Research articles
ScienceAsia 50 (2024):ID 2024037 1-4 |doi:
10.2306/scienceasia1513-1874.2024.037
Hartogs-Bochner extension theorem for L
2loc-functions on
unbounded domains
Shaban Khidra,b,*, Salomon Sambouc
ABSTRACT: We prove an L
2
loc-Hartogs-Bochner type extension theorem for unbounded domain D in a complex manifold
X of complex dimension n ? 2. More precisely, we show that if ? is a paracompactifying family of closed subsets of X
not containing X, then the ?? -cohomology group of (0, 1)-currents of class C ? on X with supports in ? is isomorphic
to the ?? -cohomology group of (0, 1)-forms with L
2
loc(X)-coefficients and with supports in ?. Moreover, we prove that
a sufficient condition for CR L2
loc-functions, defined on the boundary ? D of D, being extended holomorphically to D is
that the L
2
-
?? -cohomology groups must vanish. Similar results are given in the C
k
and L
p
-categories.
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| a |
Department of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah 21589 Saudi Arabia |
| b |
Department of Mathematics and Computer Sciences, Faculty of Science, Beni-Suef University,
Beni-Suef 62511 Egypt
|
| c |
Mathematics Department, UFR of Science and Technology, Assane Seck University of Ziguinchor, BP: 523 S?n?gal |
* Corresponding author, E-mail: skhidr@uj.edu.sa, skhidr@science.bsu.edu.eg
Received 3 Jul 2022, Accepted 11 Oct 2023
|
