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

2024/07/25 by Ryan Gibara, Gibara, Ryan, Ilmari Kangasniemi +3 · 1 citation
Engineering · Mathematics · #30L15 #53C23 #Advanced Numerical Analysis Techniques #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Approximation and Integration #Metric Geometry (math.MG) #Primary: 46E36. Secondary: 31E05 #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2407.18315

openalex publication_date 2024/07/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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) = ∞ and 0 < μ(B(x, r)) < ∞ for all x ∈ X and r ∈ (0, ∞) 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)+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 D^1,p(X) is in N^1,p(X)+ℝ in the case that the two spaces do not coincide.

Cited by

• Warped products, solid hyperbolic fillings, and the identity D^1,p = N^1,p + ℝ

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