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

By Michael Ruzhansky, Ville T. Turunen, Jens Wirth. Journal of Fourier Analysis and Applications, 2010

98 citations recorded by the source corpus.

In this paper we give several global characterisations of the Hörmander class $$\Psi ^m(G)$$Ψm(G) 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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c122f8d2253c1181ae36ec728457c7aa6b8c0fe7

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. J. Delgado
2. Niyaz Tokmagambetov
3. Michael Ruzhansky
4. Veronique Fischer
5. Ville T. Turunen

In Mathematical Physics

1. J. Delgado
2. Michael Ruzhansky
3. Niyaz Tokmagambetov
4. Veronique Fischer
5. Durvudkhan Suragan

Citations in the corpus

• Schatten classes and traces on compact groups (2013, methodology): “1] or [17], and for the complete characterisation of Hörmander classes Ψm(G) in terms of matrix-valued symbols see also [18].”
• Quantization on Nilpotent Lie Groups (2016, methodology): “One can also provide a criterion for the hypoellipticity in terms of matrixvalued symbols ([RTW14]), in analogy to the one on R given by Hörmander ([Hör67b]).”
• Nonharmonic Analysis of Boundary Value Problems (2015, background): “1 and arguing similarly to [56], we can convince ourselves that for ρ > δ the definition of the symbol class does not depend on the choice of an L-strongly admissible collection.”
• $L^{p}$ -BOUNDS FOR PSEUDO-DIFFERENTIAL OPERATORS ON COMPACT LIE GROUPS (2016, methodology): “In [RTW14] for (ρ, δ) = (1, 0), and then in [Fis15, Corollary 8.”
• Schatten classes on compact manifolds: Kernel conditions☆ (2014, unknown): “For γ = 1 it was shown in [25] that H1 + cI is globally hypoelliptic if and only if 0 / ∈ { c + ( + 1) −m(m + 1) : ∈ N, m ∈ Z, |m| } .”
• Kernel and symbol criteria for Schatten classes and r-nuclearity on compact manifolds (2014, unknown): “For γ = 1, it was shown in [19] that H1 + cI is globally hypoelliptic if and only if 0 / ∈ {c + ( + 1) − m(m + 1) : ∈ N, m ∈ Z, |m| ≤ }.”