English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In this paper, we study the behaviour at infinity of $p$-Sobolev functions inthe setting of Ahlfors $Q$-regular metric measure spaces supporting a$p$-Poincar\'e inequality. By introducing the notions of sets which are$p$-thin at infinity, we show that functions in the homogeneous space $\dotN^{1,p}(X)$ 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 atinfinity may fail for functions in $\dot N^{1,p}(X)$. While functions in $\dotN^{1,p}(X)$ may not have any reasonable limit at infinity when $p=Q$, weintroduce the notion of a $Q$-thick set at infinity, and characterize thelimits of functions in $\dot N^{1,Q}(X)$ along infinite curves in terms oflimits outside $Q$-thin sets and along $Q$-thick sets. By weakening the notionof a thick set, we show that a function in $\dot N^{1,Q}(X)$ with a limit alongsuch an almost thick set may fail to have a limit along any infinite curve.While homogeneous $p$-Sobolev functions may have infinite limits at infinitywhen $p\ge Q$, 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 theinhomogeneous spaces $N^{1,p}(X)$, the phenomenon is different. We show thatfor $1\le p\le Q$, the limit of a function $u\in N^{1,p}(X)$ is zero outside ofa $p$-thin set, whereas $\lim_{x\to+\infty}u(x)=0$ for all $u\in N^{1,p}(X)$when $p>Q$.

Large-scale behaviour of Sobolev functions in Ahlfors regular metric measure spaces