Fourier extension operator

Informally, the Fourier extension operator is an operator that takes a function defined on the surface of the unit sphere in ${\mathbb{ℝ}}^n$ and applies the inverse Fourier transform to produce a function on the entirety of ${\mathbb{ℝ}}^n$.

Definition

Formally, it is an operator $g\mapsto \widehat{gd\sigma }$ such that $$\widehat{gd\sigma }(x)=\underset{S^{n-1}}{\int }{e}^{ix\cdot \xi }g(\xi )d\sigma (\xi )$$where $d\sigma$ denotes surface measure on the unit sphere $S^{n-1}$, $x\in {\mathbb{ℝ}}^n$, and $g\in L^p({\mathbb{ℝ}}^n)$ for some $p\ge 1$. Here, the notation $\hat{f}$ denotes the fourier transform of $f$. In this Lebesgue integral, $\xi$ is a point on the unit sphere and $d\sigma (\xi )$ is the Lebesgue measure on the sphere, or in other words the Lebesgue analog of $dx$.

The Fourier extension operator is the (formal) adjoint of the Fourier restriction operator $g\mapsto \hat{f}{\mathrm{\backslash vline}}_{S^{n-1}}$, where the ${\mathrm{\backslash vline}}_X$ notation represents restriction to the set $X$.

Restriction conjecture

The restriction conjecture states that $\Vert \widehat{gd\sigma }{\Vert }_{L^q({\mathbb{ℝ}}^n)}\lesssim \Vert g{\Vert }_{L^p(S^{n-1})}$ for certain q and n, where $\Vert f{\Vert }_{L^p}$ represents the L^p norm, or ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x{)}^{p}dx$ and $f\lesssim g$ means that $f\le Cg$ for some constant $C$.

The requirements of q and n set by the conjecture are that $\frac{1}{q}<\frac{n-1}{2n}$ and $\frac{1}{q}\le \frac{n-1}{n+1}\frac{1}{p}$.

The restriction conjecture has been proved for dimension $n=2$ as of 2021.

See also

• Fourier transform § Restriction problems
• Mizohata–Takeuchi conjecture