No abstract

Pseudo-differential and Fourier series operators on the torus T n = (R/2πZ) n are analyzed by using global representations by Fourier series instead of local representations in coordinate charts. Toroidal symbols are investigated and the correspondence between toroidal and Euclidean symbols of pseudo-differential operators is established. Periodization of operators and hyperbolic partial differential equations is discussed. Fourier series operators, which are analogues of Fourier integral operators on the torus, are introduced, and formulae for their compositions with pseudo-differential operators are derived. It is shown that pseudo-differential and Fourier series operators are bounded on L 2 under certain conditions on their phases and amplitudes.

In this paper we give several global characterisations of the Hörmander class m (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.

Global quantization of pseudo-differential operators on compact Lie groups is introduced relying on the representation theory of the group rather than on expressions in local coordinates. Operators on the 3-dimensional sphere S 3 and on group SU (2) are analysed in detail. A new class of globally defined symbols is introduced giving rise to the usual Hörmander's classes of operators Ψ m (G), Ψ m (S 3 ) and Ψ m (SU(2)). Properties of the new class and symbolic calculus are analysed. 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.

No abstract