Large-scale behaviour of Sobolev functions in Ahlfors regular metric measure spaces
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| Format: | Preprint |
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2023
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| author | Kline, Josh Koskela, Pekka Nguyen, Khanh |
| author_facet | Kline, Josh Koskela, Pekka Nguyen, Khanh |
| contents | In this paper, we study the behaviour at infinity of $p$-Sobolev functions in the setting of Ahlfors $Q$-regular metric measure spaces supporting a $p$-Poincaré inequality. By introducing the notions of sets which are $p$-thin at infinity, we show that functions in the homogeneous space $\dot N^{1,p}(X)$ necessarily have limits at infinity outside of $p$-thin sets, when $1\le p<Q<+\infty$. When $p>Q$, we show by example that uniqueness of limits at infinity may fail for functions in $\dot N^{1,p}(X)$. While functions in $\dot N^{1,p}(X)$ may not have any reasonable limit at infinity when $p=Q$, we introduce the notion of a $Q$-thick set at infinity, and characterize the limits of functions in $\dot N^{1,Q}(X)$ along infinite curves in terms of limits outside $Q$-thin sets and along $Q$-thick sets. By weakening the notion of a thick set, we show that a function in $\dot N^{1,Q}(X)$ with a limit along such an almost thick set may fail to have a limit along any infinite curve. While homogeneous $p$-Sobolev functions may have infinite limits at infinity when $p\ge Q$, we provide bounds on how quickly such functions may grow: when $p=Q$, functions in $\dot N^{1,p}(X)$ have sub-logarithmic growth at infinity, whereas when $p>Q$, such functions have growth at infinity controlled by $d(\cdot, O)^{1-Q/p}$, where $O$ is a fixed base point in $X$. For the inhomogeneous spaces $N^{1,p}(X)$, the phenomenon is different. We show that for $1\le p\le Q$, the limit of a function $u\in N^{1,p}(X)$ is zero outside of a $p$-thin set, whereas $\lim_{x\to+\infty}u(x)=0$ for all $u\in N^{1,p}(X)$ when $p>Q$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_11718 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Large-scale behaviour of Sobolev functions in Ahlfors regular metric measure spaces Kline, Josh Koskela, Pekka Nguyen, Khanh Functional Analysis Analysis of PDEs Metric Geometry 46E36, 31B15, 31C15, 31B25 In this paper, we study the behaviour at infinity of $p$-Sobolev functions in the setting of Ahlfors $Q$-regular metric measure spaces supporting a $p$-Poincaré inequality. By introducing the notions of sets which are $p$-thin at infinity, we show that functions in the homogeneous space $\dot N^{1,p}(X)$ necessarily have limits at infinity outside of $p$-thin sets, when $1\le p<Q<+\infty$. When $p>Q$, we show by example that uniqueness of limits at infinity may fail for functions in $\dot N^{1,p}(X)$. While functions in $\dot N^{1,p}(X)$ may not have any reasonable limit at infinity when $p=Q$, we introduce the notion of a $Q$-thick set at infinity, and characterize the limits of functions in $\dot N^{1,Q}(X)$ along infinite curves in terms of limits outside $Q$-thin sets and along $Q$-thick sets. By weakening the notion of a thick set, we show that a function in $\dot N^{1,Q}(X)$ with a limit along such an almost thick set may fail to have a limit along any infinite curve. While homogeneous $p$-Sobolev functions may have infinite limits at infinity when $p\ge Q$, we provide bounds on how quickly such functions may grow: when $p=Q$, functions in $\dot N^{1,p}(X)$ have sub-logarithmic growth at infinity, whereas when $p>Q$, such functions have growth at infinity controlled by $d(\cdot, O)^{1-Q/p}$, where $O$ is a fixed base point in $X$. For the inhomogeneous spaces $N^{1,p}(X)$, the phenomenon is different. We show that for $1\le p\le Q$, the limit of a function $u\in N^{1,p}(X)$ is zero outside of a $p$-thin set, whereas $\lim_{x\to+\infty}u(x)=0$ for all $u\in N^{1,p}(X)$ when $p>Q$. |
| title | Large-scale behaviour of Sobolev functions in Ahlfors regular metric measure spaces |
| topic | Functional Analysis Analysis of PDEs Metric Geometry 46E36, 31B15, 31C15, 31B25 |
| url | https://arxiv.org/abs/2310.11718 |