Abstract
We study the global hypoellipticity problem for certain linear operators in Komatsu classes of Roumieu and Beurling type on compact manifolds. We present an approach by combining a characterization of these spaces via eigenfuction expansions, generated by an elliptic operator, and the analysis of matrix-symbols obtained by these expansions.
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References
Albanese, A.A., Jornet, D.: Global regularity in ultradifferentiable classes. Ann. Mat. Pura Appl. 193(2), 369–387 (2014)
Araújo, G.: Global regularity and solvability of left-invariant differential systems on compact Lie groups. Ann. Global Anal. Geom. 56(4), 631–665 (2019)
Dasgupta, A., Ruzhansky, M.: Eigenfunction expansions of ultradifferentiable functions and ultradistributions. Trans. Amer. Math. Soc. 368(12), 8481–8498 (2016)
de Lessa Victor, B.: Fourier analysis for Denjoy–Carleman classes on the torus. Ann. Fenn. Math. 46(2), 869–895 (2021)
Ferra, I.A., Petronilho, G., de Lessa Victor, B.: Global \(\cal{M}\)-hypoellipticity, global \(\cal{M}\)-solvability and perturbations by lower order ultradifferential pseudodifferential operators. J. Fourier Anal. Appl. 26(6), Paper No. 85, 40 pp. (2020)
Greenfield, S.J., Wallach, N.R.: Global hypoellipticity and Liouville numbers. Proc. Amer. Math. Soc. 31, 112–114 (1972)
Greenfield, S.J., Wallach, N.R.: Remarks on global hypoellipticity. Trans. Amer. Math. Soc. 183, 153–164 (1973)
Kirilov, A., de Moraes, W.A.A.: Global hypoellipticity for strongly invariant operators. J. Math. Anal. Appl. 486(1), 123878, 14 pp. (2020)
Kirilov, A., de Moraes, W.A.A., Ruzhansky, M.: Global properties of vector fields on compact Lie groups in Komatsu classes. Z. Anal. Anwend. 40(4), 425–451 (2021)
Yoshino, M., Gramchev, T., Popivanov, P.: Global properties in spaces of generalized functions on the torus for second order differential operators with variable coefficients. Rend. Semin. Mat. Univ. Politec. Torino 51(2), 144–174 (1993)
Acknowledgements
The authors wish to thank Alexandre Kirilov and Wagner A.A. de Moraes for useful discussions and suggestions.
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The second author was financed by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.
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de Ávila Silva, F., Machado, E.C. Global ultradifferentiable hypoellipticity on compact manifolds. Arch. Math. 118, 615–624 (2022). https://doi.org/10.1007/s00013-022-01719-z
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DOI: https://doi.org/10.1007/s00013-022-01719-z