Vol. 10, No. 2, 2017

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Operators of subprincipal type

Nils Dencker

Vol. 10 (2017), No. 2, 323–350

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 Σ2. We shall assume that the subprincipal symbol is of principal type with Hamilton vector field tangent to Σ2 at the characteristics, but transversal to the symplectic leaves of Σ2. We shall also assume that the subprincipal symbol is essentially constant on the leaves of Σ2 and does not satisfying the Nirenberg–Trèves condition (Ψ) on  Σ2. 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.

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