Global solvability for first order real linear partial differential operators

Abstract

F. Treves, in [17], using a notion of convexity of sets with respect to operators due to B. Malgrange and a theorem of C. Harvey, characterized globally solvable linear partial differential operators on C ∞ ( X ) , for an open subset X of R n . Let P = L + c be a linear partial differential operator with real coefficients on a C ∞ manifold X, where L is a vector field and c is a function. If L has no critical points, J. Duistermaat and L. Hörmander, in [2], proved five equivalent conditions for global solvability of P on C ∞ ( X ) . Based on Harvey–Treves's result we prove sufficient conditions for the global solvability of P on C ∞ ( X ) , in the spirit of geometrical Duistermaat–Hörmander's characterizations, when L is zero at precisely one point. For this case, additional non-resonance type conditions on the value of c at the equilibrium point are necessary.