“…While in reprint [36], the authors adopted this approach on a closed (compact without boundary) pointed space with a non-negative Bakry-Émery Ricci tensor.Recently, many interesting papers have been written related to symmetry, molecular cluster geometry analysis, submanifold theory, singularity theory, eigen problems, etc. [37][38][39]. In our following works, we are going to study inequalities of the Hardy type for different queries and further improve the results in this paper, combined with the technics and results in [37][38][39].…”