# Restriction conjecture

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{{Short description|Conjecture about the behaviour of the Fourier transform on curved hypersurfaces}}
In [harmonic analysis](/source/harmonic_analysis), the '''restriction conjecture''', also known as the '''Fourier restriction conjecture''', is a conjecture about the behaviour of the [Fourier transform](/source/Fourier_transform) on curved hypersurfaces.<ref>{{Cite web |last=Ansede |first=Manuel |date=2025-07-14 |title=What is the smallest space in which a needle can be rotated to point in the opposite direction? This mathematician has finally solved the Kakeya conjecture |url=https://english.elpais.com/science-tech/2025-07-14/what-is-the-smallest-space-in-which-a-needle-can-be-rotated-to-point-in-the-opposite-direction-this-mathematician-has-finally-solved-the-kakeya-conjecture.html |access-date=2025-07-20 |website=EL PAÍS English |language=en-us}}</ref><ref name="Kinnear">{{Cite web |last=Kinnear |first=George |date=7 February 2011 |title=Restriction Theory |url=https://webhomes.maths.ed.ac.uk/gkinnear/files/restriction-talk.pdf |website=webhomes.maths.ed.ac.uk}}</ref> It was first hypothesized by [Elias Stein](/source/Elias_Stein).<ref name="Stedman">{{cite web|title=The Restriction and Kakeya Conjectures|first= Richard James |last=Stedman|url=https://etheses.bham.ac.uk/id/eprint/5466/1/Stedman14MPhil.pdf|date=September 2013|publisher=University of Birmingham}}</ref> The conjecture states that two necessary conditions needed to solve a problem known as the ''restriction problem'' in that scenario are also sufficient.<ref name="Kinnear" /><ref name="Stedman" /> 

The restriction conjecture is closely related to the [Kakeya conjecture](/source/Kakeya_conjecture), [Bochner-Riesz conjecture](/source/Bochner-Riesz_conjecture) and the [local smoothing conjecture](/source/local_smoothing_conjecture).<ref>{{Cite web |date=2024-11-17 |first=Terence |last=Tao |title=Terence Tao (@tao@mathstodon.xyz) |url=https://mathstodon.xyz/@tao/113496016545909911 |access-date=2025-07-20 |website=Mathstodon |language=en}}</ref><ref>{{Cite web |last=Cepelewicz |first=Jordana |date=2023-09-12 |title=A Tower of Conjectures That Rests Upon a Needle |url=https://www.quantamagazine.org/a-tower-of-conjectures-that-rests-upon-a-needle-20230912/ |access-date=2025-07-20 |website=Quanta Magazine |language=en}}</ref>
==Statement==
The ''restriction conjecture'' states that <math display="inline">\|\widehat{g\,d\sigma}\|_{L^q(\mathbb R^n)} \lesssim \|g\|_{L^p(S^{n-1})}</math> for certain ''q'' and ''n'', where <math display="inline">\|f\|_{L^p}</math> represents the [L<sup>''p''</sup> norm](/source/Lp_norm), or <math display="inline">\int_{-\infty}^\infty f(x)^p \, dx</math> and <math display="inline">f \lesssim g</math> means that <math display="inline">f \le Cg</math> for some constant <math display="inline">C</math>.<ref name=":0">{{Cite web |last=Kinnear |first=George |date=7 February 2011 |title=Restriction Theory |url=https://webhomes.maths.ed.ac.uk/gkinnear/files/restriction-talk.pdf}}</ref>{{Clarify|reason=What does this constant depend upon?|date=August 2025}}

The requirements of ''q'' and ''n'' set by the conjecture are that <math>\frac{1}{q} < \frac{n-1}{2n}</math> and <math>\frac{1}{q} \le \frac{n-1}{n+1}\frac{1}{p}</math>.<ref name=":0" />

The restriction conjecture has been proved for dimension <math display="inline">n = 2</math> as of 2021.<ref name=":0" />

== References ==
{{reflist}}

Category:Harmonic analysis
Category:Conjectures

{{mathanalysis-stub}}

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Adapted from the Wikipedia article [Restriction conjecture](https://en.wikipedia.org/wiki/Restriction_conjecture) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Restriction_conjecture?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.