Intrinsic pseudodifferential calculi on any compact Lie group

In this paper, we define in an intrinsic way operators on a compact Lie group by means of symbols using the representations of the group. The main purpose is to show that these operators form a symbolic pseudo-differential calculus which coincides or generalises the (local) H\"ormander pseudo-differential calculus on the group viewed as a compact manifold.Comment: 48 pages, with table of content

Lower bounds for operators on graded Lie groups

In this note we present a symbolic pseudo-differential calculus on graded nilpotent Lie groups and, as an application, a version of the sharp Garding inequality. As a corollary, we obtain lower bounds for positive Rockland operators with variable coefficients as well as their Schwartz-hypoellipticity

A pseudo-differential calculus on the Heisenberg group

In this note we present a symbolic pseudo-differential calculus on the Heisenberg group. We particularise to this group our general construction [4,3,2] of pseudo-differential calculi on graded groups. The relation between the Weyl quantization and the representations of the Heisenberg group enables us to consider here scalar-valued symbols. We find that the conditions defining the symbol classes are similar but different to the ones in [1]. Applications are given to Schwartz hypoellipticity and to subelliptic estimates on the Heisenberg group.Comment: 9 page

Semiclassical analysis on compact nilmanifolds

In this paper, we develop a semi-classical calculus on compact nil-manifolds. As applications, we obtain Weyl laws for positive Rockland operators on any graded compact nil-manifolds and results on quantum ergodicity in position for sub-Laplacians on any stratified nil-manifolds.Comment: 38 page

Towards semi-classical analysis for sub-elliptic operators

We discuss the recent developments of semi-classical and micro-local analysis in the context of nilpotent Lie groups and for sub-elliptic operators. In particular, we give an overview of pseudo-differential calculi recently defined on nilpotent Lie groups as well as of the notion of quantum limits in the Euclidean and nilpotent cases.Comment: This paper summarises the main ideas discussed by the author to the Bruno Pini Mathematical Analysis Seminar of the University of Bologna in May 202