Generalization of Finite Entropy Measures in Kähler Geometry

Abstract: In this paper, we extend the concept of finite entropy measures in K\"ahler geometry. We define the finite $p$-entropy related to $\omega$-plurisubharmonic functions and demonstrate their inclusion in an appropriate energy class. Our study is anchored in the analysis of finite entropy measures on compact K\"ahler manifolds, drawing inspiration from fundamental works of Di Nezza, Guedj, and Lu. Utilizing a celebrated result by Darvas on the existence of a Finsler metric on the energy classes, we conclude this paper with a stability result for the complex Monge-Amp`ere equation.