A Dependence Domain for a Class of Micro-Differential Equations with Involutive Double Characteristics

Abstract

Let M be a real analytic manifold with a complex neighborhood X. Let P be a microdifferential operator defined in a neighborhood U in T ^* X ofq˙ ∈ T ^* _M X\M. We assume that the characteristic variety of P satisfies

$$Char(P) \subset \{ q \in U;{p_1}(q) \cdot {P_2}q = 0\} $$

with homogeneous holomorphic functions p ₁ andp ₂ on U. We assume that

$${p_1}and{p_2}arerealveluedonT_M^*X,$$

(1)

$$d{p_1} \wedge d{p_2} \wedge {\omega _X}(q) \ne 0if{p_1}(q) = {p_2}(q) = 0,$$

(2)

$$\{ {p_1},{p_2}\} (q) = 0if{p_1}(q) = {p_2}(q) = 0.$$

(3)