Fourier extension operator

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

Definition

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

The Fourier extension operator is the (formal) adjoint of the Fourier restriction operator ${\displaystyle g\mapsto {\widehat {f}}\vline _{S^{n-1}}}$, where the ${\displaystyle \vline _{X}}$ notation represents restriction to the set ${\textstyle X}$.^[1]

Restriction conjecture

The restriction conjecture states that ${\textstyle \|{\widehat {g\,d\sigma }}\|_{L^{q}(\mathbb {R} ^{n})}\lesssim \|g\|_{L^{p}(S^{n-1})}}$ for certain q and n, where ${\textstyle \|f\|_{L^{p}}}$ represents the L^p norm, or ${\textstyle \int _{-\infty }^{\infty }f(x)^{p}\,dx}$ and ${\textstyle f\lesssim g}$ means that ${\textstyle f\leq Cg}$ for some constant ${\textstyle C}$.^{[1][clarification needed]}

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

The restriction conjecture has been proved for dimension ${\textstyle n=2}$ as of 2021.^[1]

See also

• Fourier transform § Restriction problems
• Mizohata–Takeuchi conjecture