An application of symbol calculus

Abstract

Let X = ℂ _t,x ¹⁺ⁿ, t ∈ ℂ, x = (x ₁,…,x _n ) ∈ ℂⁿ, and let (t, x; τ,ξ)) be the associated symplectic coordinates in T*X. In Kashiwara and Oshima’s study of regular systems (cf [5]), the following definition occurs (with a slightly different vocabulary): a matrix of microdifferential operators A(x,D _x ) is essentially of order ≥ 0 if there exists ν > 0 such that the coefficients of any power of A are microdifferential operators of order at most v. It is shown in [5] that any regular system of microdifferential equations with regular singularities along V = t =ξ₁ = … = ξ_τ = 0, τ ≠ 0, is a quotient of a system of the form (tD _t − A(x, D _x ))u = D _x1 u = … = D _xτu = 0, with A essentially of order ≤ 0.