PreprintarXiv licencearXiv
A geometric characterization of the Maz'ya-Shaposhnikova limit on metric measure spaces
Abstract
We provide a purely geometric necessary and sufficient condition, in terms of a certain Cesàro mean of the volume function, for the existence of the Maz'ya-Shaposhnikova functional limit with respect to Gagliardo seminorms on metric measure spaces (under a mild measure condition). As an application, we establish the Maz'ya-Shaposhnikova convergence on the unbounded Sierpiński gasket. We also show that Ahlfors regularity does not imply the existence of this functional limit by constructing a concrete example.
Related
- Geometric and measure-theoretic characterizations of the measure-determining property of balls in metric spaces
- The Sunada method on Metric Measure Spaces
- Coarse geometry of metric measure spaces
- Lipschitz numbers and fine differentiability on metric measure spaces
- On one relaxation of the bounded-length-distortion condition in the context of metric measure spaces
BibTeX
@misc{gao2026geometric,
title = {{A geometric characterization of the Maz'ya-Shaposhnikova limit on metric measure spaces}},
author = {Jin Gao and Zhenyu Yu and Junda Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26666},
eprint = {2609.26666},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26666},
}
RIS
TY - UNPB
TI - A geometric characterization of the Maz'ya-Shaposhnikova limit on metric measure spaces
AU - Jin Gao
AU - Zhenyu Yu
AU - Junda Zhang
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.26666
UR - https://arxiv.org/abs/2609.26666
AB - We provide a purely geometric necessary and sufficient condition, in terms of a certain Cesàro mean of the volume function, for the existence of the Maz'ya-Shaposhnikova functional limit with respect to Gagliardo seminorms on metric measure spaces (under a mild measure condition). As an application, we establish the Maz'ya-Shaposhnikova convergence on the unbounded Sierpiński gasket. We also show that Ahlfors regularity does not imply the existence of this functional limit by constructing a concrete example.
ER -