T. Paul [J. Math. Phys. 26, 2473–2479 (1985; Zbl 0571.22021)] to the references on “wavelets” at the end of Chapter 4.

The presentation is very good – the author has taken great pains to express himself clearly and the text has been carefully typeset and proofread – an unusual occurrence in this day and age. The notation is consistent throughout, which is quite an achievement considered that the various areas represented in the book tend to talk different languages. (Perhaps this is why the interaction with complex analysis was omitted – it must have been a major task to translate much of the material on pseudodifferential operators from the Kohn-Nirenberg language into the Weyl language, and it would have required further translation in order to make the complex parametrizations consistent with the rest of the book; this may have been too formidable a prospect even for Folland to contemplate).

All in all, the author should be congratulated on a very valuable addition to the library of harmonic analysis.

Reviewer: Michael Cowling (Sydney)

### MSC:

43-02 | Research exposition (monographs, survey articles) pertaining to abstract harmonic analysis |

43A80 | Analysis on other specific Lie groups |

22E70 | Applications of Lie groups to the sciences; explicit representations |

35S05 | Pseudodifferential operators as generalizations of partial differential operators |

81-02 | Research exposition (monographs, survey articles) pertaining to quantum theory |

43A85 | Harmonic analysis on homogeneous spaces |