Math Conjectures

Fourier restriction conjecture for the sphere

OPENLandmarkConjectureProposed c. 1967 · Standard version

Canonical statement

Let n≥3n\ge3, let dσd\sigma be surface measure on Sn−1S^{n-1}, and for 1≤p,q<∞1\le p,q<\infty define
Eg(x)=∫Sn−1e2πix⋅ωg(ω) dσ(ω). Eg(x)=\int_{S^{n-1}}e^{2\pi i x\cdot\omega}g(\omega)\,d\sigma(\omega).
If
q>2nn−1andn+1q≤n−1p′,p′=pp−1 q>\frac{2n}{n-1} \quad\text{and}\quad \frac{n+1}{q}\le\frac{n-1}{p'}, \qquad p'=\frac{p}{p-1}
(with p′=∞p'=\infty for p=1p=1), then there is C=C(n,p,q)C=C(n,p,q) such that ∥Eg∥Lq(Rn)≤C∥g∥Lp(Sn−1,dσ)\|Eg\|_{L^q(\mathbb R^n)}\le C\|g\|_{L^p(S^{n-1},d\sigma)} for every g∈Lp(Sn−1,dσ)g\in L^p(S^{n-1},d\sigma).
View source LaTeX
Let \(n\ge3\), let \(d\sigma\) be surface measure on \(S^{n-1}\), and for \(1\le p,q<\infty\) define \[ Eg(x)=\int_{S^{n-1}}e^{2\pi i x\cdot\omega}g(\omega)\,d\sigma(\omega). \] If \[ q>\frac{2n}{n-1} \quad\text{and}\quad \frac{n+1}{q}\le\frac{n-1}{p'}, \qquad p'=\frac{p}{p-1} \] (with \(p'=\infty\) for \(p=1\)), then there is \(C=C(n,p,q)\) such that \(\|Eg\|_{L^q(\mathbb R^n)}\le C\|g\|_{L^p(S^{n-1},d\sigma)}\) for every \(g\in L^p(S^{n-1},d\sigma)\).
ANAL-FA-007:[Stein1993HarmonicAnalysis][Tao2003RestrictionSurvey][GuthHickmanIliopoulou2019Oscillatory]
Analysis and functional analysis

Notes

The restriction conjecture concerns the extension operator Eg(x)=∫Sn−1e2πix⋅ωg(ω) dσ(ω)Eg(x)=\int_{S^{n-1}}e^{2\pi i x\cdot\omega}g(\omega)\,d\sigma(\omega) associated with the sphere: it predicts that EE maps Lp(Sn−1)L^p(S^{n-1}) boundedly into Lq(Rn)L^q(\mathbb R^n) exactly in the range of exponents forced by stationary-phase decay and by Knapp's cap examples, the key constraint being q>2n/(n−1)q>2n/(n-1). The problem grew out of Stein's late-1960s work on restricting the Fourier transform to curved surfaces [Stein1993HarmonicAnalysis].

In two dimensions the conjecture is a classical theorem. In dimensions n≥3n\ge3 the necessary conditions are known to be sufficient on large subranges of exponents: methods including decoupling and polynomial partitioning have successively enlarged the known range [Tao2003RestrictionSurvey], with sharp oscillatory-integral estimates via polynomial partitioning among the strongest results in this direction [GuthHickmanIliopoulou2019Oscillatory].

Despite this progress, the full conjectured range is not established in any dimension n≥3n\ge3. The conjecture is closely linked to the Kakeya and Bochner–Riesz problems, and completing the remaining range is one of the central open goals of modern harmonic analysis.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.