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Global analytic, Gevrey and C ∞ hypoellipticity on the 3‐torus
Author(s) -
Himonas A. Alexandrou,
Petronilho Gerson,
Carvalho dos Santos L. A.
Publication year - 2012
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201010033
Subject(s) - mathematics , hypoelliptic operator , torus , laplace operator , pure mathematics , partial differential equation , type (biology) , manifold (fluid mechanics) , operator (biology) , mathematical analysis , linear differential equation , geometry , mechanical engineering , ecology , biochemistry , chemistry , repressor , gene , transcription factor , engineering , biology
It is proved that a class of sub‐Laplacians on the 3‐dimensional torus is globally analytic, Gevrey and C ∞ hypoelliptic if and only if either a Diophantine condition holds or there is a point of finite type for the vector fields defining the operator under consideration. This work is motivated by the desire of attaining a better understanding of necessary and sufficient conditions for the global Gevrey and C ∞ hypoellipticity of a sub‐Laplacian on a compact analytic manifold. This together with the more delicate local hypoellipticity are well‐known open problems in the theory of linear partial differential equations.