By Sigeru Mizohata

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Additional info for Hyperbolic Equations and Related Topics. Proceedings of the Taniguchi International Symposium, Katata and Kyoto, 1984

Sample text

Let lit) be a C°° function, nonnegative, equal to 1 for t^2k and to 0 for /<&/2, and put pru=nu=^x(\xm)mdmn)\ It is not difficult to prove that 77—77 maps Hs0's' into SP(s, oo) and that Pf-Pf maps SP(s, s') into 77*'°°. § 3. 1. 2-microlocal symbols. l) m '""'- In this section, we shall associate to such symbols operators defined on spaces Hfr8'. 9. We shall always assume implicitly that symbols and operators are 0 outside the unit ball, and shall not write the associated cut off functions. 2. Definition (Special symbols).

Then the Second Microlocalization and Semi-Linear Equations 21 Put Tpr=fa{2px)(px{2-pD). It is clear that one has TpT*=0 for \q—p| larger than some N0. We are going to prove now that || T ^ T J K C2~]p~q], and it is sufficient to prove this in the following two particular cases. 1st case: Assume the support of ty is contained in a small neigh• \ bourhood of 0. Then the support of Tpu is contained in a ring slightly larger than Cp. We have then (Tpu\ Tqv) = 0 for \p—q\ larger than some N0 and hence T*Tq=0 for \p-q\>N0.

In addition we Complex Vector Fields 9 assume there exist ft € C and a sector Sf in the complex plane such that Pm(c) + Re [icm\y<0 and angle ^>7r/m. 5 ([3]). 10) and £° be a characteristic covector at the origin satisfying the curve sector property then any CR distribution defined on M is hypo-analytic at (0, f°). 5 we can show that if A is a CR distribution defined on M, then its hypo-analytic wave front set is contained in a strictly convex closed cone. This fact will then imply the extendability result of Th.

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