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Lipschitz numbers and fine differentiability on metric measure spaces
Abstract
On a metric space equipped with a doubling measure supporting a Poincaré inequality, we introduce capacity-relaxed infinitesimal Lipschitz numbers and characterize Newtonian functions in N1,p, 1≤ p≤∞, and functions of bounded variation. If X is additionally Q-regular, then every BV function is $1$-finely differentiable at μ-almost every point, and its fine differential agrees with its approximate Cheeger differential. We also extend a Sobolev regularity criterion from continuous to p-quasicontinuous functions.
Related
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BibTeX
@misc{lahti2026lipschitz,
title = {{Lipschitz numbers and fine differentiability on metric measure spaces}},
author = {Panu Lahti and Yuxin Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29359},
eprint = {2610.29359},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29359},
}
RIS
TY - UNPB
TI - Lipschitz numbers and fine differentiability on metric measure spaces
AU - Panu Lahti
AU - Yuxin Li
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29359
UR - https://arxiv.org/abs/2610.29359
AB - On a metric space equipped with a doubling measure supporting a Poincaré inequality, we introduce capacity-relaxed infinitesimal Lipschitz numbers and characterize Newtonian functions in N^1,p, 1≤ p≤∞, and functions of bounded variation. If X is additionally Q-regular, then every BV function is $1$-finely differentiable at μ-almost every point, and its fine differential agrees with its approximate Cheeger differential. We also extend a Sobolev regularity criterion from continuous to p-quasicontinuous functions.
ER -