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