We prove that continuity properties of bounded analytic functions in bounded smoothly bounded pseudoconvex domains in two-dimensional affine space are determined by their behaviour near the Shilov boundary. Namely, if the function has continuous extension to an open subset of the boundary containing the Shilov boundary it extends continuously to the whole boundary. If it is e.g. Hölder continuous on such a boundary set, it is Hölder continuous on the closure of the domain. The statements may fail if the boundary is not smooth.
@article{ASNSP_2008_5_7_2_271_0, author = {Franz\'en, Salla and J\"oricke, Burglind}, title = {On propagation of boundary continuity of holomorphic functions of several variables}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, pages = {271--285}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 7}, number = {2}, year = {2008}, mrnumber = {2437028}, zbl = {1173.32004}, language = {en}, url = {http://archive.numdam.org/item/ASNSP_2008_5_7_2_271_0/} }
TY - JOUR AU - Franzén, Salla AU - Jöricke, Burglind TI - On propagation of boundary continuity of holomorphic functions of several variables JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2008 SP - 271 EP - 285 VL - 7 IS - 2 PB - Scuola Normale Superiore, Pisa UR - http://archive.numdam.org/item/ASNSP_2008_5_7_2_271_0/ LA - en ID - ASNSP_2008_5_7_2_271_0 ER -
%0 Journal Article %A Franzén, Salla %A Jöricke, Burglind %T On propagation of boundary continuity of holomorphic functions of several variables %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2008 %P 271-285 %V 7 %N 2 %I Scuola Normale Superiore, Pisa %U http://archive.numdam.org/item/ASNSP_2008_5_7_2_271_0/ %G en %F ASNSP_2008_5_7_2_271_0
Franzén, Salla; Jöricke, Burglind. On propagation of boundary continuity of holomorphic functions of several variables. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 7 (2008) no. 2, pp. 271-285. http://archive.numdam.org/item/ASNSP_2008_5_7_2_271_0/
[1] Peak points, barriers and pseudoconvex boundary points, Proc. Amer. Math. Soc. 65 (1977), 89-92. | MR | Zbl
,[2] On a generalized Dirichlet problem for plurisubharmonic functions and pseudo-convex domains. Characterization of Šilov boundaries, Trans. Amer. Math. Soc. 91 (1959), 246-276. | MR | Zbl
,[3] Stein compacts in Levi-flat hypersurfaces, Trans. Amer. Math. Soc. 360 (2008), 307-329. | MR | Zbl
, and ,[4] “Uniform Algebras”, Prentice-Hall Inc., Englewood Cliffs, N. J., 1969. | MR | Zbl
,[5] Boundary continuity of some holomorphic functions, Pacific J. Math. 80 (1979), 425-434. | MR | Zbl
,[6] Cauchy-Riemann equations in several variables, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 22 (1968), 275-314. | Numdam | MR | Zbl
,[7] Some properties of fractional integrals. II, Math. Z. 34 (1932), 403-439. | MR | Zbl
and ,[8] Holomorphic approximation on totally real submanifolds of a complex manifold, Bull. Amer. Math. Soc. 77 (1971), 824-828. | MR | Zbl
and ,[9] The relation between the solid modulus of continuity and the modulus of continuity along a Shilov boundary for analytic functions of several variables, Mat. Sb. (N.S.) 122 (164) (1983), 511-526. | MR | Zbl
,[10] The modulus of continuity of analytic functions in a domain and on its Shilov boundary, In: “Problems in the Theory of Functions of Several Complex Variables and in Infinite-Dimensional Complex Analysis”, Lecture Notes in Math., Vol. 1039, Springer, 1983, Collected and prepared by Christer O. Kiselman, 472-473. | MR
,[11] Some remarks concerning holomorphically convex hulls and envelopes of holomorphy, Math. Z. 218 (1995), 143-157. | MR | Zbl
,[12] Über polynomiale Funktionen auf Holomorphiegebieten, Math. Z. 139 (1974), 133-139. | MR | Zbl
,[13] Holomorphically convex sets in several complex variables, Ann. of Math. 74 (1961), 470-493. | MR | Zbl
,[14] Mergelyan sets and the modulus of continuity of analytic functions, J. Approx. Theory 15 (1975), 23-40. | MR | Zbl
, and ,[15] “Degree of Approximation by Polynomials in the Complex Domain”, Annals of Mathematical Studies, n. 9, Princeton University Press, Princeton, N. J., 1942. | JFM | MR | Zbl
,[16] Sur le prolongement holomorphe des fonctions C-R défines sur une hypersurface réelle de classe dans , Invent. Math. 83 (1986), 583-592. | MR | Zbl
,[17] Extension of CR-functions into a wedge, Mat. Sb. 181 (1990), 951-964. | MR | Zbl
,