Fundamental solutions for hermite and subelliptic operators

In this article, we introduce a geometric method based on multipliers to compute heat kernels for operators with potentials. Using the heat kernel, we compute the fundamental solution for the Hermite operator with singularity at an arbitrary point on Euclidean space and on Heisenberg groups. As a consequence, we obtain the fundamental solutions for the sub-laplacian □ _J in a family of quadratic submanifolds. The research is partially supported by a William Fulbright Reserch Grant and a Competitive Research Grant at Georgetown University.