An Introduction to Involutive Structures Detailing the main methods in the theory of involutive systems of complex vector fields, this book examines the major results from the last 25 years in the subject. One of the key tools of the subject-the Baouendi-Treves approximation theorem-is proved for many function spaces. This in turn is applied to questions in partial differential equations and several complex variables. Many basic problems such as regularity, unique continuation and boundary behavior of the solutions are explored. The local solvability of systems of partial differential equations is studied in some detail. The book provides a solid background for beginners in the field and also contains a treatment of many recent results which will be of interest to researchers in the subject.

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In many ways hypersurfaces with null r-th curvature function H r behave much like the minimal ones (H 1 = 0). One such manifestation is the following result to be proved in this paper, which extends to scalar-flat hypersurfaces (H 2 = 0) a well-known theorem of R. Schoen.Theorem. The only complete scalar-flat embeddings M n ⊂ R n+1 , free of flat points, which are regular at infinity and have two ends, are the hypersurfaces of revolution.

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Abstract. We prove that solutions of the homogeneous equation Lu = 0, where L is a locally integrable vector field with smooth coefficients in two variables possess the F. and M. Riesz property. That is, if Ω is an open subset of the plane with smooth boundary, u ∈ C 1 (Ω) satisfies Lu = 0 on Ω, has tempered growth at the boundary, and its weak boundary value is a measure µ, then µ is absolutely continuous with respect to Lebesgue measure on the noncharacteristic portion of ∂Ω. IntroductionConsider a Borel measure µ defined on the boundary T of the unit circle ∆ of the complex plane. A classical theorem proved in 1916 by F. and M. Riesz states that if the Fourier coefficients of µ vanish for all negative integral values, i.e.,then µ is absolutely continuous with respect to the Lebesgue measure dθ. Condition (a) is equivalent to the existence of a holomorphic function f (z) defined on ∆ whose weak boundary value is µ. The F. and M. Riesz theorem has undergone an extensive generalization in the last decades, mainly in two different directions: i) generalized analytic function algebras, which has as a starting point the fact that (a) means that µ is orthogonal to the algebra of continuous functions on T that extend holomorphically to ∆; ii) ordered groups, which emphasizes instead the role of the group structure of T in the classical result. Thus, although absolute continuity with respect to Lebesgue measure is a local property (i.e., if each point has a neighborhood where it holds then it holds everywhere), both directions focus on global objects. A remarkable exception is the paper [B] in which the author uses microlocal analysis to prove some generalizations of the theorem of F. and M. Riesz. Among other things, in [B] it is shown that if a CR measure on a hypersurface of C n is the boundary value of a holomorphic function defined on a side, then it is absolutely continuous with respect to Lebesgue measure.In view of Riemann's mapping theorem and the local character of the conclusion, another way of stating the F. and M. Riesz theorem is to say that if a holomorphic 1991 Mathematics Subject Classification. Primary 35F15, 30E25, 28A99; Secondary 42A99, 42B30, 42A38, 46F20.