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Weyl calculus on graded groups
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DOI：10.4171/dm/1074.png)
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Abstract
En 中文
The aim of this paper is to establish a pseudo-differential Weyl calculus on graded nilpotent Lie groups G which extends the celebrated Weyl calculus on R-n. To reach this goal, we develop a symbolic calculus for a very general class of quantization schemes, following the work by Mantoiu and Ruzhansky (2017), using the H & ouml;rmander symbol classes S-p,delta(m) (G) introduced in the book by Fischer and Ruzhansky (2016). We particularly focus on the so-called symmetric calculi, for which quantizing and taking the adjoint commute, among them the Euclidean Weyl calculus, but we also recover the (non-symmetric) Kohn-Nirenberg calculus, on R-n and on general graded groups (Fischer and Ruzhansky (2016)). Several interesting applications follow directly from our calculus: expected mapping properties on Sobolev spaces, the existence of one-sided parametrices and the G & aring;rding inequality for elliptic operators, and a generalization of the Poisson bracket for symmetric quantizations on stratified groups. In the particular case of the Heisenberg group H-n, we are able to answer the fundamental question of this paper: which, among all the admissible quantizations, is the natural Weyl quantization on H-n? Among other things, we discuss and investigate an analog of the symplectic invariance property of the Weyl quantization in the setting of graded groups, as well as a notion of noncommutative Poisson bracket for symbols in the setting of stratified groups.
Keywords:
PSEUDODIFFERENTIAL-OPERATORS
SEMICLASSICAL ANALYSIS
HEISENBERG GROUP
SPACES
ALGEBRA
Journal
D
IF:
0.6
Papers:
44
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0