Maximal subextension of $m$-subharmonic functions

Abstract

In this paper, we prove that given a quasi-$m$-hyperconvex domain $\hat{I}\text{c◯}\subset X$ in a compact Kähler manifold $(X,\ddot{I}‰)$, and a function $\ddot{I}†$ in the weighted energy class ${\mathcal{E}}_{\ddot{I}}{‡}^m(\hat{I}\text{c◯},\ddot{I}‰)$ with respect to a convex weight function $\ddot{I}‡:\mathbb{R}\to \mathbb{R}$, then there exists a maximal $\ddot{I}‰$-$m$-subharmonic subextension $\tilde{\ddot{I}}†$ to $X$ that preserves the weighted energy and satisfies a good control properties for its Hessian measure $1_{\hat{I}}\text{c◯}{H}_m(\tilde{\ddot{I}}†)\le 1_{\hat{I}}\text{c◯}{H}_m(\ddot{I}†)$. In the last part, we study the particular case where $(X,\ddot{I}‰)=({\mathbb{P}}^n,\ddot{I}{‰}_{FS}).$