# Solvability of subprincipal type operators

## TL;DR

This paper investigates the conditions under which certain pseudodifferential operators with vanishing principal symbols are not solvable, focusing on subprincipal type operators near involutive manifolds with specific geometric and symbolic properties.

## Contribution

It establishes non-solvability results for subprincipal type pseudodifferential operators with vanishing principal symbols under geometric and symbolic conditions.

## Key findings

- Operators are not solvable when principal symbol vanishes to order k ≥ 2.
- Solvability fails when blowup conditions are met and Nirenberg-Treves condition is not satisfied.
- Non-solvability is linked to the behavior of the refined principal symbol and its derivatives.

## Abstract

In this paper we consider the solvability of pseudodifferential operators in the case when the principal symbol vanishes of order $k \ge 2 $ at a nonradial involutive manifold $\Sigma_2$. We shall assume that the operator is of subprincipal type, which means that the $ k$:th inhomogeneous blowup at $\Sigma_2$ of the refined principal symbol is of principal type with Hamilton vector field parallel to the base $\Sigma_2$, but transversal to the symplectic leaves of $\Sigma_2$ at the characteristics. When $k = \infty $ this blowup reduces to the subprincipal symbol. We also assume that the blowup is essentially constant on the leaves of $\Sigma_2$, and does not satisfying the Nirenberg-Treves condition (${\Psi}$). We also have conditions on the vanishing of the normal gradient and the Hessian of the blowup at the characteristics. Under these conditions, we show that $P$ is not solvable.