Quantitative stability for weighted Hardy inequalities: Local and nonlocal cases
Abstract
We establish quantitative stability estimates for weighted Hardy inequalities in the local and fractional settings, showing that the Hardy deficit controls the distance of a normalized admissible function to the family of virtual extremizers. In the local case, for p>1, α∈[0,1), 0<p−αp<N, and u∈ Cc1(ℝN) normalized by ∫ℝN|u|p|x|αp−p dx=1, we prove ∫ℝN|∇ u|p|x|αp dx−((N−p+αp)/p)p∫ℝN|u|p/(|x|p−αp) dx ≥ C distα(u,ℳα)max{4,2p}, where ℳα is generated by the virtual extremizer with scale-invariant distance. We extend this to the weighted fractional Hardy inequality: for s∈(0,1), α=α1+α2≥0, α1p,α2p∈(−N,sp), and 0<sp−αp<N, the fractional Hardy deficit satisfies ∫ℝN∫ℝN(|u(x)−u(y)|p)/(|x−y|N+sp)|x|α1p|y|α2p dx dy−𝒞∫ℝN|u|p/(|x|sp−αp) dx ≥ C dists,α(u,ℳs,α)max{4,2p}. For p≥2 the distance is measured on u, while for 1<p<2 it is measured on a power-type transformation of u. Both recover the unweighted results when the weights vanish. We also prove weighted Sobolev and fractional Sobolev extension and embedding inequalities on bounded smooth domains avoiding the origin. The proof is rearrangement-free, combining these inequalities with scale-invariant Poincaré--Sobolev estimates on annuli, telescoping oscillation decompositions, and weighted Lorentz-space embeddings. The distinction between p≥2 and 1<p<2 stems from the convexity estimate used to obtain a remainder term from the deficit.
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BibTeX
@misc{sahu2026quantitative,
title = {{Quantitative stability for weighted Hardy inequalities: Local and nonlocal cases}},
author = {Vivek Sahu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25952},
eprint = {2609.25952},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25952},
}
RIS
TY - UNPB
TI - Quantitative stability for weighted Hardy inequalities: Local and nonlocal cases
AU - Vivek Sahu
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.25952
UR - https://arxiv.org/abs/2609.25952
AB - We establish quantitative stability estimates for weighted Hardy inequalities in the local and fractional settings, showing that the Hardy deficit controls the distance of a normalized admissible function to the family of virtual extremizers. In the local case, for p>1, α∈[0,1), 0<p−αp<N, and u∈ C_c^1(ℝ^N) normalized by ∫_ℝ^N|u|^p|x|^αp−p dx=1, we prove ∫_ℝ^N|∇ u|^p|x|^αp dx−((N−p+αp)/p)^p∫_ℝ^N|u|^p/(|x|^p−αp) dx ≥ C dist_α(u,ℳ_α)^max{4,2p}, where ℳ_α is generated by the virtual extremizer with scale-invariant distance. We extend this to the weighted fractional Hardy inequality: for s∈(0,1), α=α_1+α_2≥0, α_1p,α_2p∈(−N,sp), and 0<sp−αp<N, the fractional Hardy deficit satisfies ∫_ℝ^N∫_ℝ^N(|u(x)−u(y)|^p)/(|x−y|^N+sp)|x|^α_1p|y|^α_2p dx dy−𝒞∫_ℝ^N|u|^p/(|x|^sp−αp) dx ≥ C dist_s,α(u,ℳ_s,α)^max{4,2p}. For p≥2 the distance is measured on u, while for 1<p<2 it is measured on a power-type transformation of u. Both recover the unweighted results when the weights vanish. We also prove weighted Sobolev and fractional Sobolev extension and embedding inequalities on bounded smooth domains avoiding the origin. The proof is rearrangement-free, combining these inequalities with scale-invariant Poincaré--Sobolev estimates on annuli, telescoping oscillation decompositions, and weighted Lorentz-space embeddings. The distinction between p≥2 and 1<p<2 stems from the convexity estimate used to obtain a remainder term from the deficit.
ER -