Suppose we have a pseudo differential operator of principal type with a complex symbol and such that the poisson bracket of the real and imaginary parts on the characteristic set is non-negative.I would like to know if there is an analysis on the normal form of such operators. To motivate this note that for real symbols of principal type essentially there is a symplectic transformation to make the operator look like $\partial_1$ operator micro-locally. On the other hand if symbol is complex and i times the poisson bracket is strictly negative or strictly positive on the characteristic set then again micro locally operator looks like $\partial_1 + i x_1 \partial_2$. What can be done if the poisson bracket is say just non-negative rather than strictly negative? Could anything be done at least in some cases?
Thanks,
