Let Ω ⊂ C n × R be a bounded domain with smooth boundary such that ∂Ω has only nondegenerate elliptic CR singularities, and let f : ∂Ω → C be a smooth function that is CR at CR points of ∂Ω (when n = 1 we require separate holomorphic extensions for each real parameter). Then f extends to a smooth CR function on Ω, that is, an analogue of Hartogs-Bochner holds. In addition, if f and ∂Ω are real-analytic, then f is the restriction of a function that is holomorphic on a neighborhood of Ω in C n+1 . An immediate application is a (possibly singular) solution of the Levi-flat Plateau problem for codimension 2 submanifolds that are CR images of ∂Ω as above. The extension also holds locally near nondegenerate, holomorphically flat, elliptic CR singularities.
Abstract. Let M ⊂ C n+1 , n ≥ 2, be a real codimension two CR singular real-analytic submanifold that is nondegenerate and holomorphically flat. We prove that every realanalytic function on M that is CR outside the CR singularities extends to a holomorphic function in a neighborhood of M . Our motivation is to prove the following analogue of the Hartogs-Bochner theorem. Let Ω ⊂ C n × R, n ≥ 2, be a bounded domain with a connected real-analytic boundary such that ∂Ω has only nondegenerate CR singularities. We prove that if f : ∂Ω → C is a real-analytic function that is CR at CR points of ∂Ω, then f extends to a holomorphic function on a neighborhood of Ω in C n × C.
Abstract. Recently Herbig, McNeal, and Straube have showed that the Bergman projection of conjugate holomorphic functions is smooth up to the boundary on smoothly bounded domains that satisfy condition R. We show that a further smoothing property holds on a family of Reinhardt domains; namely, the Bergman projection of conjugate holomorphic functions is holomorphic past the boundary.
We solve the Levi-flat Plateau problem in the following case. Let M ⊂ C n+1 , n ≥ 2, be a connected compact real-analytic codimension-two submanifold with only nondegenerate CR singularities. Suppose M is a diffeomorphic image via a real-analytic CR map of a real-analytic hypersurface in C n × R with only nondegenerate CR singularities. Then there exists a unique compact real-analytic Levi-flat hypersurface, nonsingular except possibly for self-intersections, with boundary M . We also study boundary regularity of CR automorphisms of domains in C n × R.
ABSTRACT. Let Ω ⊂ R n be a bounded domain with C ∞ boundary. We show that a harmonic function in Ω that is Lipschitz along a family of curves transversal to bΩ is Lipschitz in Ω. The space of Lipschitz functions we consider is defined using the notion of a majorant which is a certain generalization of the power functions t α , 0 < α < 1.
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