Ryan Gibara, Ilmari Kangasniemi, Nageswari Shanmugalingam
{"title":"On homogeneous Newton-Sobolev spaces of functions in metric measure spaces of uniformly locally controlled geometry","authors":"Ryan Gibara, Ilmari Kangasniemi, Nageswari Shanmugalingam","doi":"arxiv-2407.18315","DOIUrl":null,"url":null,"abstract":"We study the large-scale behavior of Newton-Sobolev functions on complete,\nconnected, proper, separable metric measure spaces equipped with a Borel\nmeasure $\\mu$ with $\\mu(X) = \\infty$ and $0 < \\mu(B(x, r)) < \\infty$ for all $x\n\\in X$ and $r \\in (0, \\infty)$ Our objective is to understand the relationship\nbetween the Dirichlet space $D^{1,p}(X)$, defined using upper gradients, and\nthe Newton-Sobolev space $N^{1,p}(X)+\\mathbb{R}$, for $1\\le p<\\infty$. We show\nthat when $X$ is of uniformly locally $p$-controlled geometry, these two spaces\ndo not coincide under a wide variety of geometric and potential theoretic\nconditions. We also show that when the metric measure space is the standard\nhyperbolic space $\\mathbb{H}^n$ with $n\\ge 2$, these two spaces coincide\nprecisely when $1\\le p\\le n-1$. We also provide additional characterizations of\nwhen a function in $D^{1,p}(X)$ is in $N^{1,p}(X)+\\mathbb{R}$ in the case that\nthe two spaces do not coincide.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":"6 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Metric Geometry","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2407.18315","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
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.