Failure of analytic hypoellipticity for some operators of \(X^ 2+Y^ 2\) type
We show that the operator \[ {\partial^2\over \partial x^2}+ \Biggl(x^k {\partial\over \partial y}- x^l {\partial\over \partial t}\Biggr)^2 \] in \(\mathbb{R}^3\) (with nonnegative integers \(k\), \(l\) satisfying \(k< l\)) is not analytically hypoelliptic, if either of the following assumptions is satisfied: (i) \((l+ 1)/(l- k)\) is not a positive integer. (ii) Both \(l-k\) and \((l+ 1)/(l- k)\) are odd integers.
scientific article; zbMATH DE number 871809