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

Abstract

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 $p>Q$.