Local Solvability For a Class of Partial Differential Operators With Double Characteristics
Michael Christ and Georgi Karadzhov
A necessary and sufficient condition for local solvability is presented for the linear partial differential operators $-X^2-Y^2+ia(x)[X,Y]$ in $\Bbb R^3=\{(x,y,t)\}$, where $X=\partial_x,\; Y=\partial_y+x^k\partial_t$, and $a\in C^{\infty}(\Bbb R^1)$ is real valued, for each positive integer $k$.