On the notion of Laplacian bounds on $\mathrm{R}\mathrm{C}\mathrm{D}$ spaces and applications

We show that several different interpretations of the inequality $\mathrm{\Delta }f\le \eta$ are equivalent in the setting of $\mathrm{R}\mathrm{C}\mathrm{D}(K,N)$ spaces. With respect to previously available results in this direction, we improve both on the generality of the underlying space and in terms of regularity to be assumed on the function $f$. Applications are presented.