Properties for some Cegrell classes of $ m $-subharmonic function

Jawhar Hbil; Mofareh Alhazmi; Mohamed Zaway; Jawhar Hbil; Mofareh Alhazmi; Mohamed Zaway

doi:10.3934/math.2026188

AIMS Mathematics

2026, Volume 11, Issue 2: 4634-4655. doi: 10.3934/math.2026188

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Properties for some Cegrell classes of $ m $-subharmonic function

1.
Department of Mathematics College of Science Jouf University, P. O. Box 2014, Sakaka, Saudi Arabia
2.
Hatem Bettaher IReSCoMath Research Lab, University of Gabes, Gabes, Tunisia

Received: 11 December 2025 Revised: 04 February 2026 Accepted: 11 February 2026 Published: 26 February 2026
MSC : 32U05, 32U15, 32U25, 32W20

This paper investigated the properties of maximal subextensions for $ m $-subharmonic ($ m $-sh) functions in the context of complex Hessian operators. We established three main theorems that significantly advance the theory. First, we provided a complete characterization of maximal subextensions in the class $ \mathcal{F}_m^a(\Omega) $, showing that any subextension satisfying a minimality condition on its Hessian mass must coincide with the maximal subextension. Second, we proved that for functions in $ \mathcal{E}_m(\Omega) $, the maximal subextension relates to the original function through an $ m $-maximal function and preserves Lelong numbers. Third, we demonstrated convergence stability, proving that for sequences with boundary values in $ \mathcal{F}_m(\Omega, f) $ converging in $ L^1_{\text{loc}} $ and uniformly bounded below, their maximal subextensions converged to the maximal subextension of the limit function.
- $ m $-subharmonic functions,
- complex Hessian operator,
- maximal subextension,
- Cegrell classes,
- Lelong numbers,
- capacity convergence
Citation: Jawhar Hbil, Mofareh Alhazmi, Mohamed Zaway. Properties for some Cegrell classes of $ m $-subharmonic function[J]. AIMS Mathematics, 2026, 11(2): 4634-4655. doi: 10.3934/math.2026188

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Abstract

This paper investigated the properties of maximal subextensions for $ m $-subharmonic ($ m $-sh) functions in the context of complex Hessian operators. We established three main theorems that significantly advance the theory. First, we provided a complete characterization of maximal subextensions in the class $ \mathcal{F}_m^a(\Omega) $, showing that any subextension satisfying a minimality condition on its Hessian mass must coincide with the maximal subextension. Second, we proved that for functions in $ \mathcal{E}_m(\Omega) $, the maximal subextension relates to the original function through an $ m $-maximal function and preserves Lelong numbers. Third, we demonstrated convergence stability, proving that for sequences with boundary values in $ \mathcal{F}_m(\Omega, f) $ converging in $ L^1_{\text{loc}} $ and uniformly bounded below, their maximal subextensions converged to the maximal subextension of the limit function.

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