$ L^p $-theory for the $  {\partial }\overline{\partial} $-equation and isomorphisms results

Shaban Khidr; Salomon Sambou; Shaban Khidr; Salomon Sambou

doi:10.3934/era.2025004

Electronic Research Archive

2025, Volume 33, Issue 1: 68-86. doi: 10.3934/era.2025004

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Research article

$ L^p $-theory for the $ {\partial }\overline{\partial} $-equation and isomorphisms results

Shaban Khidr ^{1
,
,},
Salomon Sambou ²

1.
Department of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah, Saudi Arabia
2.
Departement of Mathematics, UFR of Sciences and Technologies, Assane Seck University of Ziguinchor, BP: 523, Senegal

Received: 07 October 2024 Revised: 25 November 2024 Accepted: 05 December 2024 Published: 09 January 2025

We establish an $ L^{p}_{\rm loc} $-existence theorem for the $ {\partial }\overline{\partial} $-equation on a half-space of $ \mathbb C^n $. The result is achieved for forms of class $ L^{p}_{\rm loc} $ as well as for those forms in the scale of $ W^{1, p}_{\rm loc} $-Sobolev spaces and admitting distributional boundary values. Some isomorphisms and regularity results in relation to de Rham, Bott–Chern, and Aeppli cohomology groups are moreover obtained.
- $ {\partial }\overline{\partial} $-problem,
- distributional boundary values,
- extensible currents,
- regularizing operators
Citation: Shaban Khidr, Salomon Sambou. $ L^p $-theory for the $ {\partial }\overline{\partial} $-equation and isomorphisms results[J]. Electronic Research Archive, 2025, 33(1): 68-86. doi: 10.3934/era.2025004

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Abstract

We establish an $ L^{p}_{\rm loc} $-existence theorem for the $ {\partial }\overline{\partial} $-equation on a half-space of $ \mathbb C^n $. The result is achieved for forms of class $ L^{p}_{\rm loc} $ as well as for those forms in the scale of $ W^{1, p}_{\rm loc} $-Sobolev spaces and admitting distributional boundary values. Some isomorphisms and regularity results in relation to de Rham, Bott–Chern, and Aeppli cohomology groups are moreover obtained.

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