On propagation of boundary continuity of holomorphic functions of several variables
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 7 (2008) no. 2, pp. 271-285.

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.

Classification : 32A40, 32E35 
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     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},
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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, Série 5, Tome 7 (2008) no. 2, pp. 271-285. http://archive.numdam.org/item/ASNSP_2008_5_7_2_271_0/

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