Failure of analytic hypoellipticity in a class of PDOs

Details Failure of analytic hypoellipticity in a class of PDOs Failure of analytic hypoellipticity in a class of PDOs

2002-12-12

arxiv.org/abs/math/0212167v1

Mathematics

Classical Analysis and ODEs

Scientific paper

For the hypoelliptic differential operators $P={\partial^2_ x}+(x^k\partial_ y -x^l{\partial_t})^2$ 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 {\em{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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