2010/04/26 by Michael Ruzhansky, Ville Turunen, Jens Wirth · 1 citation
Mathematics · #Advanced Algebra and Geometry #Class (philosophy) #Differential operator #Holomorphic and Operator Theory #Hypoelliptic operator #Invertible matrix #Lie group #Mathematical Analysis and Transform Methods #Microlocal analysis #Operator (biology) #Operator theory #Order (exchange) #Representation theory #math.AP #math.FA
paper · pdf · doi:10.1007/s00041-014-9322-9
published as J. Fourier Anal. Appl., 20 (2014), 476-499 · 20 pages
arxiv created 2010/04/26 · openalex publication_date 2014/04/07 · arxiv updated 2014/08/27 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
Abstract In this paper we give several global characterisations of the Hörmander class Ψ m(G) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>Ψ</mml:mi> <mml:mi>m</mml:mi> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> of pseudo-differential operators on compact Lie groups in terms of the representation theory of the group. The result is applied to give criteria for the ellipticity and the global hypoellipticity of pseudo-differential operators in terms of their matrix-valued full symbols. Several examples of the first and second order globally hypoelliptic differential operators are given, in particular of operators that are locally not invertible nor hypoelliptic but globally are. Where the global hypoelliptiticy fails, one can construct explicit examples based on the analysis of the global symbols.