Hessian operator on weighted m-subharmonic classes

Jawhar Hbil; Hadi Obaid Alshammari; Mohamed Zaway; Jawhar Hbil; Hadi Obaid Alshammari; Mohamed Zaway

doi:10.3934/math.2026678

AIMS Mathematics

2026, Volume 11, Issue 6: 16534-16549. doi: 10.3934/math.2026678

Previous Article Next Article

Research article

Hessian operator on weighted m-subharmonic classes

1.
Department of Mathematics, Jouf University, P.O. Box: 2014, Sakaka, Saudi Arabia
2.
Irescomath Laboratory, Gabes University, Zrig Gabes 6072, Tunisia

Received: 05 February 2026 Revised: 12 May 2026 Accepted: 19 May 2026 Published: 09 June 2026
MSC : 32W20, 32U05, 32U15, 32U40

This paper studied the characterization and existence of solutions to complex Hessian equations associated with a given weight. We provided a complete characterization of the Radon measures that could be represented as the complex Hessian measure of an $ \mathcal{F}_{m, \chi}(\Omega) $-function, where $ \chi $ was a decreasing weight function. Our main results provided both global and local characterizations of the range of the complex Hessian operator acting on these classes. Specifically, we demonstrated that the solvability of the equation $ H_m(\varphi) = \mu $ in the class $ \mathcal{F}_{m, \chi}(\Omega) $ is equivalent to a certain functional inequality involving the measure $ \mu $ and the weighted energy $ \delta_{m, \chi} $. Furthermore, we demonstrate that this global condition could be localized: a solution existed globally if, and only if, for every point in the closure $ \overline{\Omega} $ of the domain, a solution existed in some neighborhood of that point.
- Hessian operator,
- complex $ m- $subharmonic function,
- $ m- $polar sets,
- capacity,
- Hessian equations
Citation: Jawhar Hbil, Hadi Obaid Alshammari, Mohamed Zaway. Hessian operator on weighted m-subharmonic classes[J]. AIMS Mathematics, 2026, 11(6): 16534-16549. doi: 10.3934/math.2026678

Related Papers:

Abstract

This paper studied the characterization and existence of solutions to complex Hessian equations associated with a given weight. We provided a complete characterization of the Radon measures that could be represented as the complex Hessian measure of an $ \mathcal{F}_{m, \chi}(\Omega) $-function, where $ \chi $ was a decreasing weight function. Our main results provided both global and local characterizations of the range of the complex Hessian operator acting on these classes. Specifically, we demonstrated that the solvability of the equation $ H_m(\varphi) = \mu $ in the class $ \mathcal{F}_{m, \chi}(\Omega) $ is equivalent to a certain functional inequality involving the measure $ \mu $ and the weighted energy $ \delta_{m, \chi} $. Furthermore, we demonstrate that this global condition could be localized: a solution existed globally if, and only if, for every point in the closure $ \overline{\Omega} $ of the domain, a solution existed in some neighborhood of that point.

References

[1]	E. Bedford, B. A. Taylor, A new capacity for plurisubharmonic functions, Acta Math., 149 (1982), 1–40. https://doi.org/10.1007/BF02392348 doi: 10.1007/BF02392348
[2]	S. Benelkourchi, Weighted pluricomplex energy, Potential Anal., 31 (2009), 1–20. https://doi.org/10.1007/s11118-009-9119-7 doi: 10.1007/s11118-009-9119-7
[3]	S. Benelkourchi, Weighted pluricomplex energy Ⅱ, Int. J. Partial Differ. Equ., 2015 (2015), 947819. https://doi.org/10.1155/2015/947819 doi: 10.1155/2015/947819
[4]	S. Benelkourchi, V. Guedj, A. Zeriahi, Plurisubharmonic functions with weak singularities, Proceedings from the Kiselmanfest, Uppsala University, Vastra Aros, 2009, 57–74.
[5]	Z. Blocki, Weak solutions to the complex Hessian equation, Ann. Inst. Fourier, 55 (2005), 1735–1756. https://doi.org/10.5802/aif.2137 doi: 10.5802/aif.2137
[6]	U. Cegrell, Pluricomplex energy, Acta Math., 180 (1998), 187–217. https://doi.org/10.1007/BF02392899 doi: 10.1007/BF02392899
[7]	U. Cegrell, The general definition of the complex Monge–Ampère operator, Ann. Inst. Fourier, 54 (2004), 159–179. https://doi.org/10.5802/aif.2014 doi: 10.5802/aif.2014
[8]	R. Czyż, On a Monge–Ampère type equation in the Cegrell class $E_{\chi}$, Ann. Polonici Math., 99 (2010), 89–97.
[9]	A. El Gasmi, The Dirichlet problem for the complex Hessian operator in the class $\mathcal{N}_m(H)$, Math. Scand., 127 (2021), 287–316. https://doi.org/10.7146/math.scand.a-125994 doi: 10.7146/math.scand.a-125994
[10]	L. M. Hai, P. H. Hiep, Some weighted energy classes of plurisubharmonic functions, Potential Anal., 34 (2011), 43–56. https://doi.org/10.1007/s11118-010-9179-8 doi: 10.1007/s11118-010-9179-8
[11]	L. M. Hai, P. H. Hiep, N. X. Hong, N. V. Phu, The Monge–Ampère type equation in the weighted pluricomplex energy class, Int. J. Math., 25 (2014), 1450042. https://doi.org/10.1142/S0129167X14500426 doi: 10.1142/S0129167X14500426
[12]	L. M. Hai, V. V. Quan, Weak solutions to the complex $m$-Hessian equation on open subsets of $\mathbb{C}^n$, Complex Anal. Oper. Theory, 13 (2019), 4007–4025. https://doi.org/10.1007/s11785-019-00948-5 doi: 10.1007/s11785-019-00948-5
[13]	L. M. Hai, T. Van Dung, Subextension of $m$-subharmonic functions, Vietnam J. Math., 48 (2020), 47–57. https://doi.org/10.1007/s10013-019-00343-9 doi: 10.1007/s10013-019-00343-9
[14]	J. Hbil, Range of the Hessian operator on some weighted $m$-subharmonic classes, Bull. Inst. Math. Acad. Sin., 19 (2024), 15–28. https://doi.org/10.21915/BIMAS.2024102 doi: 10.21915/BIMAS.2024102
[15]	P. H. Hiep, Pluripolar sets and the subextension in Cegrell's classes, Complex Var. Elliptic Equ., 53 (2008), 675–684. https://doi.org/10.1080/17476930801966893 doi: 10.1080/17476930801966893
[16]	V. V. Hung, Local property of a class of $m$-subharmonic functions, Vietnam J. Math., 44 (2016), 603–621. https://doi.org/10.1007/s10013-015-0176-5 doi: 10.1007/s10013-015-0176-5
[17]	V. V. Hung, N. V. Phu, Hessian measures on $m$-polar sets and applications to the complex Hessian equations, Complex Var. Elliptic Equ., 62 (2017), 1135–1164. https://doi.org/10.1080/17476933.2016.1273907 doi: 10.1080/17476933.2016.1273907
[18]	H. H. Lu, A variational approach to complex Hessian equations in $\mathbb{C}^n$, J. Math. Anal. Appl., 431 (2015), 228–259. https://doi.org/10.1016/j.jmaa.2015.05.067 doi: 10.1016/j.jmaa.2015.05.067
[19]	H. N. Quy, The local properties of some subclasses of plurisubharmonic, Indian J. Pure Appl. Math., 55 (2024), 419–424. https://doi.org/10.1007/s13226-023-00402-5 doi: 10.1007/s13226-023-00402-5
[20]	H. N. Quy, The complex Monge-Ampère equation in the Cegrell's classes, Results Math., 80 (2025), 14. https://doi.org/10.1007/s00025-024-02335-9 doi: 10.1007/s00025-024-02335-9
[21]	M. Zaway, J. Hbil, Complex Hessian-type equations in the weighted $m$-subharmonic class, Ukr. Math. J., 75 (2023), 921–935. https://doi.org/10.1007/s11253-023-02237-z doi: 10.1007/s11253-023-02237-z
[22]	M. Zaway, J. Hbil, On some Weighted classes of $m$-subharmonic functions, J. Math. Phys., Anal., Geom., 20 (2024), 112–133. https://doi.org/10.15407/mag20.01.112 doi: 10.15407/mag20.01.112

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)