{{Short description|Conjecture about the behaviour of the Fourier transform on curved hypersurfaces}} In harmonic analysis, the '''restriction conjecture''', also known as the '''Fourier restriction conjecture''', is a conjecture about the behaviour of the 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.<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, Bochner-Riesz conjecture and the 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, 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
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