Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Global hypoellipticity and Liouville numbers

Authors: Stephen J. Greenfield and Nolan R. Wallach
Journal: Proc. Amer. Math. Soc. 31 (1972), 112-114
MSC: Primary 34H05; Secondary 47F05
MathSciNet review: 0296508
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [B] Robert S. Strichartz, Invariant pseudo-differential operators on a Lie group, Ann. Scuola Norm. Sup. Pisa (3) 26 (1972), 587–611. MR 0420739
  • [F] 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 0271988
  • [H] L. Hörmander, Linear partial differential operators, Die Grundlehren der math. Wissenschaften, Band 116, Academic Press, New York; Springer-Verlag, Berlin, 1963. MR 28 #4221.
  • [HW] G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 4th ed., Oxford Univ. Press, London, 1960. MR 16, 673.
  • [S] L. Schwartz, Théorie des distributions. Tomes I, II, Actualités Sci. Indust., nos. 1091, 1122, Hermann, Paris, 1950, 1951. MR 12, 31; 833.
  • [W] Nolan R. Wallach, Harmonic analysis on homogeneous spaces, Marcel Dekker, Inc., New York, 1973. Pure and Applied Mathematics, No. 19. MR 0498996
  • [Z] C. S. Herz, Functions which are divergences, Amer. J. Math. 92 (1970), 641–656. MR 0290409,

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 34H05, 47F05

Retrieve articles in all journals with MSC: 34H05, 47F05

Additional Information

Keywords: Fourier coefficients, Liouville numbers, global hypoellipticity
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society