Abstract
We consider a real analytic foliation of \({\mathbb{C}^n}\) by complex analytic manifolds of dimension m issued transversally from a CR generic submanifold \({M\subset\mathbb{C}^n}\) of codimension m. We prove that a continuous CR function f on M which has separate holomorphic extension along each leaf, is holomorphic. When the leaves are cartesian straight planes, separate holomorphic extension along suitable selections of these planes suffices and f turns out to be holomorphic in a neighbourhood of their union. If M is a hypersurface we can also specify the side of the extension, regardless the leaves are straight or not.
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References
Ajrapetyan R., Henkin G.: Analytic continuation of CR functions through the, edge of the wedge. Dokl. Acad. Nauk. SSSR 259, 777–781 (1981)
Aizenberg, L., Rea, C.: The moment condition for the free boundary problem for CR functions. Ann. SNS Pisa, 313–322 (1993)
Browder F.E.: Real analytic functions on product spaces and separate analyticity. Can. J. Math. 13, 650–656 (1961)
Baracco, L., Zampieri, G.: Separate real analyticity and CR extendibility. Forum Math. (2008)
Cameron R.H., Storvick D.A.: Analytic continuation for functions of several variables. Trans. Am. Math. Soc. 125, 7–12 (1966)
Chirka, E.: Variations of the Hartogs theorem. In: Proceedings of the Steklov Institute (2006)
Hartogs F.: Zur Theorie der analytischen Funktionen mehererer Veränderlichen. Mat. Ann. 62, 1–88 (1906)
Henkin G., Tumanov A.: Local characterization of holomorphic automorphisms of Siegel domains (Russian). Funktsional. Anal. i Prilozhen 17(4), 49–61 (1983)
Lelong P.: Fonctions plurisousharmoniques et fonctions analytiques de variables reélles. Ann. Inst. Fourier 11, 515–562 (1961)
Osgood W.F.: Note über analytische Functionen mehererer Veränderlichen. Math. Ann. 52, 462–464 (1899)
Rosay J.P.: A propos d’“edges” et d’“wedges” et de prolongement holomorphe. Trans. AMS 297, 63–72 (1986)
Siciak J.: Analyticity and separate analyticity of functions defined on lower dimensional subsets of \({\mathbb{C}^n}\). Zeszyty Nauk. U.J. 13, 53–70 (1969)
Tumanov A.: Extending CR functions on a manifold of finite type over a wedge. Math. Sb. 136, 129–140 (1988)
Tumanov A.: Connection and propagation of analyticity of CR functions. Duke J. Math. 71(1), 1–24 (1994)
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Baracco, L., Zampieri, G. Separate holomorphic extension of CR functions. manuscripta math. 128, 411–419 (2009). https://doi.org/10.1007/s00229-008-0239-y
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DOI: https://doi.org/10.1007/s00229-008-0239-y