Complex Hamiltonian mechanics and parametrices for subelliptic Laplacians. II
In this second part the system of bicharacteristic curves is considered: \[ \dot x_j(s)= \frac{\partial H}{\partial\xi_j}, \quad \dot t(s)=\frac{\partial H}{\partial\theta}, \quad \dot\xi_j(s)=- \frac{\partial H}{\partial x_j}, \quad \dot\theta(s)=- \frac{\partial H}{\partial t}, \qquad s\in[0,\tau] \tag{1} \] with boundary conditions \[ x(0)= x^0, \quad x(\tau)= x^1, \quad t(\tau)=t^1, \quad \theta(0)= \theta^0=-i. \tag{2} \] A convienable domain is put in evidence such that, applying a fixed point theorem, one obtains the existence and uniqueness of the solution of (1), (2). Furthermore, setting \(s=\sigma\tau\), \(\sigma\in [0,1]\), \(x_j(\sigma\tau; x^1,t^1,\tau; x^0,t^0)\), \(\xi_j(\sigma\tau; x^1,t^1,\tau; x^0,t^0)\); \(j=1,\dots,2n\), \(\tau(\sigma\tau; x^1,t^1,\tau; x^0,t^0)\), \(\theta(\sigma\tau; x^1,t^1,\tau; x^0,t^0)\) are differentiable functions of \(\sigma\), and are, as well as the \(\sigma\)-derivatives, analytic functions of \(x^0,t^0, x^1,t^1, \tau\). Although \(t^0\) does not appear explicitly in the boundary conditions, \(t^0\) is hidden in the reduction process of the vector fields \(X_j\) which gives the dependence on \(t^0\) of the trajectory. At the end the expression and uniform estimates for \(g\) and \(\Delta_Xg\) are given as well as the explicit solutions of the transport equations for \(v_0, v_k\), \(w_k\); \(k=1,\dots, 2n\).
scientific article; zbMATH DE number 1006822
scientific article