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A new sharp Sobolev inequality between the Q-curvature and the scalar curvature on Euclidean space
Abstract
We establish a sharp Sobolev-type inequality on ℝn, n≥ 5, that interpolates between the second-order energy and a weighted first-order term. Specifically, we determine the sharp constant S4,2 in the inequality ∫ℝn (Δu)2 dx ≥ S4,2 (∫ℝn|∇ u|2|u|4/(n−4) dx)(n−4)/(n−2), for any u∈Ḣ2(ℝn), and we classify the cases of equality. This solves the problem of minimizing the total Q-curvature of among metrics on the sphere, conformal to the standard metric, with fixed total scalar curvature.
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BibTeX
@misc{frank2026sharp,
title = {{A new sharp Sobolev inequality between the $Q$-curvature and the scalar curvature on Euclidean space}},
author = {Rupert L. Frank and Yuxin Ge and Guofang Wang and Wei Wei},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39659},
eprint = {2610.39659},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39659},
}
RIS
TY - UNPB
TI - A new sharp Sobolev inequality between the Q-curvature and the scalar curvature on Euclidean space
AU - Rupert L. Frank
AU - Yuxin Ge
AU - Guofang Wang
AU - Wei Wei
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39659
UR - https://arxiv.org/abs/2610.39659
AB - We establish a sharp Sobolev-type inequality on ℝ^n, n≥ 5, that interpolates between the second-order energy and a weighted first-order term. Specifically, we determine the sharp constant S_4,2 in the inequality ∫_ℝ^n (Δu)^2 dx ≥ S_4,2 (∫_ℝ^n|∇ u|^2|u|^4/(n−4) dx)^(n−4)/(n−2), for any u∈Ḣ^2(ℝ^n), and we classify the cases of equality. This solves the problem of minimizing the total Q-curvature of among metrics on the sphere, conformal to the standard metric, with fixed total scalar curvature.
ER -