PreprintCC BY 4.0arXiv
The Lorentz Threshold in Fourier Restriction
Abstract
Classical Stein--Tomas restriction controls a Fourier transform on a curved surface, but the estimate changes when its values lie in a Banach space. We determine the sharp Lorentz threshold at the critical exponent 4/3: for coefficients in ℓD4/3,q, the optimal Gaussian restriction constant grows like (1+log D)|3/4−1/q|. A multiscale wave-packet construction proves sharpness, including the delicate index q=2. The same threshold holds for fractional Gaussian noise, while Gaussian type and cotype alone do not distinguish the positive and negative cases. As a dispersive consequence, we establish a sharp Lorentz-channel spacetime estimate for linear capillary waves.
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BibTeX
@misc{shahmurov2026lorentz,
title = {{The Lorentz Threshold in Fourier Restriction}},
author = {R. Shahmurov and V. Shahmurov},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01232},
eprint = {2611.01232},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01232},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Lorentz Threshold in Fourier Restriction
AU - R. Shahmurov
AU - V. Shahmurov
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01232
UR - https://arxiv.org/abs/2611.01232
AB - Classical Stein--Tomas restriction controls a Fourier transform on a curved surface, but the estimate changes when its values lie in a Banach space. We determine the sharp Lorentz threshold at the critical exponent 4/3: for coefficients in ℓ_D^4/3,q, the optimal Gaussian restriction constant grows like (1+log D)^|3/4−1/q|. A multiscale wave-packet construction proves sharpness, including the delicate index q=2. The same threshold holds for fractional Gaussian noise, while Gaussian type and cotype alone do not distinguish the positive and negative cases. As a dispersive consequence, we establish a sharp Lorentz-channel spacetime estimate for linear capillary waves.
ER -