تقرير
On homogeneous Newton-Sobolev spaces of functions in metric measure spaces of uniformly locally controlled geometry
| العنوان: | On homogeneous Newton-Sobolev spaces of functions in metric measure spaces of uniformly locally controlled geometry |
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| المؤلفون: | Gibara, Ryan, Kangasniemi, Ilmari, Shanmugalingam, Nageswari |
| سنة النشر: | 2024 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Metric Geometry, Mathematics - Analysis of PDEs, Mathematics - Functional Analysis, Primary: 46E36. Secondary: 31E05, 30L15, 53C23 |
| الوصف: | 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. Comment: 57 pages |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2407.18315 |
| رقم الأكسشن: | edsarx.2407.18315 |
| قاعدة البيانات: | arXiv |
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