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

Speaker
Serge Itshak Lukasiewicz
Date
17/11/2013 - 12:00Add to Calendar 2013-11-17 12:00:00 2013-11-17 12:00:00 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 אוניברסיטת בר-אילן - Department of Mathematics mathoffice@math.biu.ac.il Asia/Jerusalem public
Abstract

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

Last Updated Date : 09/11/2013