On positivity of certain systems of partial differential equations

We extend the Fefferman-Phong inequality to N x N systems of PDEs with symbol p(x, xi) = A(x)e(xi) + B(x, xi) + C(x) = p(x, xi)* >= -cl, (x, xi) is an element of R-n x R-n, where e is a positive homogeneous quadratic form, and B(x, xi) = Sigma(n)(l=1) B-l(x)xi(l). We thus generalize a result by L.-Y. Sung that was obtained for systems of ordinary differential equations. our proof exploits a Calderon-Zygmund decomposition of the phase-space Rn x Rn of the kind introduced by C. Fefferman and D. H. Phong for studying subelliptic differential operators and goes by induction on the size N of the system.