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 μ with μ(X) = ∞ and 0 < μ(B(x, r)) < ∞ for all x ∈ X and r ∈ (0, ∞) Our objective is to understand the relationship between the Dirichlet space D1,p(X), defined using upper gradients, and the Newton-Sobolev space N1,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 D1,p(X) is in N1,p(X)+R in the case that the two spaces do not coincide.