Fourier restriction and adjoint extension operators

Assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Fix $n\ge 2$ and a compact embedded $C^{\mathrm{\infty }}$ hypersurface $S\subseteq {\mathbb{R}}^n$. Its surface measure $\sigma$ is fixed as follows.

• For $S=S^{n-1}$ it is the polar surface measure of The polar surface set function on the unit sphere, which agrees with the chart surface measure by Agreement with the existing polar sphere measure.
• For a general compact hypersurface it is the chart measure of Surface integration on compact C1 hypersurfaces, a finite Borel measure whose graph density is $\sqrt{1+\mathrm{\mid }Dh{\mathrm{\mid }}^2}$ by Chart and partition independence of surface measure.

Restriction. For a Schwartz function $f\in \mathcal{S}({\mathbb{R}}^n)$ the transform $\widehat{f}$ is again a Schwartz function, in particular an actual smooth function on ${\mathbb{R}}^n$ (Fourier transform acts continuously on Schwartz space, Schwartz space and its seminorms), so $R_{0}f:=\widehat{f}{\mathrm{\mid }}_S$ is a pointwise-defined function on $S$.

Extension. For $g\in L^1(\sigma ;\mathbb{C})$ the set function $g\sigma :E\mapsto {\int }_{E}gd\sigma$ is a complex measure on the Borel sets of $S$ with total variation $\mathrm{\mid }g\mathrm{\mid }\sigma$ (A complex L^1 density defines a complex measure whose total variation is |h| dmu, A complex measure is a finite-valued countably additive set function), and one writes $(g\sigma {)}^{\vee }(x):=\widehat{g\sigma }(-x)$ for the reflected transform of that finite measure, so that $Eg:=(g\sigma {)}^{\vee },Eg(x)={\int }_S{e}^{2\pi ix\cdot \omega }g(\omega )d\sigma (\omega )(x\in {\mathbb{R}}^n)\mathrm{.}$ The transform theorem Fourier transform of a finite complex Borel measure gives that $Eg$ is a bounded uniformly continuous function and that $\mathrm{\mid }Eg(x)\mathrm{\mid }\le \mathrm{\mid }g\sigma \mathrm{\mid }({\mathbb{R}}^n)={\int }_S\mathrm{\mid }g\mathrm{\mid }d\sigma =\mathrm{\parallel }g{\mathrm{\parallel }}_{L^1(\sigma )}(x\in {\mathbb{R}}^n)\mathrm{.}$ For $g\in L^2(\sigma ;\mathbb{C})$, finiteness of $\sigma$ gives $g\in L^1(\sigma ;\mathbb{C})$ and the additional estimate $\mathrm{\mid }Eg(x)\mathrm{\mid }\le \sigma (S{)}^{1\mathrm{/}2}\mathrm{\parallel }g{\mathrm{\parallel }}_{L^2(\sigma )}(x\in {\mathbb{R}}^n),$ where the last inequality is the $p=p^{\mathrm{\prime }}=2$ case of the complex Holder/Cauchy–Schwarz inequality Complex Holder, Minkowski, and the quotient norm together with finiteness of the surface measure; here $L^p(\sigma ;\mathbb{C})$ are the complex Lebesgue classes of Complex Lp classes and Euclidean test-function conventions. The assignment $g\mapsto Eg$ is complex-linear, since $g\mapsto g\sigma$ is additive in the density and integration against a finite measure is additive. The symbol $R$ denotes the unique bounded extension of $R_0$ when one exists.

Restriction starts on Schwartz data by necessity: for $S=S^{n-1}$ the measure $\sigma$ is carried by the Lebesgue-null set $S$ (The unit sphere is Lebesgue null), so no pointwise restriction is available for a general ambient $L^{p^{\mathrm{\prime }}}$ class; the companion page's counterexample records the explicit failure of well-definedness on equivalence classes.