Introduction This paper deals with the properties of solutions of first and second order equations in the plane. These equations are generated by a complex vector field X that is elliptic everywhere except along a simple closed curve Σ R2. The vector field X is tangent to Σ and X X vanishes to first order along Σ (and so X does not satisfy ormander’s bracket condition). Such vector fields have canonical representatives (see [8]). More precisely, there is a change of coordinates in a tubular neighborhood of Σ such that X is conjugate to a unique vector field L of the form (0.1) L = λ ∂t ir ∂r defined in a neighborhood of the circle r = 0 in R × S1, where λ R+ + iR is an invariant of the structure generated by X. We should point out that normalizations for vector fields X such that X X vanishes to a constant order n 1 along Σ are obtained in [9], but we will consider here only the case n = 1. This canon- ical representation makes it possible to study the equations generated by X in a neighborhood of the characteristic curve Σ. We would like to mention a very recent paper by F. Treves [13] that uses this normalization to study hypoellipticity and local solvability of complex vector fields in the plane near a linear singularity. The motivation for our work stems from the theory of hypoanalytic structures (see [12] and the references therein) and from the theory of generalized analytic functions (see [18]). The equations considered here are of the form Lu = F (r, t, u) and Pu = G(r, t, u, Lu), where P is the (real) second order operator (0.2) P = LL + β(t)L + β(t)L. It should be noted that very little is known, even locally, about the structure of the solutions of second order equations generated by complex vector fields with degen- eracies. The paper [5] explores the local solvability of a particular case generated by a vector field of finite type. An application to a class of second order elliptic operators with a punctual singularity in R2 is given. This class consists of operators of the form (0.3) D = a11 ∂2 ∂x2 + 2a12 ∂2 ∂xy + a22 ∂2 ∂y2 + a1 ∂x + a2 ∂y , 1
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