Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group
Abstract
We investigate the optimal Bianchi-Egnell-type quantitative stability constant for the critical nonlocal Sobolev inequality on the Heisenberg group ℍn, labelnS SHL(Q,μ)(∫ℍn∫ℍn |u(ξ)|Q∗μ|u(η)|Q∗μ/(|η−1ξ|μ) dξdη)1/Q∗μ ≤ ∫ℍn|∇ℍu|2dξ, u∈ S1,2(ℍn), where Q=2n+2, n≥1, 0<μ<Q, and Q∗μ=(2Q−μ)/(Q−2). Let M denote the manifold of Jerison-Lee bubbles, and let HNS be the infimum of the quotient between the deficit in \eqref{nS} and dist(u,M)2. The central issue is that the Euclidean nonlocal problem and the local Folland-Stein-Sobolev problem each possess their own spectral and compactness structures, whereas the present problem couples the HLS interaction with the noncommutative conformal geometry of ℍn. Under the stated conditions on (n,μ), these strict thresholds, together with a Heisenberg-group profile decomposition and a nonlocal Br{e}zis-Lieb splitting, imply that HNS is attained. The minimizer then yields the strict comparison HNS>HBE with the optimal stability constant for the local Folland-Stein-Sobolev inequality. We further prove that the sharp universal upper constant for the deficit-to-distance comparison is $1$ and characterize equality. Finally, for the associated Euler-Lagrange equation, we formulate the corresponding residual quotient and derive a strict single-bubble upper bound; this critical-point statement requires a separate expansion and does not follow from attainment of HNS.
Related
- On the Stability of Critical Points of the Hardy-Littlewood-Sobolev Inequality
- Sharp One-bubble Critical-Point Stability and Global Compactness for the Sobolev Trace Inequality
- Quantitative Stability for Fractional Yamabe minimizers
- Sobolev Minimizers for Critical Vectorial p-Laplacian Problems
- The Isoperimetric Inequality in Higher-Dimensional Heisenberg Groups
BibTeX
@misc{chen2026optimal,
title = {{Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group}},
author = {Wenjing Chen and Zexi Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.09034},
eprint = {2608.09034},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.09034},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group
AU - Wenjing Chen
AU - Zexi Wang
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2608.09034
UR - https://arxiv.org/abs/2608.09034
AB - We investigate the optimal Bianchi-Egnell-type quantitative stability constant for the critical nonlocal Sobolev inequality on the Heisenberg group ℍ^n, labelnS S_HL(Q,μ)(∫_ℍ^n∫_ℍ^n |u(ξ)|^Q^∗_μ|u(η)|^Q^∗_μ/(|η^−1ξ|^μ) dξdη)^1/Q^∗_μ ≤ ∫_ℍ^n|∇_ℍu|^2dξ, u∈ S^1,2(ℍ^n), where Q=2n+2, n≥1, 0<μ<Q, and Q^∗_μ=(2Q−μ)/(Q−2). Let M denote the manifold of Jerison-Lee bubbles, and let H_NS be the infimum of the quotient between the deficit in \eqref{nS} and dist(u,M)^2. The central issue is that the Euclidean nonlocal problem and the local Folland-Stein-Sobolev problem each possess their own spectral and compactness structures, whereas the present problem couples the HLS interaction with the noncommutative conformal geometry of ℍ^n. Under the stated conditions on (n,μ), these strict thresholds, together with a Heisenberg-group profile decomposition and a nonlocal Br{e}zis-Lieb splitting, imply that H_NS is attained. The minimizer then yields the strict comparison H_NS>H_BE with the optimal stability constant for the local Folland-Stein-Sobolev inequality. We further prove that the sharp universal upper constant for the deficit-to-distance comparison is $1$ and characterize equality. Finally, for the associated Euler-Lagrange equation, we formulate the corresponding residual quotient and derive a strict single-bubble upper bound; this critical-point statement requires a separate expansion and does not follow from attainment of H_NS.
ER -