A discussion of two arguments related to Fefferman's Fourier extension theorem in the plane
Abstract
The Fourier extension conjecture of E. Stein was proved in the plane in 1970 by C. Fefferman, see also Zygmund and Carleson and Sjölin, with simplifications given by other authors later on, in particular by L. Hörmander and T. Tao. We discuss yet two more arguments for this classical theorem on the parabola. The first argument uses C. Fefferman's decoupling together with a decomposition into Haar wavelets. This sets the stage for the second argument whose point of departure is the bilinear characterization of Tao, Vargas and Vega, and relies on the bilinear interplay with the classical wave packet constructions and discrete characterizations with an induction on scales. Each of the above two arguments rely on some form of Fefferman's convolution decoupling and the special nature of the critical planar index 4 as a positive even integer, but the second might be more amenable to weighted generalizations.
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BibTeX
@misc{sawyer2026discussion,
title = {{A discussion of two arguments related to Fefferman's Fourier extension theorem in the plane}},
author = {Eric T. Sawyer},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.20100},
eprint = {2605.20100},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.20100},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A discussion of two arguments related to Fefferman's Fourier extension theorem in the plane
AU - Eric T. Sawyer
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2605.20100
UR - https://arxiv.org/abs/2605.20100
AB - The Fourier extension conjecture of E. Stein was proved in the plane in 1970 by C. Fefferman, see also Zygmund and Carleson and Sjölin, with simplifications given by other authors later on, in particular by L. Hörmander and T. Tao. We discuss yet two more arguments for this classical theorem on the parabola. The first argument uses C. Fefferman's decoupling together with a decomposition into Haar wavelets. This sets the stage for the second argument whose point of departure is the bilinear characterization of Tao, Vargas and Vega, and relies on the bilinear interplay with the classical wave packet constructions and discrete characterizations with an induction on scales. Each of the above two arguments rely on some form of Fefferman's convolution decoupling and the special nature of the critical planar index 4 as a positive even integer, but the second might be more amenable to weighted generalizations.
ER -