Global quantization of pseudo-differential operators on compact Lie groups, SU(2), 3-sphere, and homogeneous spaces

By Michael Ruzhansky, Ville T. Turunen. 2008

114 citations recorded by the source corpus.

Global quantization of pseudo-differential operators on general compact Lie groups G is introduced relying on the representation theory of the group rather than on expressions in local coordinates. A new class of globally defined symbols is introduced and related to the usual Hormander's classes of operators Psi(m)(G). Properties of the new class and symbolic calculus are analyzed. Properties of symbols as well as L-2-boundedness and Sobolev L-2-boundedness of operators in this global quantization are established on general compact Lie groups. Operators on the three-dimensional sphere S-3 and on group SU(2) are analyzed in detail. An application is given to pseudo-differential operators on homogeneous spaces K backslash G. In particular, using the obtained global characterization of pseudo-differential operators on Lie groups, it is shown that every pseudo-differential operator in Psi(m)(K backslash G) can be lifted to a pseudo-differential operator in Psi(m)(G), extending the known results on invariant partial differential operators.

Subfields: Applied Mathematics, Mathematical Physics, Radiology, Nuclear Medicine and Imaging

View Paper: https://www.semanticscholar.org/paper/b346f4f86e87f8051133fe96b30ced00e5bf27f6

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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. Niyaz Tokmagambetov
3. Michael Ruzhansky
4. Veronique Fischer

In Radiology, Nuclear Medicine and Imaging

1. J. Delgado
2. Niyaz Tokmagambetov
3. Michael Ruzhansky

Citations in the corpus

• Nonharmonic Analysis of Boundary Value Problems (2015, methodology): “From this point of view, the analysis of pseudo-differential operators on the torus using the classical exponential bases as in [54], or further extensions using representation coefficients on compact Lie groups as in [53, 55], both fall within the realm of “harmonic” analysis.”
• Schatten classes on compact manifolds: Kernel conditions☆ (2014, background): “3) Then one has [23,24] the global quantization Af(x) = ∑”
• Kernel and symbol criteria for Schatten classes and r-nuclearity on compact manifolds (2014, methodology): “We fix three left-invariant vector fields X, Y , Z on SO(3) associated with the derivatives with respect to the Euler angles, so that we also have [X, Y ] = Z , see [17] or [18] for the detailed expressions.”
• $L^{p}$ -BOUNDS FOR PSEUDO-DIFFERENTIAL OPERATORS ON COMPACT LIE GROUPS (2016, methodology): “Our analysis will be based on the global quantization developed in [RT10] and [RT13] as a noncommutative analogue of the Kohn-Nirenberg quantization of operators on R.”