Journal of Mathematical Physics, Analysis, Geometry
2022, vol. 18, No 1, pp. 118-135   https://doi.org/10.15407/mag18.01.118     ( to contents , go back )
https://doi.org/10.15407/mag18.01.118

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
E-mail: jmhbil@ju.edu.sa

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 zaway@su.edu.sa

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

Abstract

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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