By D.C. Spencer

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**Extra info for Partial Differential Equations: Proceedings of Symposia in Pure Mathematics**

**Example text**

TREVES 34 other words, to say that P(x, D) is of principal type is to say that its real characteristics are simple. All elliptic differential operators, all hyperbolic differential opera- tors are of principal type; the parabolic ones are not. All first-order differential operators whose principal part does not vanish identically at any point are also of principal type. 2. The operator P(x, D) is said to be locally solvable at the point x0 of Q if there is an open neighborhood U Q of x0 such that, given any function such that P(x, D)u = fin U.

This is a very particular case of general phenomena. , B 0, and be a point in T*(11) where A = Pm vanishes whereas does not. , that the bicharacteristic strip of A through this point is not reduced to a single point, but also that the projection of this strip into the base space is also a true curve. The latter is called a bicharacteristic curve of A through Xo in U. The following can then be proved: (1) if Xo belongs to the singular support of a solution of Pu = fe so does a whole arc of some bicharacteristic curve of A through Xo.

M. F. Atiyah, Vecior fields on manifolds, Arbeitsgemeinschaft für Forschung des Landes Nord- I. M. SINGER 30 rhein-Westfalen, Heft 200, Westdeutscher Verlag, Cologne, 1970. MR 41 #7707. , Riemann surfaces and spin structures, Ann. Sci. École Norm. Sup. (4) 4(1971), 2. 47—62. Topology of elliptic operators, Global Analysis, Proc. Sympos. , vol. 16, Amer. Math. , 1970, pp. 101—1 19. MR 41 #9291. 4. M. F. Atiyah and R. Bott, A Lefschetz fixed point formula for elliptic complexes. I, Ann. of Math. 3.