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Endpoint weighted Fourier estimate with vanishing spherical trace
Abstract
We establish an endpoint weighted Fourier estimate under a vanishing restriction condition on the unit sphere. Let n ≥ 2 and pc = 2(n+1)/(n+3). We prove that every f ∈ Lpc(ℝn) with zero spherical Fourier restriction satisfies | |1−|ξ|2|−1/2f̂(ξ) |L2(ℝn) ≤ Cn |f|Lpc(ℝn), where the restriction condition is understood through the continuous Tomas--Stein restriction operator. This result gives a negative answer to Goldberg's counterexample-existence question and supplies a missing endpoint.
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BibTeX
@misc{li2026endpoint,
title = {{Endpoint weighted Fourier estimate with vanishing spherical trace}},
author = {Wenjuan Li and Manli Song and Hongqiang Wang and Huiju Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31353},
eprint = {2610.31353},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31353},
}
RIS
TY - UNPB
TI - Endpoint weighted Fourier estimate with vanishing spherical trace
AU - Wenjuan Li
AU - Manli Song
AU - Hongqiang Wang
AU - Huiju Wang
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31353
UR - https://arxiv.org/abs/2610.31353
AB - We establish an endpoint weighted Fourier estimate under a vanishing restriction condition on the unit sphere. Let n ≥ 2 and p_c = 2(n+1)/(n+3). We prove that every f ∈ L^p_c(ℝ^n) with zero spherical Fourier restriction satisfies | |1−|ξ|^2|^−1/2f̂(ξ) |_L^2(ℝ^n) ≤ C_n |f|_L^p_c(ℝ^n), where the restriction condition is understood through the continuous Tomas--Stein restriction operator. This result gives a negative answer to Goldberg's counterexample-existence question and supplies a missing endpoint.
ER -