Sharp radial--angular mixed norm Fourier extension and global stability on spheres
Abstract
We study the sharp Fourier extension inequality from L2(𝕊d−1) to the radial--angular mixed space LqradL2ang. Our work is motivated by and builds upon the foundational analysis of Carneiro, Oliveira e Silva and Sousa \cite{COS}, who initiated the sharp mixed-norm spherical restriction problem from the viewpoint of spherical-harmonic decomposition. Their reduction revealed that the extremal problem is governed by a competition among the individual harmonic modes, and their results established degree-zero extremizers in several important ranges of exponents, including even integer exponents and the endpoint case. The present paper addresses this question by establishing a complete modal ordering result. Building on their spherical-harmonic reduction, the sharp-constant problem is reduced to the weighted Bessel moments Iν,k(q) := ∫0∞ |Jν+k(r)|q r2ν+1−νq dr, ν:=d/2−1. For every integer d≥2 and every finite admissible exponent q>2d/(d−1), we prove Iν,k(q)<Iν,0(q), k≥1, and hence constants are the unique nonzero extremizers of the sharp mixed-norm extension inequality. In dimensions d≥3 we prove the stronger strict ordering Iν,0(q)>Iν,1(q)>Iν,2(q)>⋯. We resolve this difficulty by combining exact Hankel asymptotics with a finite remainder comparison. The modal ordering also yields the optimal quadratic global stability inequality. For every finite admissible q, Cd,q2|f|22 −|fσ̂|LqradL2ang2 ≥ Γd,q|f−P0f|22, with the best possible coefficient. When d≥3, this coefficient is explicitly Γd,q =(2π)d [ Iν,0(q)2/q−Iν,1(q)2/q ].
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BibTeX
@misc{xu2026sharp,
title = {{Sharp radial--angular mixed norm Fourier extension and global stability on spheres}},
author = {Zewei Xu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19648},
eprint = {2610.19648},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19648},
}
RIS
TY - UNPB
TI - Sharp radial--angular mixed norm Fourier extension and global stability on spheres
AU - Zewei Xu
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.19648
UR - https://arxiv.org/abs/2610.19648
AB - We study the sharp Fourier extension inequality from L^2(𝕊^d−1) to the radial--angular mixed space L^q_radL^2_ang. Our work is motivated by and builds upon the foundational analysis of Carneiro, Oliveira e Silva and Sousa \cite{COS}, who initiated the sharp mixed-norm spherical restriction problem from the viewpoint of spherical-harmonic decomposition. Their reduction revealed that the extremal problem is governed by a competition among the individual harmonic modes, and their results established degree-zero extremizers in several important ranges of exponents, including even integer exponents and the endpoint case. The present paper addresses this question by establishing a complete modal ordering result. Building on their spherical-harmonic reduction, the sharp-constant problem is reduced to the weighted Bessel moments I_ν,k(q) := ∫_0^∞ |J_ν+k(r)|^q r^2ν+1−νq dr, ν:=d/2−1. For every integer d≥2 and every finite admissible exponent q>2d/(d−1), we prove I_ν,k(q)<I_ν,0(q), k≥1, and hence constants are the unique nonzero extremizers of the sharp mixed-norm extension inequality. In dimensions d≥3 we prove the stronger strict ordering I_ν,0(q)>I_ν,1(q)>I_ν,2(q)>⋯. We resolve this difficulty by combining exact Hankel asymptotics with a finite remainder comparison. The modal ordering also yields the optimal quadratic global stability inequality. For every finite admissible q, C_d,q^2|f|_2^2 −|fσ̂|_L^q_radL^2_ang^2 ≥ Γ_d,q|f−P_0f|_2^2, with the best possible coefficient. When d≥3, this coefficient is explicitly Γ_d,q =(2π)^d [ I_ν,0(q)^2/q−I_ν,1(q)^2/q ].
ER -