Abstract
Let a, b, c ∈ ℂ2 be three non-collinear points such that their mutual joining complex lines do not intersect the unit ball \(\mathbb{B}^{2}\) and such that the line through a and b is tangent to \(\mathbb{B}^{2}\). Then the set of lines concurrent to a, b or c is a testing family for continuous functions on \(\mathbb{S}^{3}\). This improves a result by the authors and solves a case left open in the literature as described by Globevnik.
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References
M. Agranovsky, Analog of a theorem of Forelli for boundary values of holomorphic functions on the unit ball of ℂn, J. Anal. Math. 113 (2011), 293–304.
L. Baracco, Separate holomorphic extension along lines and holomorphic extension from the sphere to the ball, Amer. J. Math. 135 (2013), 493–497.
L. Baracco, Holomorphic extension from a convex hypersurface, Asian J. Math. 20 (2016), 263–266.
L. Baracco and M. Fassina, Orthogonal testing families and holomorphic extension from the sphere to the ball, Math. Z. 293 (2019), 1277–1285.
L. Baracco and S. Pinton, Testing families of complex lines for the unit ball, J. Math. Anal. Appl. 458 (2018), 1449–1455.
J. Globevnik, Meromorphic extensions from small families of circles and holomorphic extensions from spheres, Trans. Amer. Math. Soc. 364 (2012), 5857–5880.
J. Globevnik, Small families of complex lines for testing holomorphic extendibility, Amer. J. Math. 134 (2012), 1473–1490.
H. Komatsu, A local version of Bochner’s tube theorem, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 19 (1972), 201–214.
S. Lang, Complex Analysis, Springer, New York, 1999.
M. Lawrence, Hartogs’ separate analyticity theorem for CR functions, Internat. J. Math. 18 (2007), 219–229.
M. Lawrence, The LpCR Hartogs separate analyticity theorem for convex domains, Math. Z. 288 (2018), 401–414.
W. Rudin, Function Theory in the Unit Ball of ℂn, Springer, Berlin, 2008.
A. Tumanov, Propagation of extendibility of CR functions on manifolds with edges, in Multidimensional Complex Analysis and Partial Differential Equations (São Carlos, 1995), American Mathematical Society, Providence, RI, 1997, pp. 259–269.
A. Tumanov, Analytic discs and the extendibility of CR functions. Integral geometry, in Radon Transforms and Complex Analysis (Venice, 1996), Springer, Berlin, 1998, pp. 123–141.
A. Tumanov, Testing analyticity on circles, Amer. J. Math. 129 (2007), 785–790.
Acknowledgements
The authors wish to thank the anonymous referee for useful suggestions which have improved the quality of our paper.
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Baracco, L., Pinton, S. Testing families of analytic discs in the unit ball of ℂ2. JAMA 150, 383–404 (2023). https://doi.org/10.1007/s11854-023-0276-1
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DOI: https://doi.org/10.1007/s11854-023-0276-1