Failure of analytic hypoellipticity in a class of PDOs

Abstract

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 {\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.