By Woodruff S.L.

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Let E C00(11) be real-valued and have a nowhere­ vanishing gradient. To every pair of nonnegative integers J, M there is another pair of such integers, J', M', such that the following is true : (3. _ p ( a ) (x, p a <1> )9l,, ( ; p, Dx ) u } lal s J' a ! I - I PROOF. 20) when M = 0. For M > 0 the argument is almost exactly the same, with a few technicalities added, which we shall leave to the reader. R n, vanishing for Ix I > 1 and equal to one for Ix I < t. Pu) - f(x, p) . W e shall begin b y estimating lf1 (x, p )I.

N. a y ; 1 � y, 1 Since the degree of a is m and that of

k where Fk, a are operators of the kind (2. 1 ) with phase

4. 3 1 ) (d�; 11 dx ; - d71; 11 dy ; ) 5. Application to Microlocal Cauchy Problems Let IDl stand for a Ceo manifold of dimension N = n + 1 . We shall momentarily denote by y the variable in m, by T/ the variable in the cotangent spaces to IDl. Our basic datum in this section is a classical Chapter VI 338 pseudodifferential operator P of order m in IDC , with principal symbol p ( y, 71 ) . We shall make the following fundamental hypotheses: (5 . 2) at no point of Char P = {(y, 71 ) E T*IDC\O ; p (y, 71 ) = O} does the differential of p (y, 71 ) with respect to the 71 variables vanish.

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