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The following three classes of models of rigid submanifolds of higher type with CR dimension one are discussed: 1) A tube-like model that only depends on the real part of the holomorphic tangent coordinate; 2) a radial model that depends on the modulus of the holomorphic tangent coordinate and 3) a free model. The first and third models have a Lie group structure which is analyzed. A characterization of the hull of holomorphy of the first two models is presented along with a partial result on the hull of holomorphy of the third.

Abstract.An integral kernel approach is given for the proof of the theorem of Andreotti and Hill which states that the Y(q) condition of Kohn is a sufficient condition for local solvability of the tangential Cauchy Riemann equations on a real hypersurface in C". In addition, we provide an integral kernel approach to nonsolvability for a certain class of real hypersurfaces in the case when Y(q) is not satisfied.1. Let M be a smooth real hypersurface in C" and let z0 e M. If q is an integer with 0 < q < n -1, then we say that M satisfies condition Y(q) (cf. [FK]) at z0 if the Levi form of M at z0 has either max{ n -q, q + 1} eigenvalues of the same sign or min(« -q, q + 1} positive and min(« -q, q + 1} negative eigenvalues. Note that if the Levi form of M at z0 has r positive and s negative eigenvalues and r + s = n -1, then M satisfies condition Y(q) for all 0 < q < n -1 except q = r, s. . In this paper, we handle the latter half of condition Y(q) and thus we provide an entirely integral kernel approach to the proof of the result of Andreotti and Hill mentioned above. In addition, we provide an integral kernel approach to nonsolvability when condition Y{q) is not satisfied. Finally, if q = 0, our approach yields (yet) another proof of the classical Hans Lewy (2 sided) CR extension theorem with an integral kernel representation of the extension.To state our result let us introduce some notation. i\p-q(C") will denote the bundle of (p, q) forms on C. If V is an open set in C, then êp-\V) (resp. S>p-q{V)) will denote the space of smooth sections of A^^C") over F (resp. with compact support in V). Let M be a smooth real hypersurface defined by M = {z g C"; p(z) = 0}, where p: C" -* R with dp # 0 on M. Let Ap-q{M) be the subbundle of K"-q{C")\M consisting of those forms in Ap-q(C")\M which are orthogonal to the ideal generated by 3p(z), z g M (under the usual Hermitian metric for forms in C"). If Fis an open

We define a class of generic CR submanifolds of C n of real codimension d, 1 ≤ d ≤ n − 1, called the Bloom-Graham model graphs, whose graphing functions are partially decoupled in their dependence on the variables in the real directions. We prove a global version of the Baouendi-Treves CR approximation theorem, for Bloom-Graham model graphs with a polynomial growth assumption on their graphing functions.