Global hypoellipticity and Liouville numbers
HTML articles powered by AMS MathViewer
- by Stephen J. Greenfield and Nolan R. Wallach PDF
- Proc. Amer. Math. Soc. 31 (1972), 112-114 Request permission
Abstract:
We consider global hypoellipticity of constant coefficient differential operators on the $2$-torus, and prove that it is equivalent to an algebraic growth condition on the symbol. This is applied to give necessary and sufficient conditions that a constant coefficient vector field be globally hypoelliptic. Similar results are true on compact homogeneous spaces.References
- Robert S. Strichartz, Invariant pseudo-differential operators on a Lie group, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 26 (1972), 587–611. MR 420739
- André Cerezo and François Rouvière, Solution élémentaire d’un opérateur différentiel linéaire invariant à gauche sur un groupe de Lie réel compact et sur un espace homogène réductif compact, Ann. Sci. École Norm. Sup. (4) 2 (1969), 561–581 (French). MR 271988 L. Hörmander, Linear partial differential operators, Die Grundlehren der math. Wissenschaften, Band 116, Academic Press, New York; Springer-Verlag, Berlin, 1963. MR 28 #4221. G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 4th ed., Oxford Univ. Press, London, 1960. MR 16, 673. L. Schwartz, Théorie des distributions. Tomes I, II, Actualités Sci. Indust., nos. 1091, 1122, Hermann, Paris, 1950, 1951. MR 12, 31; 833.
- Nolan R. Wallach, Harmonic analysis on homogeneous spaces, Pure and Applied Mathematics, No. 19, Marcel Dekker, Inc., New York, 1973. MR 0498996
- C. S. Herz, Functions which are divergences, Amer. J. Math. 92 (1970), 641–656. MR 290409, DOI 10.2307/2373366
Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 31 (1972), 112-114
- MSC: Primary 34H05; Secondary 47F05
- DOI: https://doi.org/10.1090/S0002-9939-1972-0296508-5
- MathSciNet review: 0296508