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Gradient estimates and volume doubling for locally finite weighted graphs with $CDψ(n,-K)$ condition
Abstract
We study gradient estimates and volume growth on locally finite weighted graphs satisfying the $CDψ(n,-K)$ condition with K≥0. We establish a variational inequality for the heat semigroup and derive from it a family of Li-Yau type gradient estimates. Furthermore, under suitable assumptions on ψ, we establish a curvature dependent heat retention estimate for metric balls. Combined with a heat kernel Harnack inequality obtained from the established gradient estimate, this yields a curvature dependent exponential volume doubling estimate V(x,2r)≤ C ec√KrV(x,r). When K=0, the result reduces to a uniform volume doubling and further implies that the bottom of the spectrum of −Δ vanishes on infinite graphs.
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BibTeX
@misc{zhang2026gradient,
title = {{Gradient estimates and volume doubling for locally finite weighted graphs with $CDψ(n,-K)$ condition}},
author = {Qianwei Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.30409},
eprint = {2608.30409},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30409},
}
RIS
TY - UNPB
TI - Gradient estimates and volume doubling for locally finite weighted graphs with $CDψ(n,-K)$ condition
AU - Qianwei Zhang
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2608.30409
UR - https://arxiv.org/abs/2608.30409
AB - We study gradient estimates and volume growth on locally finite weighted graphs satisfying the $CDψ(n,-K)$ condition with K≥0. We establish a variational inequality for the heat semigroup and derive from it a family of Li-Yau type gradient estimates. Furthermore, under suitable assumptions on ψ, we establish a curvature dependent heat retention estimate for metric balls. Combined with a heat kernel Harnack inequality obtained from the established gradient estimate, this yields a curvature dependent exponential volume doubling estimate V(x,2r)≤ C e^c√KrV(x,r). When K=0, the result reduces to a uniform volume doubling and further implies that the bottom of the spectrum of −Δ vanishes on infinite graphs.
ER -