Mixed Hessian inequalities and uniqueness in the class $\mathcal{E}(X,ω,m)$

We prove a general inequality for mixed Hessian measures by global arguments. Our method also yields a simplification for the case of complex Monge-Ampère equation. Exploiting this and using Kołodziej's mass concentration technique we also prove the uniqueness of the solutions to the complex Hessian equation on compact Kähler manifolds in the case of probability measures vanishing on $m$-polar sets.

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