Phase-space analysis and pseudodifferential calculus on the Heisenberg group.

*(English)*Zbl 1246.35003
Astérisque 342. Paris: Société Mathématique de France (SMF) (ISBN 978-2-85629-334-8/pbk). vi, 128 p. (2012).

Let \({\mathcal F}\) be the Fourier transform of a space of functions defined on the additive group of reals \(G={\mathbb R}\) to functions on \(\hat{G}={\mathbb R}\) – the dual group of \(G\). The important classes of operators are

\(\bullet\) operators of multiplication \(f \mapsto bf\) by a function \(b\) on \(G\);

\(\bullet\) Fourier multipliers \(f \mapsto {\mathcal F}^{-1}a{\mathcal F}f\) for a function \(a\) on \(\hat{G}\).

An algebra that naturally encompasses these two types of operators is the algebra of pseudo-differential operators. An efficient calculus, which captures both the algebraic and metric aspects, can be developed in term of symbols – functions on \(G\times \hat{G}\). This construction can be efficiently carried out for other Abelian groups if the appropriate Fourier transform is defined within the Pontryagin duality between \(G\) and \(\hat{G}\), see [M. Ruzhansky and V. Turunen, Pseudo-differential operators and symmetries. Background analysis and advanced topics. Pseudo-Differential Operators. Theory and Applications 2. Basel: Birkhäuser (2010; Zbl 1193.35261)].

The paper under review follows the above scheme for the case of the non-commutative Heisenberg group. The obvious complication is that the dual of the Heisenberg group does not possess a group structure itself. The authors are able to construct an efficient symbolic calculus for a large algebra of operators. The technique is based on microlocal analysis on the Heisenberg group. Among the obtained results are the proof of some fundamental properties of pseudo-differential operators, their comparison with Littlewood-Paley operators and the continuity on Sobolev spaces.

\(\bullet\) operators of multiplication \(f \mapsto bf\) by a function \(b\) on \(G\);

\(\bullet\) Fourier multipliers \(f \mapsto {\mathcal F}^{-1}a{\mathcal F}f\) for a function \(a\) on \(\hat{G}\).

An algebra that naturally encompasses these two types of operators is the algebra of pseudo-differential operators. An efficient calculus, which captures both the algebraic and metric aspects, can be developed in term of symbols – functions on \(G\times \hat{G}\). This construction can be efficiently carried out for other Abelian groups if the appropriate Fourier transform is defined within the Pontryagin duality between \(G\) and \(\hat{G}\), see [M. Ruzhansky and V. Turunen, Pseudo-differential operators and symmetries. Background analysis and advanced topics. Pseudo-Differential Operators. Theory and Applications 2. Basel: Birkhäuser (2010; Zbl 1193.35261)].

The paper under review follows the above scheme for the case of the non-commutative Heisenberg group. The obvious complication is that the dual of the Heisenberg group does not possess a group structure itself. The authors are able to construct an efficient symbolic calculus for a large algebra of operators. The technique is based on microlocal analysis on the Heisenberg group. Among the obtained results are the proof of some fundamental properties of pseudo-differential operators, their comparison with Littlewood-Paley operators and the continuity on Sobolev spaces.

Reviewer: Vladimir V. Kisil (Leeds)

##### MSC:

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

35R03 | PDEs on Heisenberg groups, Lie groups, Carnot groups, etc. |

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

43A80 | Analysis on other specific Lie groups |