Holomorphic extension of solutions of elliptic linear partial differential operators with analytic coefficients

We start with a simple fact: the fundamental solutions of the Laplacian in Rn can be continued as multi-valued

analytic functions in Cn up to the complex bicharacteristic conoid. This extension ramies around the complex

isotropic cone: z2

1 +


z2n

= 0 and has "moderate growth".

For an elliptic linear partial dierential operator of the second order with analytic coe-cients and simple complex characteristics in an open set  Rn, this may be generalized: every fundamental solution can be continued at least locally as a multi-valued analytic function in Cn up to the complex bicharacteristic conoid.

This holomorphic extension is ramied around the bicharacteristic conoid and belongs to the so-called Nilsson

class ("moderate growth").

In fact, those results remain true for such operators with degree bigger than 4 , but the proofs are different due to the lack of natural geodesic distance associated to the operators

Those results may be connected with D-module theory, and more precisely with regular holonomic D-Modules.

We'll explain this link and state a general conjecture