PreprintCC BY 4.0arXiv
On the dichotomy of p-walk dimensions on metric measure spaces II
Abstract
On a volume doubling metric measure space endowed with a family of p-energies such that the Poincaré inequality and the capacity upper bound with p-walk dimension βp hold, for p in an open interval I⊆(1,+∞), we prove the following dichotomy: either βp=p for all p∈ I, or the function p↦ βp/p takes values in (1,+∞) and is strictly decreasing on I. Moreover, if the metric measure space is dh-Ahlfors regular, we prove the following dichotomy: either βp=dh+p−1 for all p∈ I, or the function p↦ (βp−dh)/(p−1) takes values in (−∞,1) and is strictly increasing on I.
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BibTeX
@misc{yang2026dichotomy,
title = {{On the dichotomy of $p$-walk dimensions on metric measure spaces II}},
author = {Meng Yang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02422},
eprint = {2611.02422},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02422},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the dichotomy of p-walk dimensions on metric measure spaces II
AU - Meng Yang
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.02422
UR - https://arxiv.org/abs/2611.02422
AB - On a volume doubling metric measure space endowed with a family of p-energies such that the Poincaré inequality and the capacity upper bound with p-walk dimension β_p hold, for p in an open interval I⊆(1,+∞), we prove the following dichotomy: either β_p=p for all p∈ I, or the function p↦ β_p/p takes values in (1,+∞) and is strictly decreasing on I. Moreover, if the metric measure space is d_h-Ahlfors regular, we prove the following dichotomy: either β_p=d_h+p−1 for all p∈ I, or the function p↦ (β_p−d_h)/(p−1) takes values in (−∞,1) and is strictly increasing on I.
ER -