Math Conjectures

Diagonal Fourier extension conjecture for compact paraboloids

OPENMajorConjectureProposed c. 1967 · Standard version

Canonical statement

Let d≥3d\ge3, let Pd−1={(ξ,∣ξ∣2):ξ∈Rd−1}⊂Rd\mathcal P^{d-1}=\{(\xi,|\xi|^2):\xi\in\mathbb R^{d-1}\}\subset\mathbb R^d, and let dσ=η dSd\sigma=\eta\,dS, where dSdS is surface measure and 0≤η∈Cc∞(Pd−1)0\le\eta\in C_c^\infty(\mathcal P^{d-1}) is nonzero. Define
Eσf(x)=∫Pd−1e2πix⋅ωf(ω) dσ(ω). E_\sigma f(x)=\int_{\mathcal P^{d-1}}e^{2\pi i x\cdot\omega}f(\omega)\,d\sigma(\omega).
For every q>2d/(d−1)q>2d/(d-1) there is a constant C=C(d,q,σ)C=C(d,q,\sigma) such that
∥Eσf∥Lq(Rd)≤C∥f∥Lq(dσ) \|E_\sigma f\|_{L^q(\mathbb R^d)}\le C\|f\|_{L^q(d\sigma)}
for every f∈Lq(dσ)f\in L^q(d\sigma).
View source LaTeX
Let \(d\ge3\), let \(\mathcal P^{d-1}=\{(\xi,|\xi|^2):\xi\in\mathbb R^{d-1}\}\subset\mathbb R^d\), and let \(d\sigma=\eta\,dS\), where \(dS\) is surface measure and \(0\le\eta\in C_c^\infty(\mathcal P^{d-1})\) is nonzero. Define \[ E_\sigma f(x)=\int_{\mathcal P^{d-1}}e^{2\pi i x\cdot\omega}f(\omega)\,d\sigma(\omega). \] For every \(q>2d/(d-1)\) there is a constant \(C=C(d,q,\sigma)\) such that \[ \|E_\sigma f\|_{L^q(\mathbb R^d)}\le C\|f\|_{L^q(d\sigma)} \] for every \(f\in L^q(d\sigma)\).
ANAL-FA-016:[Stein1993HarmonicAnalysis][Tao2003RestrictionSurvey][Stovall2020ParaboloidExtremizers][RiosSawyer2026ParaboloidCharacterization]
Analysis and functional analysisPartial differential equations

Notes

This is the diagonal form of the Fourier extension conjecture for compact pieces of the paraboloid Pd−1⊂Rd\mathcal P^{d-1}\subset\mathbb R^d: for every q>2d/(d−1)q>2d/(d-1), the extension operator f↦f dσ^f\mapsto\widehat{f\,d\sigma} should be bounded from Lq(dσ)L^q(d\sigma) to Lq(Rd)L^q(\mathbb R^d). The problem belongs to Stein's restriction program, which took shape in the late 1960s, so the proposal date is approximate; the standard formulation and background appear in Stein's treatise [Stein1993HarmonicAnalysis].

The two-dimensional case is classical, and in higher dimensions successive methods have established the estimate on progressively larger ranges of qq, as surveyed by Tao [Tao2003RestrictionSurvey]; complementary work has studied extremizers for the paraboloid extension inequality [Stovall2020ParaboloidExtremizers]. Nevertheless, in every dimension d≥3d\ge3 the full range down to the critical exponent 2d/(d−1)2d/(d-1) is unproved. A recent claimed proof by Rios and Sawyer was explicitly retracted because of an uncontrolled error term; the current version instead establishes an equivalent smooth-Alpert testing characterization of the conjecture [RiosSawyer2026ParaboloidCharacterization].

A resolution must control the extension operator throughout the full supercritical range; the conjecture therefore remains open in every dimension d≥3d\ge3.

References (4)

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.