The fractional Sobolev space on a compact $C^1$ boundary

Assume Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) and the boundary conventions of Bounded C1 domains and their outward normals. Let $\mathrm{\Omega }\subset {\mathbb{R}}^n$, $n\ge 2$, be a bounded $C^1$ domain, let $0<s<1$, $1\le p<\mathrm{\infty }$ and $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$. Fix a finite family of boundary charts: for each $j$ let $W_j\subseteq {\mathbb{R}}^n$ be open and ${\mathrm{\Phi }}_j:W_j\to {\mathrm{\Phi }}_j(W_j)\subseteq B_j\times \mathbb{R}$ a flattening chart as in Bounded C^k domains and boundary charts, so that ${\mathrm{\Phi }}_j(\mathrm{\Omega }\cap W_j)={\mathrm{\Phi }}_j(W_j)\cap \{t<0\}$ and the boundary corresponds to the graph $\{t=0\}$; write ${\mathrm{\Psi }}_j:=\pi \circ {\mathrm{\Phi }}_j{\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }\cap W_j}$ for the induced parametrisation of $\mathrm{\partial }\mathrm{\Omega }\cap W_j$ by an open subset $V_j:={\mathrm{\Psi }}_j(\mathrm{\partial }\mathrm{\Omega }\cap W_j)\subseteq {\mathbb{R}}^{n-1}$, and assume the $\mathrm{\partial }\mathrm{\Omega }\cap W_j$ cover $\mathrm{\partial }\mathrm{\Omega }$. Fix a subordinate finite ambient partition: nonnegative ${\chi }_j\in C_c^{\mathrm{\infty }}({\mathbb{R}}^n)$ with $\mathrm{supp}{\chi }_j\subseteq W_j$ and ${\sum }_j{\chi }_j=1$ on a neighbourhood of $\mathrm{\partial }\mathrm{\Omega }$ (Finite ambient partitions near compact sets). Finally let $\mathrm{\partial }\mathrm{\Omega }$ carry the chart-independent surface measure of Surface integration on compact C1 hypersurfaces, which fixes the meaning of almost-everywhere equality and of $L^p(\mathrm{\partial }\mathrm{\Omega };\mathbb{K})$ (The space $L^p(\mu )$ as the quotient by null functions).

For a Borel function $g:\mathrm{\partial }\mathrm{\Omega }\to \mathbb{K}$ put $\mathrm{\parallel }g{\mathrm{\parallel }}_{W^{s,p}(\mathrm{\partial }\mathrm{\Omega })}:={\sum }_j\parallel ({\chi }_{j}g)\circ {\mathrm{\Psi }}_j^{-1}{\parallel }_{W^{s,p}({\mathbb{R}}^{n-1})},$ where the chart representation $({\chi }_{j}g)\circ {\mathrm{\Psi }}_j^{-1}$ is read in graph coordinates and extended by zero off $V_j$, and the norm on the right is that of The Gagliardo--Slobodeckij space on Euclidean space. The fractional Sobolev space of the boundary is $W^{s,p}(\mathrm{\partial }\mathrm{\Omega };\mathbb{K}):=\{g\in L^p(\mathrm{\partial }\mathrm{\Omega };\mathbb{K}):\mathrm{\parallel }g{\mathrm{\parallel }}_{W^{s,p}(\mathrm{\partial }\mathrm{\Omega })}<\mathrm{\infty }\}\mathrm{.}$ The right-hand side is a finite sum of finite norms of compactly supported chart representations, so $W^{s,p}(\mathrm{\partial }\mathrm{\Omega };\mathbb{K})$ is exactly the set of $L^p(\mathrm{\partial }\mathrm{\Omega })$ classes whose chart representations all lie in the Euclidean space of The Gagliardo--Slobodeckij space on Euclidean space; that the resulting space and the topology of the norm do not depend on the choices of atlas and partition, up to equivalence of norms, is proved in Chart independence of the fractional boundary norm ↗ and is not assumed here.

Three conventions belong to the definition. First, membership is a property of the $L^p(\mathrm{\partial }\mathrm{\Omega })$ class: $g$ is an almost-everywhere class with respect to the surface measure, and the chart representations are classes in $W^{s,p}({\mathbb{R}}^{n-1})$; representative independence for the Euclidean factor is established together with the well-definedness lemma on this page and is not presupposed here. Second, the norm is a finite sum over a finite atlas, and each ${\chi }_{j}g$ is supported in the interior of the chart, so the localisations are compactly supported and no boundary behaviour of ${\mathrm{\Psi }}_j^{-1}$ outside $V_j$ enters. Third, on this page the exponent used is the trace exponent $s=\theta =1-1\mathrm{/}p$ for $1<p<\mathrm{\infty }$.

Schikorra, Section V.1, printed pp. 96-97, patches the local Slobodeckij norms over the boundary through finitely many charts. Gagliardo, printed pp. 286-289, defines the boundary norm through finitely many local representations and their incremental quotients, and Kampanou, Theorems 3.4-3.5, printed pp. 27-31, carries out the same localisation on $C^l$ domains with a partition of unity. The finite atlas and partition used below are exactly the data fixed by these constructions.