In this paper we consider the solvability of pseudodifferential operators when the
principal symbol vanishes of at least second order at a nonradial involutive
manifold .
We shall assume that the subprincipal symbol is of principal type with Hamilton vector field
tangent to
at the characteristics, but transversal to the symplectic leaves of
. We
shall also assume that the subprincipal symbol is essentially constant on the leaves
of
and does not satisfying the Nirenberg–Trèves
condition ()
on .
In the case when the sign change is of infinite order, we also need a condition
on the rate of vanishing of both the Hessian of the principal symbol and
the complex part of the gradient of the subprincipal symbol compared
with the subprincipal symbol. Under these conditions, we prove that
is not
solvable.
Keywords
solvability, pseudodifferential operator, subprincipal
symbol
