Complex Hessian-type equations in the weighted $m$-subharmonic class

  • Mohamed Zaway Department of Mathematics, Faculty of Sciences of Sfax, University of Sfax, Tunisia and Irescomath Laboratory, Gabes University, Zrig Gabes, Tunisia
  • Jawhar Hbil Department of Mathematics, Jouf University, Sakaka, Saudi Arabia and Irescomath Laboratory, Gabes University, Zrig Gabes, Tunisia
Keywords: m-subharmonic function, Capacity, Hessian operator.


UDC 517.5

We study the existence of a solution to a general type of complex Hessian equation on some Cegrell classes. For a given measure $\mu$ defined on an $m$-hyperconvex domain  $\Omega \subset \mathbb{C}^n,$  under  suitable conditions, we prove that the equation $\chi(.) H_m(.)=\mu$ has a solution that belongs to the class $\mathcal{E}_{m,\chi}(\Omega).$


H. Amal, S. Asserda, A. El Gasmi, Weak solutions to the complex Hessian type equations for arbitrary measures, Complex Anal. and Oper. Theory, 14, (2020). DOI:

S. Benelkourchi, V. Guedj, A. Zeriahi, Plurisubharmonic functions with weak singularities, Complex Analysis, Digital Geometry, Proc. Kiselmanfest, Uppsala Univ. (2007), p. 5773.

Z. Błocki, Weak solutions to the complex Hessian equation, Ann. Inst. Fourier (Grenoble), 55, № 5, 1735–1756 (2005). DOI:

U. Cegrell, Pluricomplex energy, Acta Math., 180, 187–217 (1998). DOI:

U. Cegrell, The general definition of the comlex Monge–Amp`ere operator, Ann. Inst. Fourier (Grenoble), 54, 159–179 (2004). DOI:

D. T. Chuyen, V. T. Thanh, H. T. Lam, D. T. Duong, Some relation and comparison principle between the classes $mathcal{F}_{m,chi},$ $mathcal{E}_m$ and $mathcal{N}_m,$} Tap chi Khoa hoc – Dai hoc Tay bac, 20, 95–103 (2020).

R. Czyz, On a Monge–Amp`ere type equation in the Cegrell class $mathcal{E}_{chi}$, Ann. Polon. Math., 99, 89–97 (2010). DOI:

A. El Gasmi, The Dirichlet problem for the complex Hessian operator in the class $N_m(Omega,f)$, Math. Scand., 127, 287–316 (2021). DOI:

L. M. Hai, P. H. Hiep, N. X. Hong, N. V. Phu, The Monge–Amp'ere type equation in the weighted pluricomplex energy class, Int. J. Math., 25, № 5, Article 1450042 (2014). DOI:

L. M. Hai, V. Van Quan, Weak solutions to the complex $m$-Hessian equation on open subsets of $mathbb{C}^n$, Complex Anal. and Oper. Theory, 13, 4007–4025 (2019). DOI:

J. Hbil, M. Zaway, Some results on complex $m$-subharmonic classes}; ArXiv:2201.06851.

V. V. Hung, Local property of a class of m-subharmonic functions, Vietnam J. Math., 44, № 3, 621–630 (2016). DOI:

V. V. Hung, N. V. Phu, Hessian measures on $m$-polar sets and applications to the complex Hessian equations, Complex Var. and Elliptic Equat., 8, 1135–1164 (2017). DOI:

C. H. Lu, A variational approach to complex Hessian equations in $C^n$, J. Math. Anal. and Appl., 431, № 1, 228–259 (2015). DOI:

C. H. Lu, Equations Hessiennes complexes, Ph. D. Thesis, Univ. Paul Sabatier, Toulouse, France (2012);

A. S. Sadullaev, B. I. Abdullaev, Potential theory in the class of $m$-subharmonic functions, Tr. Mat. Inst. Steklova, 279, 166–192 (2012). DOI:

N. V. Thien, Maximal $m$-subharmonic functions and the Cegrell class $mathcal{N}_m,$} Indag. Math., 30, 717–739 (2019).

How to Cite
Zaway, M., and J. Hbil. “Complex Hessian-Type Equations in the Weighted $m$-Subharmonic Class”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 75, no. 6, June 2023, pp. 805 -16, doi:10.37863/umzh.v75i6.7122.
Research articles