Extension Operators for Fractional Sobolev Spaces on Lipschitz Submanifolds
Philipp Weder
TL;DR
The paper addresses extending $W^{s,p}$-functions from a Lipschitz subset $\Lambda$ of a compact Lipschitz submanifold $M$ to the ambient manifold by adapting the Euclidean extension construction to manifolds via local parameterizations. It proves the existence of a linear continuous extension operator $\mathsf{E}_{\Lambda}^{M}$ for $s\in(0,1)$ and $p\in[1,\infty)$ with an explicit size-dependent bound $\|\mathsf{E}_{\Lambda}^{M}(u)\|_{s,p,M}^p \le C^{(0)}\left(1 + |\Lambda|^{-\frac{sp}{k}} + |\Lambda|^{\frac{(1-s)p}{k}}\right) \|u\|_{0,p,\Lambda}^p + C^{(s)}|u|_{s,p, \Lambda}^p$, where constants depend on the ambient data but not on $|\Lambda|$. The construction uses a partition of unity to localize the problem and carefully analyzes scaling via local charts, yielding explicit, size-aware continuity constants. This has practical implications for numerical analysis in geometry processing, such as geometry simplification and error estimation, and points toward extensions to $s\ge 1$ under stronger regularity assumptions. Overall, the work extends fractional Sobolev extension theory to subsets of Lipschitz submanifolds with transparent geometric scaling.
Abstract
A well-known result is that any Lipschitz domain is an extension domain for $W^{s,p}$. This paper extends this result to Lipschitz subsets of compact Lipschitz submanifolds of $\mathbb{R}^n$. We adapt the construction of an extension operator for Lipschitz domains in arXiv:1104.4345v3 to manifolds via local coordinate charts. Furthermore, the dependence on the size of the extension domain is explicit in all estimates. This result is motivated by applications in numerical analysis, most notably geometry simplification, where the explicit dependence of the continuity constant on the domain size is essential.