#### Vol. 10, No. 2, 2017

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Operators of subprincipal type

### Nils Dencker

Vol. 10 (2017), No. 2, 323–350
##### Abstract

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 ${\Sigma }_{2}$. 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 satisfying the Nirenberg–Trèves 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.

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##### Keywords
solvability, pseudodifferential operator, subprincipal symbol
##### Mathematical Subject Classification 2010
Primary: 35S05
Secondary: 35A01, 58J40, 47G30
##### Milestones
Received: 23 November 2015
Revised: 18 September 2016
Accepted: 1 November 2016
Published: 23 February 2017
##### Authors
 Nils Dencker Centre for Mathematical Sciences University of Lund Box 118 SE-221 00 Lund Sweden