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Global hypoellipticity of a Mathieu operator

Author: Masafumi Yoshino
Journal: Proc. Amer. Math. Soc. 111 (1991), 717-720
MSC: Primary 35H05; Secondary 58G99
MathSciNet review: 1042277
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Abstract: We give a necessary and sufficient condition for the global hypoellipticity of a Mathieu operator on the torus $ {\mathbb{T}^d}$ in terms of continued fractions. It is not hypoelliptic, nor does it satisfy a controllability condition, a Hörmander condition, or a Siegel condition. But it is still globally hypoelliptic (cf. [1, 3]).

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  • [1] S. Greenfield and N. Wallach, Global hypoellipticity and Liouville numbers, Proc. Amer. Math. Soc. 31 (1972), 112-114. MR 0296508 (45:5568)
  • [2] W. B. Jones and W. J. Thron, Continued fractions, Analytic Theory and Applications, Addison-Wesley, Reading, MA, 1980. MR 595864 (82c:30001)
  • [3] K. Taira, Le principe du maximum et l'hypoellipticité globale, Seminaire Bony-Sjöstrand-Meyer, no. 1, 1984/85.
  • [4] M. Yoshino, A class of globally hypoelliptic operators on the torus, Math. Z. 201 (1989), 1-11. MR 990183 (90m:35046)

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Keywords: Hypoellipticity, global hypoellipticity, continued fractions
Article copyright: © Copyright 1991 American Mathematical Society

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