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Global regularity and solvability of left-invariant differential systems on compact Lie groups. (English) Zbl 1437.35685
The problem of the global hypoellipticity in a compact manifold is extremely difficult due to the interplay of symbols of operators and geometrical aspects. The scholars are then mainly addressed to manifolds with special structure. In particular S. J. Greenfield [Proc. Am. Math. Soc. 31, 115–118 (1972; Zbl 0229.35024)] observed that for operators with constant coefficients on a torus global hypoellipticity implies global analytic-hypoellipticity. In the present paper the author extends this result to systems of left-invariant differential operators on compact Lie groups, considering also the case of the Gevrey classes. In short, the main theorem is the following: if the complex at a given degree has closed range when acting between smooth spaces, then it also has closed range when acting between Gevrey spaces. Moreover the dimensions of the cohomological spaces are the same in the smooth and Gevrey setting.

35R03 PDEs on Heisenberg groups, Lie groups, Carnot groups, etc.
35S05 Pseudodifferential operators as generalizations of partial differential operators
35A01 Existence problems for PDEs: global existence, local existence, non-existence
Full Text: DOI
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