Some results on the convergence of Hessian operator and $ m-$subharmonic functions

Jawhar Hbil; Mohamed Zaway; Jawhar Hbil; Mohamed Zaway

doi:10.3934/math.2022502

AIMS Mathematics

2022, Volume 7, Issue 5: 9023-9038. doi: 10.3934/math.2022502

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Research article

Some results on the convergence of Hessian operator and $ m-$subharmonic functions

Jawhar Hbil ^{1
,
,},
Mohamed Zaway ²

1.
Department of Mathematics, College of science, Jouf University, P.O. Box 2014, Sakaka, Saudi Arabia
2.
Department of mathematics, College of science, Shaqra University, P.O. Box 1040, Ad-Dwadimi 1191, Saudi Arabia

Received: 19 October 2021 Revised: 17 February 2022 Accepted: 28 February 2022 Published: 08 March 2022
MSC : 32U40, 32U05, 32U20

In this paper we treat the problem of connection between the convergence in $ m- $capacity and the convergence of the Hessian measure for a sequence$ f_j $ of $ m- $subharmonic functions. We prove first that, under some conditions, the convergence of $ f_j $ in capacity $ Cap_m $ implies the weak convergence of the Hessian measures $ H_m(f_j) $. Then we show that the converse sense of convergence is also true in some particular cases.
- $ m- $subharmonic function,
- $ m- $Capacity,
- Hessian operator
Citation: Jawhar Hbil, Mohamed Zaway. Some results on the convergence of Hessian operator and $ m-$subharmonic functions[J]. AIMS Mathematics, 2022, 7(5): 9023-9038. doi: 10.3934/math.2022502

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Abstract

In this paper we treat the problem of connection between the convergence in $ m- $capacity and the convergence of the Hessian measure for a sequence$ f_j $ of $ m- $subharmonic functions. We prove first that, under some conditions, the convergence of $ f_j $ in capacity $ Cap_m $ implies the weak convergence of the Hessian measures $ H_m(f_j) $. Then we show that the converse sense of convergence is also true in some particular cases.

References

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