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

TL;DR

This paper investigates the relationship between Dirichlet and Newton-Sobolev spaces on certain metric measure spaces, revealing conditions under which they coincide or differ, especially in hyperbolic spaces.

Contribution

It characterizes when Newton-Sobolev and Dirichlet spaces coincide or differ on spaces with controlled geometry, including hyperbolic spaces, under various conditions.

Findings

Spaces do not coincide under broad geometric conditions.

In hyperbolic space, spaces coincide if and only if 1 ≤ p ≤ n-1.

Provides criteria for membership in Newton-Sobolev spaces.

Abstract

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.