where II }l (~) is the usual Sobolev norm of derivatives of order < s in L 2. For operators of principal type one has fairly complete results concerning (1.1) so our main interest is to study double characteristics. When Pm is non-negative and /~m-1 is purely imaginary at the characteristics, Radkevi~ [ 11] gave necessary and sufficient conditions for P to have the properties in question. (For a proof see also Melin [13].) Starting from an example of Gru~in [5], Sj6strand [13] obtained such conditions when the characteristics form a symplectic manifold and P= vanishes precisely to the second order there. A part of these results were also found independently by Boutet de Monvel and Trhves [2]. A very interesting point here is that ~ _ t just has to avoid a discrete set of values, essentially the eigenvalues of a harmonic oscillator. When the characteristics form a manifold which is either symplectic or involutive a construction of a parametrix in a class of pseudodifferential operators of type 1/2, 1/2 has been given by Boutet de Monvel [1], and Grigis [4] has extended the construction to characteristic manifolds where the symplectic form has constant rank, The case where the characteristic manifold is of codimension 2 is exceptional since the range of the Hessian of P,, at the characteristics may then be the whole complex plane. We shall discuss this situation briefly in Section 6 just to show that it is not possible then to relax the hypotheses made by Sj/istrand [13, Theorem 1.2] that the manifold is symplectic and that the index of the Hessian is 2 . In all the other results referred to above there is a proper closed convex cone (angle) FCtl; such that