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 $\mu$ with $\mu (X)=\mathrm{\infty }$ and $0<\mu (B(x,r))<\mathrm{\infty }$ for all $x\in X$ and $r\in (0,\mathrm{\infty })$ 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 $1\le p<\mathrm{\infty }$. 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.