The following question is intended for a person more acquainted with the works of Yves Laurent.
see: http://archive.numdam.org/article/ASENS_1987_4_20_3_391_0.pdf (French)
http://link.springer.com/article/10.1023%2FA%3A1000791329695 (English)
for a review.
Specifically, I understand that in the case of ordinary differential equations \begin{eqnarray*} y^{(n)}(x)=\sum_{i=0}^{n-1}a_i(x)y^{(i)}(x) \end{eqnarray*} $a_i(x)$ meromorphic, the singular points of $y(x)$ can be categorized as either regular or irregular, and further, the irregular singular points can be labeled according to their Gevrey class. The works of Ramis and Malgrange showed that a Newton polygon can be constructed from $a_i(x)$ and it's vertices define the jumps in the index of the differential operator over $C[[x]]_s$, the formal power series of Gevrey class $s$.
Going to multidimensions $x=(x_1,...,x_n)$, the partial differential operator \begin{eqnarray*} P(x,D)=\sum_\alpha^m P_\alpha(x)D^\alpha \end{eqnarray*} also defines a Newton polygon, which is related in some way to a pair of filtrations defined by Laurent. One cannot define an irregularity index but it is possible to define an irregularity sheaf, which in some way encodes the singularity information, though I remain fuzzy on the details.
My question is, for the case of the operator $P(x,D)$ above, how much can Laurent say about its solution space, and its singularities, compared to the one-dimensional case? I understand that the first thing to go is the formal series $C[[x]]$, which are replaced by microfunctions, but can he give a complete description of the growth of solutions in the same way that Gevrey asymptotics determine how fast a solution diverges near a singular point? I want to emphasize that this is not the holonomic case, which I realize most of his work is about. However, in his a review paper he eludes to this case and omits the details, pointing to original sources either behind a pay wall or published in French! Any help deciphering this is much appreciated.
