Mascolo, Raffaella (2008) Separately CR functions and peak interpolation manifolds. [Tesi di dottorato pre - 2008]
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In this thesis two different themes are analysed: the first concerns the theory of separately CR functions on CR manifolds and the second deals with the problem of characterizing peak interpolation manifolds on boundaries of pseudoconvex domains of .
For the first theme, let be a CR manifold of , with boundary , foliated by a family of CR manifolds of CR-dimension , issued from , such that the following transversality condition is satisfied at any common point of : . By using the approximation of CR functions by polynomials, we prove that if and is CR and along each leaf , then is CR in a neighbourhood of the boundary in .
Assuming to be connected, we also prove that the function comes to be CR all over , thus reaching the global result.
The use of the technique of polynomial approximation by integration with the heat kernel enables us to reprove a result by Henkin and Tumanov; moreover, a generalization of their result is given for foliations by CR manifolds of CR-dimension 1 instead of foliations by complex curves.
Our problem, concerning separately CR functions, reminds the well-known result by Hartogs on separately holomorphic functions. We present a simplified proof of Hartogs Theorem, using a ``propagation'' argument; further applications and variations of this theorem, using different techniques, are also proved in the present work.
For the second theme of the thesis, let be a bounded domain in with smooth boundary ; we denote by the algebra of continuous functions on , that are holomorphic in . A submanifold of is an interpolation manifold for if, for every and every compact set , there exists a function such that , while is a peak manifold for if, for every compact set , there exists a function such that and on . The problem of characterizing peak interpolation manifolds on boundaries of strictly pseudoconvex domains has been completely solved by Henkin and Tumanov, as well as by Rudin, with different techniques: it turns out that, in order for a smooth submanifold of to be a peak interpolation manifold, it is necessary and sufficient that satisfies a certain directional condition, which is the one of being complex tangential ().
For a complex tangential submanifold of , with strictly pseudoconvex, we present a geometric and easy proof of the property of being totally real; then, we generalize such a result, proving that, also in the case of a (weakly) pseudoconvex domain of finite type, a complex tangential submanifold of is totally real.
We analyse in details the techniques of Henkin-Tumanov and Rudin for strictly pseudoconvex domains, with the aim at extending their characterization to weakly pseudoconvex domains.
Following the first of these technique, we generalize some steps of Henkin-Tumanov proof and get some conclusions for the case of (weakly) pseudoconvex domains of type 4 in .
The second technique, based on the construction of suitable integrals and an application of a Theorem by Bishop, admits a generalization, proposed by Bharali, for weakly convex domains having real analytic boundary. Analysing the main tools of Bharali's proof, we focus our attention on his local stratification for submanifolds of the boundary; we present a different technique to stratify real analytic boundaries of weakly pseudoconvex domains, such that on each strata the Levi form is non degenerate.
Finally, we add a remark on the idea of extending the notion of peaking: even if the natural generalization of holomorphic functions is given by -closed complex differential forms, it turns out that these forms always peak inside the domain of definition, so the notion of peaking does not serve any purpose.
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