An application of symbol calculus

Let X = ℂ t,x 1+n , t ∈ ℂ, x = (x 1,…,x n ) ∈ ℂ n , 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 = … = ξτ = 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.