Failure of analytic hypoellipticity in a class of differential operators

Costin, Ovidiu; Costin, Rodica D.

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 2 (2003) no. 1, pp. 21-45.

For the hypoelliptic differential operators $P={\partial }_x^2+{\left(x^k{\partial }_y-x^l{\partial }_t\right)}^2$ introduced by T. Hoshiro, generalizing a class of M. Christ, in the cases of $k$ and $l$ left open in the analysis, the operators $P$ also fail to be analytic hypoelliptic (except for $(k,l)=(0,1)$), in accordance with Treves’ conjecture. The proof is constructive, suitable for generalization, and relies on evaluating a family of eigenvalues of a non-self-adjoint operator.

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