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Examples of globally hypoelliptic operator on special dimensional spheres without infinitesimal transitivity. (English) Zbl 1034.35015

Summary: This paper gives examples of globally hypoelliptic operator on \(S^3\), or on \(S^7\), or on \(S^{15}\) which is sum of squares of real vector fields. These operators fail to satisfy the infinitesimal transitivity condition (the Hörmander bracket condition) at every point and therefore they are not hypoelliptic in any subdomain.

MSC:

35H10 Hypoelliptic equations
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References:

[1] Hörmander, L.: Hypoelliptic second order differential equations. Acta Math., 119 , 147-171 (1967). · Zbl 0156.10701
[2] Omori, H., and Kobayashi, T.: Global hypoellipticity of subelliptic operators on closed manifolds. Hokkaido Math. J., 28 , 613-633 (1999). · Zbl 0942.35050
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