The restriction conjecture for the paraboloid

About 2 years old · traced to 2024

Let $\mathbb P^{n-1}\subseteq\mathbb{R}^n$ be the truncated paraboloid

\mathbb P^{n-1}=\{(\omega',|\omega'|^2):|\omega'|\leq 1\}.

Define its extension operator by

Ef(x):=\int_{\mathbb P^{n-1}}e^{i\omega x}f(\omega)\,d\mathrm{vol}_{\mathbb P^{n-1}}(\omega).

Restriction conjecture. The estimate

\|Ef\|_{L^p(\mathbb{R}^n)}\lesssim\|f\|_{L^p(\mathbb P^{n-1})}

holds for $p>\frac{2n}{n-1}$ . This is presented as the restriction conjecture in dimension $n$ , arising from polynomial-partitioning methods and intended to describe the sharp range of exponents for the paraboloid extension estimate. The source gives no resolution status or further account of what remains open.

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

The restriction conjecture for the paraboloid
Let $f\colon[-1,1]^{d-1}\to\mathbb{C}$ , and let $Ef$ denote the extension operator for the paraboloid. Paraboloid restriction conjecture. For each $q>\frac{2d}{d-1}$ and each $f\colon[-1,1]^d\to\mathbb{C}$ , we have
$\Vert Ef\Vert_q\lesssim_{q,d}\Vert f\Vert_\infty.$
This is a central conjecture in Fourier restriction theory. The supplied text gives no resolution status, so its status is left open.
source: Joshua Zahl, “A discretized Severi-type theorem with applications to harmonic analysis”, arXiv:1801.05106 (2021).

References

Primary source

John Green, Terry Harris, Kaiyi Huang and Arian Nadjimzadah, “A Study Guide for "A Restriction Estimate using Polynomial Partitioning"”, arXiv:2402.03470 (2024).

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