×

zbMATH — the first resource for mathematics

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.

MSC:
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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Araújo, G., Regularity and solvability of linear differential operators in Gevrey spaces, Math. Nachr., 291, 729-758 (2018) · Zbl 1392.35087
[2] Bergamasco, AP; Petronilho, G., Closedness of the range for vector fields on the torus, J. Differ. Equ., 154, 132-139 (1999) · Zbl 0926.35030
[3] Bolley, P., Camus, J., Mattera, C.: Analyticité microlocale et itérés d’opérateurs. In: Séminaire Goulaouic-Schwartz (1978/1979), pages Exp. No. 13, 9. École Polytech., Palaiseau (1979) · Zbl 0406.35017
[4] Caetano, PAS; Cordaro, PD, Gevrey solvability and Gevrey regularity in differential complexes associated to locally integrable structures, Trans. Amer. Math. Soc., 363, 185-201 (2011) · Zbl 1217.35042
[5] Chavel, I.: Eigenvalues in Riemannian Geometry, Pure and Applied Mathematics (Including a Chapter by Burton Randol, With an Appendix by Jozef Dodziuk), vol. 115. Academic Press Inc., Orlando (1984) · Zbl 0551.53001
[6] Chevalley, C.; Eilenberg, S., Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc., 63, 85-124 (1948) · Zbl 0031.24803
[7] Dasgupta, A.; Ruzhansky, M., Gevrey functions and ultradistributions on compact Lie groups and homogeneous spaces, Bull. Sci. Math., 138, 756-782 (2014) · Zbl 1327.46041
[8] Dasgupta, A.; Ruzhansky, M., Eigenfunction expansions of ultradifferentiable functions and ultradistributions, Trans. Amer. Math. Soc., 368, 8481-8498 (2016) · Zbl 1366.46024
[9] Fujita, K.; Morimoto, M., Gevrey classes on compact real analytic Riemannian manifolds, Tokyo J. Math., 18, 341-355 (1995) · Zbl 0928.58032
[10] Greenfield, SJ, Hypoelliptic vector fields and continued fractions, Proc. Amer. Math. Soc., 31, 115-118 (1972) · Zbl 0229.35024
[11] Greenfield, SJ; Wallach, NR, Global hypoellipticity and Liouville numbers, Proc. Amer. Math. Soc., 31, 112-114 (1972) · Zbl 0229.35023
[12] Greenfield, SJ; Wallach, NR, Remarks on global hypoellipticity, Trans. Amer. Math. Soc., 183, 153-164 (1973) · Zbl 0274.35018
[13] Gunning, R.C., Rossi, H.: Analytic Functions of Several Complex Variables. Prentice-Hall Inc, Englewood Cliffs (1965) · Zbl 0141.08601
[14] Hilgert, J., Neeb, K.-H.: Structure and Geometry of Lie Groups. Springer Monographs in Mathematics. Springer, New York (2012) · Zbl 1229.22008
[15] Hochschild, G.; Serre, J-P, Cohomology of Lie algebras, Ann. Math., 2, 591-603 (1953) · Zbl 0053.01402
[16] Jahnke, M.R.: Top-Degree Solvability for Hypocomplex Structures and the Cohomology of Left-invariant Involutive Structures on Compact Lie Groups. University of São Paulo, PhD thesis (2018)
[17] Knapp, A.W.: Lie Groups Beyond an Introduction, Progress in Mathematics, vol. 140. Birkhäuser Boston Inc, Boston (1996) · Zbl 0862.22006
[18] Komatsu, H., Projective and injective limits of weakly compact sequences of locally convex spaces, J. Math. Soc. Japan, 19, 366-383 (1967) · Zbl 0168.10603
[19] Kotake, T.; Narasimhan, MS, Regularity theorems for fractional powers of a linear elliptic operator, Bull. Soc. Math. France, 90, 449-471 (1962) · Zbl 0104.32503
[20] Köthe, G.: Topological Vector Spaces. II. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 237. Springer, New York (1979)
[21] Lions, J.-L., Magenes, E.: Non-homogeneous Boundary Value Problems and Applications (Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 183), vol. III. Springer, New York (1973) · Zbl 0251.35001
[22] Malaspina, F.; Nicola, F., Gevrey local solvability in locally integrable structures, Ann. Mat. Pura Appl., 193, 1491-1502 (2014) · Zbl 1319.35321
[23] Ragognette, LF, Ultradifferential operators in the study of Gevrey solvability and regularity, Math. Nachr., 292, 409-427 (2019) · Zbl 1416.35169
[24] Schapira, P., Solutions hyperfonctions des équations aux dérivées partielles du premier ordre, Bull. Soc. Math. France, 97, 243-255 (1969) · Zbl 0184.31701
[25] Seeley, RT, Eigenfunction expansions of analytic functions, Proc. Amer. Math. Soc., 21, 734-738 (1969) · Zbl 0183.10102
[26] Suzuki, H., On the semi-local existence of real analytic solutions of a partial differential equation, Sci. Rep. Tokyo Kyoiku Daigaku Sect. A, 11, 245-252 (1972) · Zbl 0255.35004
[27] Treves, F.: Hypo-Analytic Structures, Princeton Mathematical Series (Local Theory), vol. 40. Princeton University Press, Princeton (1992) · Zbl 0787.35003
[28] Wallach, N.R.: Harmonic Analysis on Homogeneous Spaces. Pure and Applied Mathematics, No. 19. Marcel Dekker Inc, New York (1973) · Zbl 0265.22022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.