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Weak Solutions to the Complex Hessian Type Equations for Arbitrary Measures

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Abstract

We study the complex equations of Hessian type

$$\begin{aligned} (dd^c u)^m\wedge \beta ^{n-m}=F(u,.)d\mu , \end{aligned}$$

where \(\mu \) is a positive Borel measure defined on an m-hyperconvex domain of \({\mathbb {C}}^{n}\), m is an integer such that \(1\le m\le n\) and \(\beta :=dd^{c}\vert z\vert ^{2}\) is the standard kähler form in \({\mathbb {C}}^{n}. \) We show that, under some regularity conditions on the density F, if this equation admits a (weak) subsolution in \(\Omega \), then it admits a (weak) solution with a prescribed least maximal m-subharmonic majorant in \(\Omega \).

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Acknowledgements

We would like to thank Ahmed Zeriahi, for very useful discussions and comments. We would also like to thank the referees for their careful reading as well as for their comments and suggestions that have contributed to improving the readability and quality 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. Faculty of sciences, Department of mathematics, Ibn tofail university, Kenitra, Morocco

    Saïd Asserda & Ayoub El Gasmi

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  1. Hichame Amal
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  2. Saïd Asserda
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  3. Ayoub El Gasmi
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Correspondence to Ayoub El Gasmi.

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Communicated by Ronen Peretz.

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Amal, H., Asserda, S. & El Gasmi, A. Weak Solutions to the Complex Hessian Type Equations for Arbitrary Measures. Complex Anal. Oper. Theory 14, 80 (2020). https://doi.org/10.1007/s11785-020-01044-9

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