On homogeneous Newton-Sobolev spaces of functions in metric measure spaces of uniformly locally controlled geometry
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908333663846400 |
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| author | Gibara, Ryan Kangasniemi, Ilmari Shanmugalingam, Nageswari |
| author_facet | Gibara, Ryan Kangasniemi, Ilmari Shanmugalingam, Nageswari |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_18315 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On homogeneous Newton-Sobolev spaces of functions in metric measure spaces of uniformly locally controlled geometry Gibara, Ryan Kangasniemi, Ilmari Shanmugalingam, Nageswari Metric Geometry Analysis of PDEs Functional Analysis Primary: 46E36. Secondary: 31E05, 30L15, 53C23 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. |
| title | On homogeneous Newton-Sobolev spaces of functions in metric measure spaces of uniformly locally controlled geometry |
| topic | Metric Geometry Analysis of PDEs Functional Analysis Primary: 46E36. Secondary: 31E05, 30L15, 53C23 |
| url | https://arxiv.org/abs/2407.18315 |