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On the Converse of Shimoda's Theorem
Abstract
We study Shimoda's theorem on the holomorphicity of a function f(z,w) defined in a bidisc U× V={ | z |<1 }× { | w |<1 }, which is holomorphic with respect to w∈ V for each fixed z∈ U and is holomorphic with respect to z∈ U for each fixed w∈ E, where E⊂ V is a countable set with at least one limit point in V. Shimoda proved that in this case f(z,w) is holomorphic in U× V except for a nowhere dense closed subset of U× V. We prove a converse: for any relatively closed nowhere dense set S⊂ U such that ∂ U⋃ S is connected, there exists a function satisfying the conditions of Shimoda's theorem that is holomorphic in (U∖ S)× V and doesn't extend holomorphically to any point of S× V.
Related
BibTeX
@misc{imomkulov2026converse,
title = {{On the Converse of Shimoda's Theorem}},
author = {Sevdiyor Imomkulov and Kamol Rasulov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40381},
eprint = {2610.40381},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40381},
}
RIS
TY - UNPB
TI - On the Converse of Shimoda's Theorem
AU - Sevdiyor Imomkulov
AU - Kamol Rasulov
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40381
UR - https://arxiv.org/abs/2610.40381
AB - We study Shimoda's theorem on the holomorphicity of a function f(z,w) defined in a bidisc U× V={ | z |<1 }× { | w |<1 }, which is holomorphic with respect to w∈ V for each fixed z∈ U and is holomorphic with respect to z∈ U for each fixed w∈ E, where E⊂ V is a countable set with at least one limit point in V. Shimoda proved that in this case f(z,w) is holomorphic in U× V except for a nowhere dense closed subset of U× V. We prove a converse: for any relatively closed nowhere dense set S⊂ U such that ∂ U⋃ S is connected, there exists a function satisfying the conditions of Shimoda's theorem that is holomorphic in (U∖ S)× V and doesn't extend holomorphically to any point of S× V.
ER -