PreprintCC BY 4.0arXiv
Probabilistic Fourier extension and the Schrödinger equation
Abstract
We give applications of the probabilistic Fourier extension theorem for the paraboloid in d dimensions (see [Saw7] for the sphere, and the proof there holds for a smooth compact piece of the paraboloid) to the global initial value problem for the Schrödinger equation with initial data f, in which we obtain sharp averaged Sobolev regularity results over certain `martingale transforms' of the intial data f. The deterministic analogue in three dimensions has been obtained recently by OpenAI [Ope].
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BibTeX
@misc{korobenko2026probabilistic,
title = {{Probabilistic Fourier extension and the Schrödinger equation}},
author = {Lyudmila Korobenko and Eric T. Sawyer},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22634},
eprint = {2610.22634},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22634},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Probabilistic Fourier extension and the Schrödinger equation
AU - Lyudmila Korobenko
AU - Eric T. Sawyer
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.22634
UR - https://arxiv.org/abs/2610.22634
AB - We give applications of the probabilistic Fourier extension theorem for the paraboloid in d dimensions (see [Saw7] for the sphere, and the proof there holds for a smooth compact piece of the paraboloid) to the global initial value problem for the Schrödinger equation with initial data f, in which we obtain sharp averaged Sobolev regularity results over certain `martingale transforms' of the intial data f. The deterministic analogue in three dimensions has been obtained recently by OpenAI [Ope].
ER -