Sharp Nonlocal Poincaré Inequalities and Liminf Lower Bounds for Sobolev and BV Energies on Metric Measure Spaces
Abstract
Let p∈[1,∞), and let (X,ρ,μ) be a complete doubling q-Poincaré space for some q∈[1,p] with a lower dimension d and an upper dimension D. In this article, we establish Poincaré inequalities for two families of nonlocal nonconvex functionals defined using distances or volumes. More precisely, for any ball B with its radius rB∈(0,diam X), any λ∈(0,∞), and any measurable function f:X→ℝ, &1/μ(B)2∫B∫B|f(x)−f(y)|p dμ(x) dμ(y) & ≲λprBpϕ(B)γp+λprBp/μ(B)mathop∫ΛB∫ΛBgenfrac2|f(x)−f(y)|ρ(x,y)ϕ(x,y)γ>λϕ(x,y)γp/V(x,y) dμ(x) dμ(y), where V(x,y):=μ(B(x,ρ(x,y))) and either ϕ=ρ with ϕ(B)=rB and γ∈(−∞,−min{q,D}/p] or ϕ=V with ϕ(B)=μ(B) and γ∈(−∞,−q/pmax{q,d}]. Here both ranges of γ are sharp over the class of q-Poincaré spaces with the prescribed lower and upper dimensions, and the constant Λ∈[1,∞) and the implicit positive constant depend only on p, γ, and the geometric constants of X. We also obtain lower bounds for Sobolev energy and total variation in terms of the limit inferior of these functionals, for γ in either the above negative ranges or (0,∞). Our proofs use the curve characterization of Poincaré spaces and an adaptive dyadic refinement argument based on medians.
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BibTeX
@misc{dai2026sharp,
title = {{Sharp Nonlocal Poincaré Inequalities and Liminf Lower Bounds for Sobolev and BV Energies on Metric Measure Spaces}},
author = {Feng Dai and Tuomas P. Hytönen and Dachun Yang and Wen Yuan and Yirui Zhao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40743},
eprint = {2610.40743},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40743},
}
RIS
TY - UNPB
TI - Sharp Nonlocal Poincaré Inequalities and Liminf Lower Bounds for Sobolev and BV Energies on Metric Measure Spaces
AU - Feng Dai
AU - Tuomas P. Hytönen
AU - Dachun Yang
AU - Wen Yuan
AU - Yirui Zhao
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40743
UR - https://arxiv.org/abs/2610.40743
AB - Let p∈[1,∞), and let (X,ρ,μ) be a complete doubling q-Poincaré space for some q∈[1,p] with a lower dimension d and an upper dimension D. In this article, we establish Poincaré inequalities for two families of nonlocal nonconvex functionals defined using distances or volumes. More precisely, for any ball B with its radius r_B∈(0,diam X), any λ∈(0,∞), and any measurable function f:X→ℝ, &1/μ(B)^2∫_B∫_B|f(x)−f(y)|^p dμ(x) dμ(y) & ≲λ^pr_B^pϕ(B)^γp+λ^pr_B^p/μ(B)mathop∫_ΛB∫_ΛB_genfrac2|f(x)−f(y)|ρ(x,y)ϕ(x,y)^γ>λϕ(x,y)^γp/V(x,y) dμ(x) dμ(y), where V(x,y):=μ(B(x,ρ(x,y))) and either ϕ=ρ with ϕ(B)=r_B and γ∈(−∞,−min{q,D}/p] or ϕ=V with ϕ(B)=μ(B) and γ∈(−∞,−q/pmax{q,d}]. Here both ranges of γ are sharp over the class of q-Poincaré spaces with the prescribed lower and upper dimensions, and the constant Λ∈[1,∞) and the implicit positive constant depend only on p, γ, and the geometric constants of X. We also obtain lower bounds for Sobolev energy and total variation in terms of the limit inferior of these functionals, for γ in either the above negative ranges or (0,∞). Our proofs use the curve characterization of Poincaré spaces and an adaptive dyadic refinement argument based on medians.
ER -