Fourier Bessel-potential and restriction-quotient Sobolev spaces

Assume the Axiom of Choice when using the well-definedness results below; it is inherited conservatively from the published $L^2$ completeness supplier. Let $d\ge 1$ and $s>d\mathrm{/}2$. Write $L_s^2({\mathbb{R}}^d)$ for the equivalence classes modulo Lebesgue-almost-everywhere equality of measurable complex functions $g$ satisfying $\mathrm{\parallel }g{\mathrm{\parallel }}_{L_s^2}^2={\int }_{{\mathbb{R}}^d}(1+\mathrm{\mid }\xi {\mathrm{\mid }}^2{)}^s\mathrm{\mid }g(\xi ){\mathrm{\mid }}^{2}d\xi <\mathrm{\infty }\mathrm{.}$ For such $g$, the inverse Fourier representative is the function $U_g(x)={\int }_{{\mathbb{R}}^d}g(\xi )e^{2\pi ix\cdot \xi }d\xi \mathrm{.}$ The whole-space Bessel-potential space $H^s({\mathbb{R}}^d)$ consists of these representatives, with norm $\mathrm{\parallel }{U}_g{\mathrm{\parallel }}_{H^s({\mathbb{R}}^d)}=\mathrm{\parallel }g{\mathrm{\parallel }}_{L_s^2}$. For the admitted domain $\mathrm{\Omega }$ of Radial kernels, polynomial augmentation, and site geometry, define $H^s(\mathrm{\Omega })$ to be the restrictions $U{\mathrm{\mid }}_{\mathrm{\Omega }}$ of whole-space members, with the quotient norm $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^s(\mathrm{\Omega })}=\inf \{\mathrm{\parallel }U{\mathrm{\parallel }}_{H^s({\mathbb{R}}^d)}:U{\mathrm{\mid }}_{\mathrm{\Omega }}=u\}\mathrm{.}$ The whole-space inner product is $⟨U_g,U_h{⟩}_{H^s}=\int (1+\mathrm{\mid }\xi {\mathrm{\mid }}^2{)}^{s}g(\xi )\overline{h(\xi )}d\xi$, linear in the first variable as in Real and complex inner-product spaces and their induced length. The inverse Fourier map is injective, whole-space representatives are continuous, the whole-space model is Hilbert, and Schwartz functions are dense in that norm; these facts are supplied by The weighted Fourier model is an injective Hilbert representation ↗ and Schwartz functions are dense in the Fourier Bessel-potential norm ↗. The restriction quotient is a norm by bounded point evaluation; its Hilbert realization as the restricted native space is proved in The restricted Matérn native space is the Sobolev quotient space ↗. No extension operator or identification with an intrinsic fractional Sobolev norm on $\mathrm{\Omega }$ is part of this definition.