On the notion of Laplacian bounds on 𝑅𝐶𝐷 spaces and applications

Abstract

We show that several different interpretations of the inequality Δ f ≤ η \Delta f\leq \eta are equivalent in the setting of RCD ⁡ ( K , N ) \operatorname {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 f . Applications are presented.