On the notion of Laplacian bounds on $\mathrm{RCD}$ spaces and applications

Abstract

We show that several different interpretations of the inequality $\Delta f\le \eta$ are equivalent in the setting of $\mathrm{RCD}(K,N)$ 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.

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