The analysis of linear partial differential operators. III: Pseudo-differential operators. Reprint of the 1994 ed.

*(English)*Zbl 1115.35005
Classics in Mathematics. Berlin: Springer (ISBN 978-3-540-49937-4/pbk). xii, 525 p. (2007).

The third and fourth volumes of the author’s impressive monograph “The Analysis of Partial Differential Operators” are a detailed and unified approach of pseudo-differential and Fourier integral operators. The present book is a paperback edition of the third volume of this monograph.

The theory of singular integral operators, as they appeared in the study of elliptic operators, boundary value and Cauchy problems gave rise to the theory of pseudo-differential and Fourier integral operators. The present book gives the author’s powerful techniques and results in a presentation that incorporates these connections in an elegant, homogeneous and most natural manner.

For a detailed description of the contents of the book we refer the reader to the review of the first edition [The analysis of linear partial differential operators. III: Pseudo-differential operators. (Grundlehren der Mathematischen Wissenschaften 274, Springer-Verlag, Berlin) (1985; Zbl 0601.35001)].

Although the first edition appeared some twenty years ago, the book remains one of the most inspiring text in mathematics, and it would be an impossible task to list the developments that rely on techniques and results contained in it. We would only mention here that the Weyl calculus developed in Chapter XVIII lies at the very foundation of the proof of the sufficiency of the \(\Psi\)-condition for the local solvability of principal type pseudo-differential operators [see N. Denker, Ann. Math. (2) 163, No. 2, 405–444 (2006; Zbl 1104.35080)].

See the joint “Looking back”-review by Niels Jacob in Zbl 0712.35001.

The theory of singular integral operators, as they appeared in the study of elliptic operators, boundary value and Cauchy problems gave rise to the theory of pseudo-differential and Fourier integral operators. The present book gives the author’s powerful techniques and results in a presentation that incorporates these connections in an elegant, homogeneous and most natural manner.

For a detailed description of the contents of the book we refer the reader to the review of the first edition [The analysis of linear partial differential operators. III: Pseudo-differential operators. (Grundlehren der Mathematischen Wissenschaften 274, Springer-Verlag, Berlin) (1985; Zbl 0601.35001)].

Although the first edition appeared some twenty years ago, the book remains one of the most inspiring text in mathematics, and it would be an impossible task to list the developments that rely on techniques and results contained in it. We would only mention here that the Weyl calculus developed in Chapter XVIII lies at the very foundation of the proof of the sufficiency of the \(\Psi\)-condition for the local solvability of principal type pseudo-differential operators [see N. Denker, Ann. Math. (2) 163, No. 2, 405–444 (2006; Zbl 1104.35080)].

See the joint “Looking back”-review by Niels Jacob in Zbl 0712.35001.

Reviewer: Viorel Iftimie (Bucureşti)

##### MSC:

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

35Sxx | Pseudodifferential operators and other generalizations of partial differential operators |

58J40 | Pseudodifferential and Fourier integral operators on manifolds |

53D05 | Symplectic manifolds, general |

47G30 | Pseudodifferential operators |