An extension theorem for separately meromorphic functions with pluripolar singularities
| dc.creator | Jarnicki, Marek | |
| dc.creator | Pflug, Peter | |
| dc.date | 2002-09-17 | |
| dc.date | 2002-10-16 | |
| dc.date.accessioned | 2026-07-07T04:50:56Z | |
| dc.date.available | 2026-07-07T04:50:56Z | |
| dc.description | Let $D_j\subset\mathbb C^{n_j}$ be a pseudoconvex domain and let $A_j\subset D_j$ be a locally pluriregular set, $j=1,...,N$. Put $$ X:=\bigcup_{j=1}^N A_1\times...\times A_{j-1}\times D_j\times A_{j+1}\times...\times A_N. $$ Let $M\subset X$ be relatively closed. For any $j\in\{1,...,N\}$ let $Σ_j$ be the set of all $(z',z'')\in(A_1\times...\times A_{j-1})\times(A_{j+1}\times...\times A_N)$ such that the fiber $M_{(z',\cdot,z'')}:=\{z_j\in\mathbb C^{n_j}: (z',z_j,z'')\in M\}$ is not pluripolar. Assume that $Σ_1,...,Σ_N$ are pluripolar. Put ${multline*} X':=\bigcup_{j=1}^N\{(z',z_j,z'')\in(A_1\times...\times A_{j-1})\times D_j \times(A_{j+1}\times...\times A_N): (z',z'')\notinΣ_j\}$. Then there exists a relatively closed pluripolar subset $\widetilde M\subset\widetilde X$ of the `envelope of holomorphy' $\widetilde X$ of $X$ such that: $\bullet$ $\widetilde M\cap X'\subset M$, $\bullet$ every function $f$ separately meromorphic on $X\setminus M$ extends to a (uniquely determined) function $\widetilde f$ meromorphic on $\widetilde X\setminus\widetilde M$, $\bullet$ if $f$ is separately holomorphic on $X\setminus M$, then $\widetilde f$ is holomorphic on $\widetilde X\setminus\widetilde M$, and $\bullet$ $\widetilde M$ is singular with respect to the family of all functions $\widetilde f$. \noindent In the case where N=2, $M=\varnothing$, the above result may be strengthened. | |
| dc.description | 11 pages | |
| dc.identifier | https://arxiv.org/abs/math/0209207 | |
| dc.identifier | http://arxiv.org/abs/math/0209207 | |
| dc.identifier | Kyushu J. Math. 57 (2003), 291-302. | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/64970 | |
| dc.subject | Complex Variables | |
| dc.subject | 32D15; 32D10 | |
| dc.title | An extension theorem for separately meromorphic functions with pluripolar singularities | |
| dc.type | text |