Existence and Uniqueness of Limits at Infinity for Bounded Variation Functions

Panu Lahti, Khanh Nguyen
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Abstract

In this paper, we study the existence of limits at infinity along almost every infinite curve for the upper and lower approximate limits of bounded variation functions on complete unbounded metric measure spaces. We prove that if the measure is doubling and supports a 1-Poincaré inequality, then for every bounded variation function f and for 1-a.e. infinite curve \(\gamma \), for both the upper approximate limit \(f^\vee \) and the lower approximate limit \(f^\wedge \) we have that

$$\begin{aligned} \lim _{t\rightarrow +\infty }f^\vee (\gamma (t)) \mathrm{\ \ and\ \ }\lim _{t\rightarrow +\infty }f^\wedge (\gamma (t)) \end{aligned}$$

exist and are equal to the same finite value. We give examples showing that the conditions of the doubling property of the measure and a 1-Poincaré inequality are needed for the existence of limits. Furthermore, we establish a characterization for strictly positive 1-modulus of the family of all infinite curves in terms of bounded variation functions. These generalize results for Sobolev functions given in Koskela and Nguyen (J Funct Anal 285(11):110154, 2023).

有界变分函数无穷大极限的存在性和唯一性
在本文中,我们研究了完全无界度量空间上有界变化函数的上近似极限和下近似极限沿几乎每条无限曲线的无穷大极限的存在性。我们证明,如果度量是加倍的,并且支持 1-Poincaré 不等式,那么对于每个有界变化函数 f 和 1-a.e. 无限曲线 \(\gamma \),对于上近似极限 \(f^\vee \)和下近似极限 \(f^\wedge \),我们都有 $$(开始{对齐})。\f^\vee (\gamma (t))\lim _{t\rightarrow +\infty }f^\wedge (\gamma (t))\end{aligned}$$存在并且等于同一个有限值。我们举例说明了极限的存在需要度量的加倍性质和 1-Poincaré 不等式这两个条件。此外,我们用有界变函数为所有无限曲线族的严格正 1 模建立了一个特征。这些概括了 Koskela 和 Nguyen (J Funct Anal 285(11):110154, 2023) 中给出的 Sobolev 函数的结果。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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