Maximal subextension of $m$-subharmonic functions

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