Microdifferential Operators

Summary

After recalling the principal notions concerning the Ring¹⁾ D_X of holomorphic differential operators on a complex manifold X, we introduce the Ring ${\hat E_x}$ of formal microdifferential operators on the cotangent bundle T*X, assuming for a while that X is an open set in a complex (finite dimensional) vector space. This Ring is defined as a sheaf of formal series of homogeneous holomorphic functions, the composition law being obtained by extending the Leibniz formula. Then by means of the quasi-norms of L. Boutet de Monvel and P. Kree [1], we construct the sub-Ring ℰ_X of ${\hat E_x}$ of microdifferential operators of M. Sato, M. Kashiwara, T. Kawaï [1], and prove the invertibility of operators whose principal symbols do not vanish.

We construct various Banach algebras of zero order microdifferential operators, and use these algebras to obtain the extension to microdifferential operators of the classical division theorems of Spath and Weierstrass, as stated in Sato-Kashiwara-Kawaï [1], and also an extension of the Cauchy-Kowalewski theorem.

The division theorem asserts that if the principal symbol of an operator P has a zero exactly of order p in some direction $\frac{\partial }{{\partial {\xi _1}}},$, then any operator Q may be divided by P with a remainder which is a polynomial of order at most p – 1 in ${D_{{x_1}}}$, having as coefficients microdifferential operators independent of ${D_{{x_1}}}$.

Roughly speaking the Cauchy-Kowalewski theorem asserts that if (x)=(x₁,…,x_n) are coordinates on X, the Cauchy Problem with values in the sheaf of operators which do not depend on ${D_{{x_1}}},...,{D_{{x_p}}},$, is well posed for operators P which are polynomials in ${D_{{x_1}}},...,{D_{{x_p}}},$, with zero order microdifferential operators as coefficients (and of course, with data on a non characteristic hypersurface). The proof is an application of the abstract Cauchy-Kowalewski theorem in scales of Banach spaces. We recall here this theorem with proof, following F. Trèves [1].

The left D_X-module B_Z|X, associated to a submanifold Z of X, and its microlocalisation C_Z|X, are constructed in § 4. We avoid the cohomological tools, by associating a left Ideal of D_X to a system of functions defining Z, and proving the intrinsic character of the constructions.

Then, following Sato-Kashiwara-Kawaï [1] and Kashiwara [5], we are ready to prove that given a complex contact transformation φ from an open set $U \subset T*X$ to an open set $U' \subset T*X',\phi$ can be locally “quantized”, that is extended to an isomorphism $\hat \phi$ of filtered Rings from ${E_x}{|_u}$ to ${\phi ^{ - 1}}({E_{x'}}{|_{u'}})$. In fact if we set $\Lambda _\phi ^a = \{ (x,x';\xi ,\xi ');(x'; - \xi ') = \phi (x,\xi )\}$, we have to find an Ideal ℐ of –_X×X’, whose symbol Ideal coincides with the defining Ideal of $\Lambda _\phi ^a$. Then we must prove, by successive applications of the division theorem, that given a section P of ℰ_X there exists a unique section Q of ℰ_X’ such that P—Q belongs to ℐ. When φ is the identity, ℐ is naturally associated to a volume element dx on X, and the anti-isomorphism P ↦ Q is nothing but the adjoint with respect to dx. In the general case $\hat \phi$ is the composite of an antiisomorphism associated to $\Lambda _\phi ^a$ and of the adjoint, for a volume element. We discuss some examples of quantized contact transformations, and in particular the action on ℰ_X of a complex change of coordinates. This allows us to define now ℰ_X when X is a complex manifold.

Once we are able to make use of quantized contact transformations, the theory of systems with “simple characteristics” becomes transparent, as shown in Sato-Kashiwara-Kawaï [1] Such a system ℳ is a left ℰ_X-module endowed with a generator u such that, if ℐ denotes the left Ideal of ℰ_X annihilating u, the symbol Ideal ℐ of ℐ coincides with J_V, the defining Ideal of a smooth conic involutive manifold V. When V is regular involutive, V is isomorphic by a contact transformation to the manifold $V' = \{ (x,\xi ) \in T*{\mathbb{C}^n};{\xi _1} = ... = {\xi _p} = 0,{\xi _n} \ne 0\}$, and successive applications of the Cauchy-Kowalewski theorem permit us to prove that ℳ is then locally isomorphic to a module ℰ_X/J, where J is the left Ideal generated by ${D_{{x_1}}},...,{D_{{x_p}}}$. When V is Lagrangean, it is isomorphic to $V' = \{ (x,\xi ) \in T*{\mathbb{C}^n};{x_1} = {\xi _2} = ... = {\xi _n} = 0,{\xi _1} \ne 0\}$, the conormal bundle to the hypersurface {x₁=0{, and there exists a complex number α, unique modulo ℤ, such that ℳ=ℰ_X/ J, the Ideal J being generated by ${x_1}{D_{{x_1}}} - a,{D_{{x_2}}},...,{D_{{x_n}}}$.

The notions of symplectic geometry used in this Chapter are recalled in Appendix A.