On positivity of certain systems of partial differential equations

摘要

We extend the Fefferman–Phong inequality to N × N systems of PDEs with symbol (x, ξ) ∈ ℝⁿ × ℝⁿ, where e is a positive homogeneous quadratic form, and B(x, ξ) = Σℓ=1ⁿ Bℓ(x)ξℓ. We thus generalize a result by L.-Y. Sung that was obtained for systems of ordinary differential equations. Our proof exploits a Calderón–Zygmund decomposition of the phase-space ℝⁿ × 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.