Fine properties of metric space-valued mappings of bounded variation in metric measure spaces

Abstract

Here we consider two notions of mappings of bounded variation (BV) from the metric measure space into the metric space; one based on relaxations of Newton-Sobolev functions, and the other based on a notion of AM-upper gradients. We show that when the target metric space is a Banach space, these two notions coincide with comparable energies, but for more general target metric spaces, the two notions can give different function-classes. We then consider the fine properties of BV mappings (based on the AM-upper gradient property), and show that when the target space is a proper metric space, then for a BV mapping into the target space, co-dimension $1$-almost every point in the jump set of a BV mapping into the proper space has at least two, and at most $k_0$, number of jump values associated with it, and that the preimage of balls around these jump values have lower density at least $\gamma$ at that point. Here $k_0$ and $\gamma$ depend solely on the structural constants associated with the metric measure space, and jump points are points at which the map is not approximately continuous.