Complex Hamiltonian mechanics and parametrices for subelliptic Laplacians. III (Q1357491)

This last part (III) concerns the construction of a parametrix for \(\Delta_X\) starting with the case when \(X_j\); \(j=1,\dots,2n\) are real-analytic coefficients. In this case the authors consider also the modified operator \(\Delta= \Delta_X\) plus lower order terms \(\Delta_0, \Delta_1\) and for systems with appropriately modified \(v_k\)s and \(w_k\)s. For the general case when the \(X_j\) are only \(C^\infty\) coefficients when one cannot construct a Hamiltonian path, an analytic approximation is considered for \(X_j\) and, corresponding to this approximation, one obtains approximate solutions for the Hamilton-Jacobi equations and for the transport equations. Finally, one proves that a similar expression of the parametrix, as that in the case when the \(X_j\) are real analytic coefficients, holds also in the general case.