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