Operators of subprincipal type

TL;DR

This paper investigates the solvability of certain pseudodifferential operators with degenerate principal symbols, establishing conditions under which such operators are not solvable, especially when the subprincipal symbol exhibits specific tangential and constancy properties.

Contribution

It introduces new conditions on the subprincipal symbol and principal symbol's Hessian that determine non-solvability of operators with second-order vanishing principal symbols.

Findings

Operators are not solvable under specified conditions.

Conditions involve the behavior of subprincipal and principal symbols.

Results extend understanding of solvability in degenerate cases.

Abstract

In this paper we consider the solvability of pseudodifferential operators when the principal symbol vanishes of at least second order at a non-radial involutive manifold $\Sigma_2$. We shall assume that the subprincipal symbol is of principal type with Hamilton vector field tangent to $\Sigma_2$ at the characteristics, but transversal to the symplectic leaves of $\Sigma_2$. We shall also assume that the subprincipal symbol is essentially constant on the leaves of $\Sigma_2$ and does not satisfy the Nirenberg-Treves condition (${\Psi}$) on $\Sigma_2$. 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 $P$ is not solvable.