Periodic Sobolev domain for a circle

Assume the Axiom of Choice and Countable Choice. Let $S^1=\mathbb{R}\mathrm{/}(2\pi \mathbb{Z})$ with angular coordinate $\varphi$, and identify $L^2(S^1;\mathbb{C})$ with $L^2((0,2\pi ),d\varphi ;\mathbb{C})$; the omitted endpoint has measure zero. Let $H^2(0,2\pi )$ be the interval Sobolev space whose elements and first two weak derivatives are in $L^2$. For $f\in H^2(0,2\pi )$ use the continuous endpoint representatives of $f$ and $f^{\mathrm{\prime }}$, and denote their traces by $f^{\ast }(0),f^{\ast }(2\pi )$ and $(f^{\mathrm{\prime }}{)}^{\ast }(0),(f^{\mathrm{\prime }}{)}^{\ast }(2\pi )$. Define the periodic second-order domain $H_{\mathrm{p}\mathrm{e}\mathrm{r}}^2(S^1)=\{f\in H^2(0,2\pi ):f^{\ast }(0)=f^{\ast }(2\pi ),(f^{\mathrm{\prime }}{)}^{\ast }(0)=(f^{\mathrm{\prime }}{)}^{\ast }(2\pi )\}\mathrm{.}$ It is the domain used for the periodic realization of the second derivative; it imposes no boundary at the angular seam beyond periodic matching.

The Axiom of Choice

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

One-dimensional Sobolev spaces on intervals

Endpoint trace bound and decay of H1 functions