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

TL;DR

This paper establishes the equivalence of various interpretations of Laplacian bounds on RCD spaces, extending previous results by broadening the setting and reducing regularity assumptions, with applications demonstrated.

Contribution

It proves the equivalence of different Laplacian bound interpretations on RCD spaces under more general conditions and lower regularity assumptions than prior work.

Findings

Multiple interpretations of Laplacian bounds are shown to be equivalent in RCD spaces.

The results apply to a broader class of spaces and functions with less regularity.

Applications of these equivalences are provided in the paper.

Abstract

We show that several different interpretations of the inequality $\Delta f\leq\eta$ 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.