The Dirichlet problem for the complex Hessian operator in the class $\mathcal{N}_m(H)$

Ayoub El-Gasmi

The Dirichlet problem for the complex Hessian operator in the class $\mathcal{N}_m(H)$

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arxiv 1712.06911 v1 pith:2RJSLXAL submitted 2017-12-19 math.CV

Ayoub El-Gasmi This is my paper · ORCID

classification math.CV

keywords complexhessianmeasureclassmathcalacchborelcegrell

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abstract

We prove that, in a $m$-hyperconvex domain in $\mathbb{C}^{n},$ if a non-negative Borel measure is dominated by a complex Hessian measure, then it is a complex Hessian measure of a function in the class $\mathcal{N}_m(H)$. This is an extension of P. {\AA}hg, U. Cegrell, R. Czy\.z and P.H. Hiep's result in \cite{ACCH}.

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Complex Hessian equations with prescribed singularity on compact K\"ahler manifolds
math.CV 2019-09 conditional novelty 7.0 of 10

The total Hessian mass is monotone in singularity type, and Hessian equations H_m(u)=µ have unique solutions in prescribed singularity classes on compact Kähler manifolds.

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