A pointwise characterization of functions of bounded variation on metric spaces
This paper gives a characterization of the space of functions of bounded variation in terms of a pointwise inequality connected to the maximal function of a measure. The result hold in doubling metric measure spaces supporting a 1-Poincaré inequality. In this setting, functions of bounded variation can be defined via total variation \[ \|Du\|(X)=\inf\left\{\liminf_{i\rightarrow\infty} \int_{X}\text{Lip}\, u_{i}\,d\mu\,:\, u_{i}\in \text{Lip}_{loc}(X),u_{i}\rightarrow u\text{ in } L^{1}_{loc}(X)\right\}, \] where \(\text{Lip}\, u\) is a local Lipschitz constant of \(u\).
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