m-Generalized Lelong Numbers and Capacity Associated to a Class of m-Positive Closed Currents

Elkhadhra, Fredj

doi:10.1007/s00025-018-0933-3

Cited by 5 publications

(3 citation statements)

References 10 publications

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“…Thanks to the argument before Theorem 4, the m-superHessian operator T ∧β n−m ∧(dd # ϕ) m+p−n is a well defined positive measure provided that ϕ is m-convex and bounded near ∂Ω ∩ Supp T . Hence, inspired by the work of Elkhadhra [7], we state the following definition:…”

Section: Proof (1) Using Induction It Suffices To Prove Thatmentioning

confidence: 99%

“…Observe that ν m T (ϕ) measure the asymptotic behaviour of the current T ∧β n−m ∧(dd # ϕ) m+p−n near the m-polar set {ϕ = −∞}. Moreover, this notion is a real Hessian version of the definition of generalized Lelong number given by Elkhadhra [7] in the complex Hessian setting. Proposition 6.…”

Section: Proof (1) Using Induction It Suffices To Prove Thatmentioning

confidence: 99%

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$m$-potential theory and $m$-generalized Lelong numbers associated with $m$-positive supercurrents

Elkhadhra¹,

Zahmoul²

2020

Preprint

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In this study, we first define the local potential associated to a positive closed supercurrent in analogy to the one investigated by Ben Massaoud and El Mir. Next, we study the definition and the continuity of the m-superHessian operator for unbounded m-convex functions. As an application, we generalize our work on Demailly-Lelong numbers and several related results in the superformalism setting. Furthermore, strongly inspired by the complex hessian theory, we introduce the Cegrell-type classes as well as a generalization of some m-potential results in the class of m-convex functions.

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Section: Proof (1) Using Induction It Suffices To Prove Thatmentioning

confidence: 99%

Section: Proof (1) Using Induction It Suffices To Prove Thatmentioning

confidence: 99%

$m$-potential theory and $m$-generalized Lelong numbers associated with $m$-positive supercurrents

Elkhadhra¹,

Zahmoul²

2020

Preprint

Self Cite

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show abstract

“…Pluripotential theory for m-subharmonic functions developed rapidly in last two decades, and there are vast literatures (cf. [1,2,8,10,12,13,15,17,19,20,22,23,25,26,30] and references therein).…”

Section: Introductionmentioning

confidence: 99%

On quaternionic pluripotential theory associated to quaternionic $m$-subharmonic functions

Liu¹,

Wang²

2022

Preprint

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Many aspects of pluripotential theory are generalized to quaternionic m-subharmonic functions. We introduce quaternionic version of notions of the m-Hessian operator, m-subharmonic functions, m-Hessian measure, m-capapcity, the relative m-extremal function and the m-Lelong number, and show various propositions for them, based on d 0 and d 1 operators, the quaternionic counterpart of ∂ and ∂, and quaternionic closed positve currents. The definition of quaternionic m-Hessian operator can be extended to locally bounded quaternionic m-subharmonic functions and the corresponding convergence theorem is proved. The comparison principle and the quasicontinuity of quaternionic m-subharmonic functions are established. We also find the fundamental solution of the quaternionic m-Hessian operator.

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Complex Hessian Operator Associated to an m-Positive Closed Current and Weighted m-Capacity

Elaini

Elkhadhra

2021

Complex Anal. Oper. Theory

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m-Generalized Lelong Numbers and Capacity Associated to a Class of m-Positive Closed Currents

Cited by 5 publications

References 10 publications

$m$-potential theory and $m$-generalized Lelong numbers associated with $m$-positive supercurrents

$m$-potential theory and $m$-generalized Lelong numbers associated with $m$-positive supercurrents

On quaternionic pluripotential theory associated to quaternionic $m$-subharmonic functions

Complex Hessian Operator Associated to an m-Positive Closed Current and Weighted m-Capacity

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