Hormander Class of Pseudo-Differential Operators on Compact Lie Groups and Global Hypoellipticity

Michael Ruzhansky*, Ville Turunen, Jens Wirth

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

39 Citations (Scopus)


In this paper we give several global characterisations of the Hormander class 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.

Original languageEnglish
Pages (from-to)476-499
Number of pages24
JournalJournal of Fourier Analysis and Applications
Issue number3
Publication statusPublished - Jun 2014
MoE publication typeA1 Journal article-refereed


  • Pseudo-differential operators
  • compact Lie groups
  • microlocal analysis
  • elliptic operators
  • global hypoellipticity
  • Leibniz formula


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