Restriction conjecture for the hyperbolic paraboloid

Let $d\geq 2$ and let $\vec{v}=(v_1,\ldots,v_{d-1})\in\{-1,1\}^{d-1}$ . Consider the hyperbolic paraboloid

\mathcal{H}_{\vec{v}}^{d-1}:=\left\{(\xi_1,\ldots,\xi_{d-1},\sum_{i=1}^{d-1}v_i\xi_i^2):\xi_i\in[0,1]^{d-1}\right\}\subset\mathbb{R}^d.

For $f$ on $[0,1]^{d-1}$ , define its extension operator by

\widehat{f\,d\sigma}(x):=\int_{[0,1]^{d-1}}f(\xi)e\left(x\cdot\left(\xi_1,\ldots,\xi_{d-1},\sum_{i=1}^{d-1}v_i\xi_i^2\right)\right)\,d\xi,

where $e(t)=e^{2\pi i t}$ . Restriction conjecture. For $p>2+\frac{2}{d-1}$ , there is a constant $C_p$ such that

\|\widehat{f\,d\sigma}\|_{L^p(\mathbb{R}^d)}\leq C_p\|f\|_{L^p([0,1]^{d-1})}.

This is a central conjecture in restriction theory for the hyperbolic paraboloid. It is solved when $d=2$ by Fefferman, but it remains open in higher dimensions.

Sources & referencesView supporting material

Primary source

Changkeun Oh, “On decoupling and restriction estimates”, arXiv:2503.14276 (2025).

Additional references

4 papers in this index state this conjecture (2017–2025). The statement above is taken from the most recent of them; the others are arXiv:2503.15760, arXiv:2307.06445, arXiv:1701.03523.

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Progress summary

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Partially solved

The conjecture is proved in two dimensions, but higher-dimensional cases remain open despite several recent improvements to the range of exponents.

The conjecture asks for the predicted extension estimate for hyperbolic paraboloids when $p>2+\frac{2}{d-1}$ . The two-dimensional case is known, while the full statement remains unresolved in higher dimensions.

Known results

The two-dimensional case was proved independently by Fefferman and Zygmund.
In 2020, bilinear estimates gave bounds under signature restrictions, including $p>\frac{2(d+2)}{d}$ after $\varepsilon$ -removal.
In 2024, polynomial partitioning produced estimates in selected higher-dimensional signatures, including $p>2.75$ for a hyperbolic paraboloid in $\mathbb{R}^5$ .

2025–2026 partial progress

A 2025 preprint improved the three-dimensional bound to $p>\frac{22}{7}$ , still short of the conjectural threshold $p>3$ . A 2026 paper studied maximizers conditionally on boundedness and reported the three-dimensional off-scaling-line range below $p=\frac{11}{4}$ ; it did not prove the conjecture. No retrieved source reports a full proof, counterexample, or AI-generated solution.

Current status (as of August 2026): The two-dimensional case is settled and higher-dimensional partial estimates are known, but the full restriction conjecture remains open.

Sources

No solutions have been posted yet.