We study the large-scale behavior of Newton-Sobolev functions on complete, connected, proper, separable metric measure spaces equipped with a Borel measure \(\mu \) with \(\mu (X) = \infty \) and \(0< \mu (B(x, r)) < \infty \) for all \(x \in X\) and \(r \in (0, \infty )\) . Our objective is to understand the relationship between the Dirichlet space \(D^{1,p}(X)\) , defined using upper gradients, and the Newton-Sobolev space \(N^{1,p}(X)+\mathbb {R}\) , for \(1\le p<\infty \) . 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 \(\mathbb {H}^n\) with \(n\ge 2\) , these two spaces coincide precisely when \(1\le p\le n-1\) . We also provide additional characterizations of when a function in \(D^{1,p}(X)\) is in \(N^{1,p}(X)+\mathbb {R}\) in the case that the two spaces do not coincide.
