{"title":"有界变分函数无穷大极限的存在性和唯一性","authors":"Panu Lahti, Khanh Nguyen","doi":"10.1007/s12220-024-01788-2","DOIUrl":null,"url":null,"abstract":"<p>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 <i>f</i> and for 1-a.e. infinite curve <span>\\(\\gamma \\)</span>, for both the upper approximate limit <span>\\(f^\\vee \\)</span> and the lower approximate limit <span>\\(f^\\wedge \\)</span> we have that </p><span>$$\\begin{aligned} \\lim _{t\\rightarrow +\\infty }f^\\vee (\\gamma (t)) \\mathrm{\\ \\ and\\ \\ }\\lim _{t\\rightarrow +\\infty }f^\\wedge (\\gamma (t)) \\end{aligned}$$</span><p>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).</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":"22 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Existence and Uniqueness of Limits at Infinity for Bounded Variation Functions\",\"authors\":\"Panu Lahti, Khanh Nguyen\",\"doi\":\"10.1007/s12220-024-01788-2\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>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 <i>f</i> and for 1-a.e. infinite curve <span>\\\\(\\\\gamma \\\\)</span>, for both the upper approximate limit <span>\\\\(f^\\\\vee \\\\)</span> and the lower approximate limit <span>\\\\(f^\\\\wedge \\\\)</span> we have that </p><span>$$\\\\begin{aligned} \\\\lim _{t\\\\rightarrow +\\\\infty }f^\\\\vee (\\\\gamma (t)) \\\\mathrm{\\\\ \\\\ and\\\\ \\\\ }\\\\lim _{t\\\\rightarrow +\\\\infty }f^\\\\wedge (\\\\gamma (t)) \\\\end{aligned}$$</span><p>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).</p>\",\"PeriodicalId\":501200,\"journal\":{\"name\":\"The Journal of Geometric Analysis\",\"volume\":\"22 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-17\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"The Journal of Geometric Analysis\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1007/s12220-024-01788-2\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"The Journal of Geometric Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s12220-024-01788-2","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Existence and Uniqueness of Limits at Infinity for Bounded Variation Functions
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
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).