Failure of analytic hypoellipticity in a class of differential operators

Costin, Ovidiu; Costin, Rodica D.

Geodesic

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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

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Résumé

For the hypoelliptic differential operators $P = \partial_{x}^{2} + {(x^{k} \partial_{y} - x^{l} \partial_{t})}^{2}$ 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.

MR Zbl

Classification : 35B65

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     title = {Failure of analytic hypoellipticity in a class of differential operators},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {21--45},
     publisher = {Scuola normale superiore},
     volume = {Ser. 5, 2},
     number = {1},
     year = {2003},
     mrnumber = {1990973},
     zbl = {1150.35018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ASNSP_2003_5_2_1_21_0/}
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Costin, Ovidiu; Costin, Rodica D. Failure of analytic hypoellipticity in a class of differential operators. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 2 (2003) no. 1, pp. 21-45. http://geodesic.mathdoc.fr/item/ASNSP_2003_5_2_1_21_0/