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m-Potential Theory and m-Generalized Lelong Numbers Associated with m-Positive Supercurrents

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Abstract

In this study, we first define the local potential associated to a weakly positive closed supercurrent in analogy to the one investigated by Ben Messaoud and El Mir in the complex setting. Next, we study the definition and the continuity of the m-superHessian operator for unbounded m-convex functions. As an application, we generalize our previous 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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Acknowledgements

The second named author likes to express his gratitude towards Professor Jean-Pierre Demailly for his hospitality and for many stimulating discussions and comments during his visit to “Institut Fourier”. The authors would like to thank the referee for the helpful remarks and suggestions that enhanced the presentation of the paper.

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Authors and Affiliations

  1. Higher School of Sciences and Technology of Hammam Sousse, University of Sousse, MaPSFA LR11ES35, 4011, Hammam Sousse, Tunisia

    Fredj Elkhadhra & Khalil Zahmoul

Authors
  1. Fredj Elkhadhra
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  2. Khalil Zahmoul
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Correspondence to Khalil Zahmoul.

Additional information

Communicated by Irene Sabadini.

Dedicated to the late Professor Jean-Pierre Demailly.

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This article is part of the topical collection “Higher Dimensional Geometric Function Theory and Hypercomplex Analysis” edited by Irene Sabadini, Michael Shapiro and Daniele Struppa.

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Elkhadhra, F., Zahmoul, K. m-Potential Theory and m-Generalized Lelong Numbers Associated with m-Positive Supercurrents. Complex Anal. Oper. Theory 17, 16 (2023). https://doi.org/10.1007/s11785-022-01318-4

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