×

Some remarks concerning holomorphically convex hulls and envelopes of holomorphy. (English) Zbl 0816.32011

Let \(\Omega\) be a bounded strictly convex domain in \(\mathbb{C}^ 2\) with smooth boundary, and let \(K\) be a compact subset of the boundary \(\partial \Omega\). Stout, Lupacciolu and Alexander proved that each continuous \(CR\)-function on \(\partial \Omega \backslash K\) has uniquely determined analytic extension to \(\Omega \backslash \widehat K\), \(\widehat K\) being the polynomial hull of \(K\).
There is a heuristic way to see this, namely by observing the connection between Oka’s characterization principle for polynomial hulls on the one hand and analytic continuation along continuous one-parameter families of one-dimensional analytic varieties on the other hand. Monodromy considerations for proving that the envelope of holomorphy is schlicht are based on the fact that in \(\mathbb{C}^ 2\) the complex codimension of the considered varieties is equal to one. These observations are managed into a rigorous proof, which differs from the original one.
The result is generalized to arbitrary domains of holomorphy with \(C^ 2\) boundary, the polynomial hull in that case to be replaced by the hull with respect to the space of functions which are holomorphic in \(\Omega\) and continuous in the closure \(\overline \Omega\). Previous results of this kind concern the case of domains \(\Omega\) with \(\overline \Omega\) having a Stein neighbourhood basis and the (in general larger) hull with respect to functions which are analytic in a neighbourhood of \(\overline \Omega\).

MSC:

32D10 Envelopes of holomorphy
32E20 Polynomial convexity, rational convexity, meromorphic convexity in several complex variables
32D20 Removable singularities in several complex variables
32V05 CR structures, CR operators, and generalizations
32E35 Global boundary behavior of holomorphic functions of several complex variables
PDF BibTeX XML Cite
Full Text: DOI EuDML

References:

[1] Alexander, H.: A note on polynomial hulls. Proc. Am. Math. Soc.33, 389–391 (1972) · Zbl 0239.32013
[2] Alexander, H., Stout, E.L.: A note on hulls. Bull. Lond. Math. Soc.22, 258–260 (1990) · Zbl 0668.32015
[3] Catlin, D.: Boundary behaviour of holomorphic functions on pseudoconvex domains. J. Differ. Geom.15, 605–625 (1980) · Zbl 0484.32005
[4] Diederick, K., Formæss, J.E.: An example with nontrivial nebenhuelle. Math. Ann.225, 275–292 (1977) · Zbl 0333.32018
[5] Gamelin, Th.: Uniform algebras. Prentice-Hall, Englewood Cliffs, NJ 1969 · Zbl 0213.40401
[6] Greenfield, S.J.: Cauchy-Riemann equations in several variables. Ann. Sc. Norm. Super. Pisa22, 275–314 (1968) · Zbl 0159.37502
[7] Hakim, M., Sibony, N.: Spectre de \(A(\bar \Omega )\) pour des domaines bornés faiblement pseudoconvexes réguliers. J. Funct. Anal.37, 127–135 (1980) · Zbl 0441.46044
[8] Hanges, N., Treves, F.: Propagation of holomorphic extendability of CR-functions. Math. Ann.263, 157–177 (1983) · Zbl 0507.32009
[9] Hartman, Ph.: Ordinary differential equations. John Wiley & Sons: New York-London-Sydney 1964 · Zbl 0125.32102
[10] Hörmander, L.: An introduction to complex analysis in several variables. D. van Nostrand, Princeton, NJ-Toronto-New York-London 1966 · Zbl 0138.06203
[11] Lupacciolu, G.: A theorem on holomorphic extension of CR-functions. Pac. J. Math.124, 177–191 (1986) · Zbl 0597.32014
[12] Rosay, J.P., Stout, E.L.: Radó’s theorem for CR-functions. Proc. Am. Math. Soc.106, 1017–1026 (1989) · Zbl 0674.32007
[13] Slodkowski, Z.: Analytic set-valued functions and spectra. Math. Ann.256, 363–386 (1981) · Zbl 0458.46036
[14] Stolzenberg, G.: Polynomially and rationally convex sets. Acta Math.109, 259–289 (1963) · Zbl 0122.08404
[15] Stolzenberg, G.: Uniform approximation on smooth curves. Acta Math.115, 185–198 (1966) · Zbl 0143.30005
[16] Stout, E.L.: Analytic continuation and boundary continuity of functions of several complex variables. Proc. R. Soc. Edinb.89 A, 63–74 (1981) · Zbl 0491.32007
[17] Stout, E.L.: Removable singularities for the boundary values of holomorphic functions. Proc. Mittag-Leffler special year in complex variables: Princcton Univ. Press · Zbl 0772.32011
[18] Sussmann, H.J.: Orbits of families of vector fields and integrability of distributions. Trans. Am. Math. Soc.180, 171–188 (1973) · Zbl 0274.58002
[19] Trépreau, J.-M.: Sur le prolongement holomorphe des fonctions CR définies sur une hypersurface réelle de classeC 2 dans \(\mathbb{C}\)2. Invent. Math.83, 583–592 (1986) · Zbl 0586.32016
[20] Trépreau, J.-M.: Sur la propagation des singularités dans les variétés CR. Bull. Soc. Math. Fr.118, 403–450 (1990)
[21] Tumanov, A.E.: Extending CR-functions on manifolds of finite type to a wedge. Math. Sb., Nov. Ser.136, 128–139 (1988)
[22] Vladimirov, V.S.: Fonctions de plusieurs variables complexes. Dunod Paris 1971
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.