Sharp Capacity Estimates and Volume Growth for Fractional Sobolev Spaces
Abstract
We obtain sharp two-sided estimates for the relative capacity of fat and thin annuli in the fractional Sobolev spaces Wsp,q, where s∈(0,1), p∈(0,∞), q∈(0,∞], in terms of an explicit volume growth quantity. The underlying space is a general metric measure space equipped with a doubling and reverse doubling measure. In particular, whenever the capacity is finite, the estimates are independent of the parameter q, and in the Ahlfors regular case with p=q they recover Costea's logarithmic law at sp=Q. For p<q, we also introduce a sharp condition on the measure that is necessary for the existence of admissible functions in the space Wsp,q. For thin annuli (Br,Br+ε r) under an η-annular decay property, we uncover a purely nonlocal phenomenon: for sp <η, the capacity remains bounded as ε→ 0, because the limiting competitor χBr already has finite energy, while if sp≥η it blows up as $ \varepsilon\to0$.
Related
- Weighted and Sobolev Estimates for Fractional Schrödinger Operators with Reverse Hölder Potentials
- Optimal Global Sobolev Estimates for Fractional Dirichlet Problems in Bounded Lipschitz Domains
- Solutions of the space-fractional heat equation with polynomial growth: Liouville theorems and Schauder estimates
- New thoughts on fractional Sobolev spaces
- Infinitesimal Hilbertianity for manifolds with Sobolev regular Riemannian metrics through uniform lower capacity bounds
BibTeX
@misc{agarwal2026sharp,
title = {{Sharp Capacity Estimates and Volume Growth for Fractional Sobolev Spaces}},
author = {Angha Agarwal},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24752},
eprint = {2610.24752},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24752},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sharp Capacity Estimates and Volume Growth for Fractional Sobolev Spaces
AU - Angha Agarwal
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.24752
UR - https://arxiv.org/abs/2610.24752
AB - We obtain sharp two-sided estimates for the relative capacity of fat and thin annuli in the fractional Sobolev spaces W^s_p,q, where s∈(0,1), p∈(0,∞), q∈(0,∞], in terms of an explicit volume growth quantity. The underlying space is a general metric measure space equipped with a doubling and reverse doubling measure. In particular, whenever the capacity is finite, the estimates are independent of the parameter q, and in the Ahlfors regular case with p=q they recover Costea's logarithmic law at sp=Q. For p<q, we also introduce a sharp condition on the measure that is necessary for the existence of admissible functions in the space W^s_p,q. For thin annuli (B_r,B_r+ε r) under an η-annular decay property, we uncover a purely nonlocal phenomenon: for sp <η, the capacity remains bounded as ε→ 0, because the limiting competitor χ_B_r already has finite energy, while if sp≥η it blows up as $ \varepsilon\to0$.
ER -