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Extremals for Sharp Trace Gagliardo--Nirenberg--Sobolev Inequalities on Convex Cones
Abstract
Zugmeyer~\cite{Zug} established sharp trace Gagliardo--Nirenberg--Sobolev inequalities on convex cones for a≥ n>p>1. We classify all nonzero real-valued extremals of these inequalities. Let the cone be the epigraph of a finite convex function that is positively homogeneous of degree one, and let H be any norm. We prove that equality holds precisely for functions of the form A H(x+te+ℓ)−(a−p)/(p−1), where A∈ℝbackslash{0}, t>0, e=(0,…,0,1), and ℓ belongs to the largest vector subspace contained in the cone. The result includes the critical Sobolev trace inequality at a=n, as well as nonsmooth norms and cones, half-spaces, and cones containing lines.
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BibTeX
@misc{zhou2026extremals,
title = {{Extremals for Sharp Trace Gagliardo--Nirenberg--Sobolev Inequalities on Convex Cones}},
author = {Yuxuan Zhou and Yingfang Zhang and Wenming Zou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28503},
eprint = {2610.28503},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28503},
}
RIS
TY - UNPB
TI - Extremals for Sharp Trace Gagliardo--Nirenberg--Sobolev Inequalities on Convex Cones
AU - Yuxuan Zhou
AU - Yingfang Zhang
AU - Wenming Zou
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28503
UR - https://arxiv.org/abs/2610.28503
AB - Zugmeyer~\cite{Zug} established sharp trace Gagliardo--Nirenberg--Sobolev inequalities on convex cones for a≥ n>p>1. We classify all nonzero real-valued extremals of these inequalities. Let the cone be the epigraph of a finite convex function that is positively homogeneous of degree one, and let H be any norm. We prove that equality holds precisely for functions of the form A H(x+te+ℓ)^−(a−p)/(p−1), where A∈ℝbackslash{0}, t>0, e=(0,…,0,1), and ℓ belongs to the largest vector subspace contained in the cone. The result includes the critical Sobolev trace inequality at a=n, as well as nonsmooth norms and cones, half-spaces, and cones containing lines.
ER -