Boundary points, minimal L² integrals and concavity property II: Weakly pseudoconvex Kähler manifolds

In this paper, we consider minimal L² integrals on the sublevel sets of plurisubharmonic functions on weakly pseudoconvex Kähler manifolds with Lebesgue measurable gain related to modules at boundary points of the sublevel sets, and establish a concavity property of the minimal L² integrals. As applications, we present a necessary condition for the concavity degenerating to linearity, a concavity property related to modules at inner points of the sublevel sets, an optimal support function related to modules, a strong openness property of modules and a twisted version, and an effectiveness result of the strong openness property of modules.