On homogeneous Newton-Sobolev spaces of functions in metric measure spaces of uniformly locally controlled geometry

We study the large-scale behavior of Newton-Sobolev functions on complete, connected, proper, separable metric measure spaces equipped with a Borel measure with (X) = and 0 < (B (x, r) ) < for all x X and r (0, ) Our objective is to understand the relationship between the Dirichlet space D^1, p (X), defined using upper gradients, and the Newton-Sobolev space N^1, p (X) +R, for 1 p<. We show that when X is of uniformly locally p-controlled geometry, these two spaces do not coincide under a wide variety of geometric and potential theoretic conditions. We also show that when the metric measure space is the standard hyperbolic space Hⁿ with n 2, these two spaces coincide precisely when 1 p n-1. We also provide additional characterizations of when a function in D^1, p (X) is in N^1, p (X) +R in the case that the two spaces do not coincide.