A Note on Global Solvability of Vector Fields

We consider global solvability of complex vector fields on noncompact manifolds. The case of real vector fields had been considered by Malgrange, and Hörmander studied the complex case, assuming that the real and imaginary parts are everywhere linearly independent. 1. We consider a smooth complex vector field L = X + iY without zeros on a smooth, paracompact, noncompact manifold M. The real vector fields X and Y do not vanish simultaneously. The pair of vectors X and Y defines an everywheredefined group of local diffeomorphisms in M [8, p. 175]. The equivalence classes defined by this group (two points, a and b, are related if there is a local diffeomorphism in the group that takes a into b) will be called the orbits of L in M. The orbits are connected submanifolds of M with a natural differentiable structure. If U Ç M is open, the orbit of L in U through the point p is contained in, but not necessarily equal to, the intersection of U with the orbit of L in M through p. We shall deal with the global solvability of the equation (L + a)u = f with u and / smooth, and a smooth and fixed. If L + a is surjective in C°°(M), it follows that L should verify condition (P) in M [5]. Condition (P) implies [3] that the orbits of L have dimension one or two. We shall assume that: (1.1) No orbit is relatively compact in M. This condition is not necessary for the solvability of L + a, but it implies the sufficient conditions for semiglobal solvability given in [4]. Assume furthermore that: (1.2) For each compact set K of M there exists a compact set K' such that if B is an orbit of L in M\K which is relatively compact in M, then B Ç K'. If (1.1) holds, (1.2) can be reformulated for one-dimensional orbits as follows: (1.2)' For each compact set K of M there exists a compact set K' Ç M such that if 7 is a one-dimensional orbit of L in M\K with endpoints in K, it follows that IQK'. THEOREM 1.1. If L verifies (P), (1.1) and (1.2), then (L + a)C°0(M) = C°°(M) VaeC7°°(M). In the special cases where (i) X and Y are linearly independent everywhere, or (ii) M is an open subset of the plane, Theorem 1.1 follows from [2, §7.1, and 6]. When dim M — 2 it is possible to make these results more precise. THEOREM 1.2. Assume that L satisfies (1.1) and dim M = 2. Then (L + a)C°°{M) = C°°(M) if and only if L satisfies (P) and (1.2)'. Received by the editors April 18, 1984. 1980 Mathematics Subject Classification. Primary 33A05, 35F05. xThe author was partially supported by CNPq (Brazil). ©1985 American Mathematical Society 0002-9939/85 $1.00 + $.25 per page 61 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use