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

Ryan Gibara, Ilmari Kangasniemi, Nageswari Shanmugalingam

TL;DR

This work analyzes when the homogeneous Dirichlet space $D^{1,p}(X)$, defined via upper gradients, coincides with the tame Newton-Sobolev class $N^{1,p}(X)+\mathbb{R}$ on unbounded metric measure spaces with uniformly locally $p$-controlled geometry. It develops a unifying framework based on $p$-modulus, MEC$_p$ property, and uniformization to establish broad non-coincidence results (including globally doubling and Ahlfors-regular settings) and to pinpoint a sharp dichotomy in the hyperbolic setting. A central highlight is the complete classification on standard hyperbolic space $\mathbb{H}^n$: $D^{1,p}(\mathbb{H}^n)=N^{1,p}(\mathbb{H}^n)+\mathbb{R}$ if and only if $1\le p\le n-1$, with counterexamples when $p>n-1$. The paper also develops trace theory and end-analytic characterizations that identify when a $D^{1,p}$ function actually lies in $N^{1,p}$ via behavior along curves to infinity, providing practical criteria for boundary data at infinity.

Abstract

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 $μ(X) = \infty$ and $0 < μ(B(x, r)) < \infty$ 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.

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

TL;DR

This work analyzes when the homogeneous Dirichlet space , defined via upper gradients, coincides with the tame Newton-Sobolev class on unbounded metric measure spaces with uniformly locally -controlled geometry. It develops a unifying framework based on -modulus, MEC property, and uniformization to establish broad non-coincidence results (including globally doubling and Ahlfors-regular settings) and to pinpoint a sharp dichotomy in the hyperbolic setting. A central highlight is the complete classification on standard hyperbolic space : if and only if , with counterexamples when . The paper also develops trace theory and end-analytic characterizations that identify when a function actually lies in via behavior along curves to infinity, providing practical criteria for boundary data at infinity.

Abstract

We study the large-scale behavior of Newton-Sobolev functions on complete, connected, proper, separable metric measure spaces equipped with a Borel measure with and for all and Our objective is to understand the relationship between the Dirichlet space , defined using upper gradients, and the Newton-Sobolev space , for . We show that when is of uniformly locally -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 with , these two spaces coincide precisely when . We also provide additional characterizations of when a function in is in in the case that the two spaces do not coincide.
Paper Structure (20 sections, 48 theorems, 202 equations)

This paper contains 20 sections, 48 theorems, 202 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Geometry and potential theory related to curves in $X$
  4. Newton-Sobolev and Dirichlet classes of functions in nonsmooth settings
  5. Large-scale geometry and potential theory of $X$
  6. Gromov hyperbolicity, uniformization
  7. The case of globally doubling measures
  8. Sobolev-Poincaré inequalities and unbounded Ahlfors regular spaces
  9. The case of Gromov hyperbolic spaces with uniformly locally $p$-controlled geometry
  10. Gromov hyperbolic spaces $X$ with at least two points in $\partial_\varepsilon X$.
  11. $p$-parabolic Gromov hyperbolic spaces
  12. Results based on ends at infinity
  13. Identifying functions in Dirichlet spaces that belong to $N^{1,p}(X)$ via behavior along paths
  14. The case of the standard hyperbolic space
  15. The polar coordinate model of the hyperbolic space
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $1\le p<\infty$ and let $(X,d,\mu)$ be a complete, connected, proper, separable metric measure space of uniformly locally $p$-controlled geometry with $\mu(X)=\infty$ and $0 < \mu (B(x, r)) < \infty$ for all $x \in X, r \in (0, \infty)$. Then $D^{1,p}(X)\ne N^{1,p}(X)+{\mathbb R}$ if any one of One the other hand, the standard hyperbolic space $X=\mathbb{H}^n$, which is a Gromov hyperbolic sp

Theorems & Definitions (112)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Lemma 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.6
  • Definition 2.7
  • Definition 2.10
  • Remark 2.11
  • ...and 102 more