A discussion of two new proofs of Fefferman's Fourier extension theorem in the plane
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866911698702565376 |
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| author | Sawyer, Eric T. |
| author_facet | Sawyer, Eric T. |
| contents | 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 proofs of this classical theorem on the parabola. The first proof uses C. Fefferman's decoupling together with a decomposition into Haar wavelets. This sets the stage for the second proof in Rios and Sawyer, that uses smooth Alpert wavelets and a new decoupling method, which exploits averaging smooth Alpert projections over grids, the extraction of Dirichlet kernels, and periodic stationary phase, all of which was extended to higher dimensions in arXiv:2512.24990v7. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_20100 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A discussion of two new proofs of Fefferman's Fourier extension theorem in the plane Sawyer, Eric T. Classical Analysis and ODEs math.CA 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 proofs of this classical theorem on the parabola. The first proof uses C. Fefferman's decoupling together with a decomposition into Haar wavelets. This sets the stage for the second proof in Rios and Sawyer, that uses smooth Alpert wavelets and a new decoupling method, which exploits averaging smooth Alpert projections over grids, the extraction of Dirichlet kernels, and periodic stationary phase, all of which was extended to higher dimensions in arXiv:2512.24990v7. |
| title | A discussion of two new proofs of Fefferman's Fourier extension theorem in the plane |
| topic | Classical Analysis and ODEs math.CA |
| url | https://arxiv.org/abs/2605.20100 |