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 Hörmander’s classes of operators Ψ^m(G). Properties of the new class and symbolic calculus are analyzed. Properties of symbols as well as L²-boundedness and Sobolev L²-boundedness of operators in this global quantization are established on general compact Lie groups. Operators on the three-dimensional sphere and on group SU(2) are analyzed in detail. An application is given to pseudo-differential operators on homogeneous spaces K\G. In particular, using the obtained global characterization of pseudo-differential operators on Lie groups, it is shown that every pseudo-differential operator in Ψ^m(K\G) can be lifted to a pseudo-differential operator in Ψ^m(G), extending the known results on invariant partial differential operators.

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In this paper we give criteria on integral kernels ensuring that integral operators on compact manifolds belong to Schatten classes. A specific test for nuclearity is established as well as the corresponding trace formulae. In the special case of compact Lie groups, kernel criteria in terms of (locally and globally) hypoelliptic operators are also given

This article is distributed under the terms of the Creative Commons Attribution\ud License which permits any use, distribution, and reproduction in any medium, provided\ud the original author(s) and the source are credited.In this paper we study weakly hyperbolic second order equations with time dependent irregular coefficients. This means assuming that the coefficients are less regular than Hölder. The characteristic roots are also allowed to have multiplicities. For such equations, we describe the notion of a ‘very weak solution’ adapted to the type of solutions that exist for regular coefficients. The construction is based on considering Friedrichs-type mollifiers of coefficients and corresponding classical solutions, and their quantitative behaviour in the regularising parameter. We show that even for distributional coefficients, the Cauchy problem does have a very weak solution, and that this notion leads to classical or to ultradistributional solutions under conditions when such solutions also exist. In concrete applications, the dependence on the regularising parameter can be traced explicitly