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**Harmonic analysis on the Heisenberg group.**
*(English)*
Zbl 0892.43001

Progress in Mathematics (Boston, Mass.). 159. Boston, MA: Birkhäuser. xii, 191 p. (1998).

This monograph deals with various aspects of harmonic analysis on the Heisenberg group. The aim of this work is to develop several topics from classical Fourier analysis, such as Plancherel and Paley-Wiener theorems, Wiener-Tauberian theorems, Bochner-Riesz means and multiplier theorems for the Fourier transform and so on, in the noncommutative setup of the Heisenberg group.

In Chapter 1, the Heisenberg group is introduced. Basic results about the representation theory on the Heisenberg group are studied. Then the group Fourier transform is defined, and the Plancherel theorem and the inversion formula are proved. Further, properties of the group Fourier transform, the Hermite, special Hermite and Laguerre functions are introduced. The author proves analogues of Paley-Wiener theorems and Hardy’s theorem for the group Fourier transform on the Heisenberg group.

In Chapter 2 the spectral theory of the sub-Laplacian is studied as developed by R. Strichartz in [J. Funct. Anal. 87, 51-148 (1989; Zbl 0694.43008); 96, 350-406 (1991; Zbl 0734.43004)]. The eigenfunctions of the sub-Laplacian are given in terms of the special Hermite functions. The author obtains an Abel summability result for the expansions in terms of the eigenfunctions of the sub-Laplacian. For the spectral projection operators associated to the expansions, he establishes a Paley-Wiener theorem and some restriction theorem. Using the restriction theorem, he studies the Bochner-Riesz means for the sub-Laplacian. The author also develops a Littlewood-Paley-Stein theory for the sub-Laplacian and proves a multiplier theorem for the sub-Laplacian.

The Heisenberg group \(H^n\) and the unitary group \(U(n)\) form a Gelfand pair. The group algebra \(L^1(H^n/U(n))\) forms a commutative Banach algebra under convolution. In Chapter 3 the author studies the Gelfand transform on this algebra. The Gelfand spectrum is identified with the set of all bounded \(U(n)\)-spherical functions which are given by Bessel and Laguerre functions. The author also considers the Banach algebra generated by surface measures and obtains optimal estimates for its character. Further, the author studies Wiener-Tauberian theorems and spherical means on the Heisenberg group. Using the summability result proved in Chapter 2, he studies the injectivity of the spherical mean value operators. He also proves a maximal theorem for the spherical means on the Heisenberg group.

In Chapter 4 the author considers the reduced Heisenberg group and in that context improves some theorems treated in previous chapters, such as the restriction theorem for spectral projections, the Wiener-Tauberian theorem for \(L^p\) functions and the maximal theorem for spherical means. Some problems behave better when studied on the reduced Heisenberg group. The author also proves results concerning mean periodic functions on the reduced Heisenberg group.

In Chapter 1, the Heisenberg group is introduced. Basic results about the representation theory on the Heisenberg group are studied. Then the group Fourier transform is defined, and the Plancherel theorem and the inversion formula are proved. Further, properties of the group Fourier transform, the Hermite, special Hermite and Laguerre functions are introduced. The author proves analogues of Paley-Wiener theorems and Hardy’s theorem for the group Fourier transform on the Heisenberg group.

In Chapter 2 the spectral theory of the sub-Laplacian is studied as developed by R. Strichartz in [J. Funct. Anal. 87, 51-148 (1989; Zbl 0694.43008); 96, 350-406 (1991; Zbl 0734.43004)]. The eigenfunctions of the sub-Laplacian are given in terms of the special Hermite functions. The author obtains an Abel summability result for the expansions in terms of the eigenfunctions of the sub-Laplacian. For the spectral projection operators associated to the expansions, he establishes a Paley-Wiener theorem and some restriction theorem. Using the restriction theorem, he studies the Bochner-Riesz means for the sub-Laplacian. The author also develops a Littlewood-Paley-Stein theory for the sub-Laplacian and proves a multiplier theorem for the sub-Laplacian.

The Heisenberg group \(H^n\) and the unitary group \(U(n)\) form a Gelfand pair. The group algebra \(L^1(H^n/U(n))\) forms a commutative Banach algebra under convolution. In Chapter 3 the author studies the Gelfand transform on this algebra. The Gelfand spectrum is identified with the set of all bounded \(U(n)\)-spherical functions which are given by Bessel and Laguerre functions. The author also considers the Banach algebra generated by surface measures and obtains optimal estimates for its character. Further, the author studies Wiener-Tauberian theorems and spherical means on the Heisenberg group. Using the summability result proved in Chapter 2, he studies the injectivity of the spherical mean value operators. He also proves a maximal theorem for the spherical means on the Heisenberg group.

In Chapter 4 the author considers the reduced Heisenberg group and in that context improves some theorems treated in previous chapters, such as the restriction theorem for spectral projections, the Wiener-Tauberian theorem for \(L^p\) functions and the maximal theorem for spherical means. Some problems behave better when studied on the reduced Heisenberg group. The author also proves results concerning mean periodic functions on the reduced Heisenberg group.

Reviewer: K.Saka (Akita)

### MSC:

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

43A30 | Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc. |

43A85 | Harmonic analysis on homogeneous spaces |

43A90 | Harmonic analysis and spherical functions |

22E27 | Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.) |

22D15 | Group algebras of locally compact groups |

22D20 | Representations of group algebras |

42Bxx | Harmonic analysis in several variables |