Description

Nilpotent Lie groups appear naturally in the analysis of manifolds and provide an abstract setting for many notions of Euclidean analysis. As is generally the case when studying analysis on nilpotent Lie groups, we restrict ourselves to the very large subclass of homogeneous (nilpotent) Lie groups, that is, Lie groups equipped with a family of dilations compatible with the group structure. They are the groups appearing ‘in practice’ in the applications (some of them are described below). From the point of view of general harmonic analysis, working in this setting also leads to the distillation of the results of the Euclidean harmonic analysis depending only on the group and dilation structures.

In order to motivate the work presented in this monograph, we focus our attention in this introduction on three aspects of the analysis on nilpotent Lie groups: the use of nilpotent Lie groups as local models for manifolds, questions regarding hypoellipticity of differential operators, and the development of pseudodifferential operators in this setting. We only outline the historical developments of ideas and results related to these topics, and on a number of occasions we refer to other sources for more complete descriptions. We end this introduction with the main topic of this monograph: the development of a pseudo-differential calculus on homogeneous Lie groups.

Nilpotent Lie groups by themselves and as local models

It has been realised for a long time that the analysis on nilpotent Lie groups can be effectively used to prove subelliptic estimates for operators such as ‘sums of squares’ of vector fields on manifolds. Such ideas started coming to light in the works on the construction of parametrices for the Kohn-Laplacian b (the Laplacian associated to the  angential CR complex on the boundary X of a strictly pseudoconvex domain), which was shown earlier by J. J. Kohn to be hypoelliptic (see e.g. an exposition by Kohn [Koh73] on the analytic and smooth hypoellipticities). Thus, the corresponding parametrices and subsequent subelliptic estimates have been obtained by Folland and Stein in [FS74] by first establishing a version of the results for a family of sub-Laplacians on the Heisenberg group, and then for the Kohn-Laplacian b by replacing X locally by the Heisenberg group. These ideas soon led to powerful generalisations. The general techniques for approximat-ing vector fields on a manifold by left-invariant operators on a nilpotent Lie group have been developed by Rothschild and Stein in [RS76]. Here the dimension of the nilpotent Lie group is normally larger than that of the manifold, and a first step of such a construction is to perform the ‘lifting’ of vector fields to the group. Consequently, this approach allowed one to produce parametrices for the original differential operator on the manifold by using the analysis on homogeneous Lie groups. A more geometric version of these constructions has been carried out by Folland in [Fol77b], see also Goodman [Goo76] for the presentation of nilpotent Lie algebras as tangent spaces (of sub-Riemannian manifolds). The functional analytic background for the analysis in the stratified setting was laid down by Folland in [Fol75]. A general approach to studying geometries appearing from systems of vector fields has been developed by Nigel, Stein and Wainger [NSW85].

Thus, one of the motivations for carrying out the analysis and the calculus of operators on nilpotent Lie groups comes from the study of differential operators on CR (Cauchy Riemann) or contact manifolds, modelling locally the operators there on homogeneous invariant convolution operators on nilpotent groups. In ‘practice’ and from this motivation, only nilpotent Lie groups endowed with some compatible structure of dilations, i.e. homogeneous Lie groups, are considered. This will be also the setting of our present exposition.