On a class of unsolvable operators

The authors consider the following problem: For a given nonsolvable differential operator $P$, characterize the data $f(x)$ for which the equation $Pu=f$ has a solution (locally or microlocally near a fixed point). The authors give a complete answer for the microlocal solvability of linear analytic partial differential operators $P$ of the form $P=A^l+$ lower-order-terms in the setting of Gevrey classes $G^s$, $1<s<l/(l-1)$ if $l>1$ and $1<s\leq\infty$ when $l=1$ $(G^\infty\coloneq C^\infty)$. Here $A$ is an analytic pseudodifferential operator of complex principal type which is microlocally modelled by the Mizohata operator $M$, $M=D_{r_n}+ix^h_n D_{x_1}$, $h$ being an odd positive integer. In particular, it is shown that the operator $P$ is not solvable in the space of distributions and $s$-ultradistributions for every $1<s\leq\infty$. For the case of dimension 2, i.e. $x\in\bold R^2$, the authors propose a necessary and sufficient condition on the $s$-ultradistribution $g$ in order to solve locally the equation $Pu=g$ (roughly speaking, the equation must be microlocally solvable in any direction). If the integer $h$ is even, the constructions in the proofs yield the local solvability of $P$ in $G^s$, $1<s<l/(l-1)$.