Boundary Forelli conjecture for bundles of complex lines

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Let Bn⊂CnB^n\subset\mathbb C^n be the unit ball, and let a1,…,an+1∈Bna_1,\ldots,a_{n+1}\in B^n be vertices in general position. An n+1n+1-bundle of complex lines consists of the complex lines containing at least one of these vertices. Let f∈C(∂Bn)f\in C(\partial B^n) be a continuous function on the unit sphere. For each such complex line LL, consider the restriction of ff to L∩∂BnL\cap\partial B^n. Boundary Forelli conjecture. Any n+1n+1 bundles of complex lines with vertices a1,…,an+1a_1,\ldots,a_{n+1} in general position test boundary values of holomorphic functions in BnB^n: if ff holomorphically extends in every disc determined by these lines, then ff is the boundary value of a holomorphic function in BnB^n. The conjecture identifies vertices lying in a hyperplane as the only obstruction to the continuous boundary version of the Forelli theorem; the supplied source does not indicate whether the conjecture has been resolved.

References

Primary source

Mark Agranovsky, “Boundary Forelli theorem for the sphere in C^n and n+1 bundles of complex lines”, arXiv:1003.6125 (2010).

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