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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.

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