---
tag: 9CE11G48JY
kind: conjecture
name: Restriction conjecture
msc: [42B10, 42B20]
uses: [DPVM34ADZV, SASP14MA1Z, T62369RFH6, CG9JEATW3Q]
about: [DPVM34ADZV]
sources:
  - url: https://en.wikipedia.org/wiki/Restriction_conjecture
history:
  posed_by: "Elias Stein"
---

## Statement

Let $n \ge 2$, let $S^{n-1} \subseteq \R^n$ be the unit [[sphere|T62369RFH6]], and let $\sigma$ be its surface measure.
Let $\hat f$ be the [[Fourier transform|DPVM34ADZV]] of $f$.
Let $1 \le p < \frac{2n}{n+1}$, let $p'$ be given by $1/p + 1/p' = 1$, and let $1 \le q \le \frac{n-1}{n+1} p'$.
Then there is a constant $C$ such that
$$\Big( \int_{S^{n-1}} \abs{\hat f(\xi)}^q \, d\sigma(\xi) \Big)^{1/q} \le C \norm{f}_{L^p(\R^n)}$$
for every [[Schwartz function|CG9JEATW3Q]] $f$ on $\R^n$.

## Remarks

The conditions on $p$ and $q$ are necessary.
The condition $p < 2n/(n+1)$ comes from the function $\widehat{d\sigma}$.
It decays exactly like $\abs{x}^{-(n-1)/2}$, so it lies in $L^{p'}$ only for $p' > 2n/(n-1)$.
By duality, the conjecture requires $\widehat{d\sigma}$ to lie in $L^{p'}$.
The condition $q \le \frac{n-1}{n+1}p'$ comes from the Knapp example, a function whose Fourier transform is concentrated on a small cap of the sphere.

Status: the conjecture holds for $n = 2$ and is open for every $n \ge 3$.
The [[Stein–Tomas restriction theorem|ZZVWHNFY6F]] proves the case $q = 2$, where the range is $p \le 2(n+1)/(n+3)$.
The conjecture implies the Kakeya conjecture that every [[Kakeya set|6PEN7AKC7M]] in $\R^n$ has Hausdorff dimension $n$.

## Explanations

### L1 human

The conjecture says exactly when the Fourier transform of a function in $L^p$ can be restricted to the sphere, with a bound in an $L^q$ norm on the sphere.
The two conditions on $p$ and $q$ are known to be needed, and the question is whether they are enough.
It is one of the main open problems of harmonic analysis, linked to the geometry of thin tubes in Kakeya sets.