Journal of Mathematical Physics, Analysis, Geometry
2022, vol. 18, No 1, pp. 118-135     ( to contents , go back )

A Weak Solution to the Complex Hessian Equation Associated to an m-Positive Closed Current

Jawhar Hbil

Department of Mathematics, College of science, Jouf University, P.O. Box 2014, Sakaka, Saudi Arabia

Mohamed Zaway

Department of Mathematics, Faculty of Sciences and Humanities in Ad-Dawadmi, Shaqra University, 11911, Saudi Arabia
Irescomath Laboratory, Gabes University, 6072 Zrig Gabes, Tunisia
E-mail: m

Received October 15, 2020, revised December 21, 2020.


The aim of this paper is to study the existence of a solution to the complex Hessian equation associated to an $m$-positive closed current $T$. We give a suffcient condition on $T$ and the measure $\mu$ so that the equation $T\wedge\beta^{n-m}\wedge (dd^c .)^{m-p}=\mu$ has a solution on the set of $m$-subharmonic functions. For this we establish a connection between the convergence in $cap_{m,T}$ of a sequence of $m$-subharmonic functions and the weak convergence of the associated Hessian measure.

Mathematics Subject Classification 2010: 32U40; 32U05; 32U20
Key words: m-positive closed current, m-subharmonic function, Capacity, Hessian operator

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