479 research outputs found
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
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