Let n≥ 2, let P^n-1=(xi,|xi|^2):xi∈ R^n-1⊂ R^n, and let f satisfy operatornamesuppwidehat f⊂ N_R^-1( P^n-1). Decompose f=∑_thetaf_theta, where the Fourier supports of the f_theta lie in caps of diameter R^-1/2 on P^n-1. The reverse square-function conjecture asserts that, for every varepsilon>0, [|(∑_theta|f_theta|^2)^1/2|_L^2n/(n-1)( R^n)lesssim_varepsilonR^varepsilon|f|_L^2n/(n-1)( R^n).] The estimate is known in dimension n=2; the conjecture remains open for n≥ 3.

Reverse square-function conjecture

Let $n\ge 2$ , let $\mathcal P^{n-1}=\{(\xi,|\xi|^2):\xi\in\mathbb R^{n-1}\}\subset\mathbb R^n$ , and let $f$ satisfy $\operatorname{supp}\widehat f\subset N_{R^{-1}}(\mathcal P^{n-1})$ . Decompose $f=\sum_{\theta}f_\theta$ , where the Fourier supports of the $f_\theta$ lie in caps of diameter $R^{-1/2}$ on $\mathcal P^{n-1}$ . The reverse square-function conjecture asserts that, for every $\varepsilon>0$ ,

\left\|\left(\sum_{\theta}|f_\theta|^2\right)^{1/2}\right\|_{L^{2n/(n-1)}(\mathbb R^n)}\lesssim_{\varepsilon}R^{\varepsilon}\|f\|_{L^{2n/(n-1)}(\mathbb R^n)}.

The estimate is known in dimension $n=2$ ; the conjecture remains open for $n\ge 3$ .

References

Primary source

arXiv

Additional references

Strip Decoupling Inequalities for the Paraboloid in R^3 — arXiv — Jacob Glidewell

fourier-restriction-theory harmonic-analysis

Posted 5 days ago · edited 5 days ago

Progress summary

Refreshed 5 days ago

Claimed progress

A new sharp special case improves related estimates, but the conjecture remains open in three or more dimensions.

The reverse square-function conjecture predicts sharp bounds for functions Fourier-supported near a paraboloid. Terence Tao described the higher-dimensional conjecture as open in 2020; the two-dimensional case is known.

Known results

$n=2$ : follows from the Córdoba--Fefferman argument, with the logarithmic loss removable.
$n\ge 3$ : the conjecture remains open; it is at least as difficult as related restriction and Kakeya conjectures (Tao, 2020).
The known decoupling estimate has been shown sharp in several supercritical regimes, but this does not prove the conjecture (Gan and later work, 2024).

October 2026 strip-decoupling advance

Jacob Glidewell's paper claims a sharp decoupling exponent for strips near the three-dimensional paraboloid, using two-ends Furstenberg incidence bounds. This is a specific partial result, not a solution of the general reverse square-function conjecture.

Current status (as of October 2026): The two-dimensional case is settled, while the general conjecture for $n\ge 3$ remains open; the reported three-dimensional strip estimate is claimed progress, not a verified resolution.

Sources

No solutions have been posted yet.