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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