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Continuous Solutions for Degenerate Complex Hessian Equation

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Abstract

Let (X,ω) be an n-dimensional compact Kähler manifold and fix an integer m such that 1 ≤ m ≤ n. Let μ be a finite Borel measure on X satisfying the conditions \({\mathscr{H}}_{m}(\delta , A,\omega )\). We study degenerate complex Hessian equations of the form (ω + ddcφ)m ∧ ωn−m = F(φ,.)dμ. Under some natural conditions on F, we prove that if \(0<\delta <\frac {m}{n-m}\), then this equation has a unique continuous solution.

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Acknowledgements

The authors would like to thank the referees very much for suggestions and valuable remarks which led to the improvement of the exposition of the paper.

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Authors and Affiliations

  1. Department of Mathematics, Laboratory LaREAMI, Regional Centre of Trades of Education and Training, Kenitra, Morocco

    Hichame Amal

  2. Ibn Tofail University, Faculty of Sciences, Department of Mathematics, PO 242, Kenitra, Morocco

    Saïd Asserda & Manar Bouhssina

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  1. Hichame Amal
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  2. Saïd Asserda
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  3. Manar Bouhssina
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Correspondence to Manar Bouhssina.

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Amal, H., Asserda, S. & Bouhssina, M. Continuous Solutions for Degenerate Complex Hessian Equation. Acta Math Vietnam 48, 371–386 (2023). https://doi.org/10.1007/s40306-023-00498-1

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