Degree-independent Sobolev extension on locally uniform domains

by Luke G. Rogers

We consider the problem of constructing extensions $$L^{p}_{k}(\Omega)\rightarrow L^{p}_{k}(\mathbb{R}^{n})$$, where $$L^{p}_{k}$$ is the Sobolev space of functions with $$k$$ derivatives in $$L^{p}$$ and $$\Omega\subset\mathbb{R}^{n}$$ is a domain. In the case of Lipschitz $$\Omega$$, Calderón gave a family of extension operators depending on $$k$$, while Stein later produced a single ($$k$$-independent) operator. For the more general class of locally-uniform domains, which includes examples with highly non-rectifiable boundaries, a $$k$$-dependent family of operators was constructed by Jones. In this work we produce a $$k$$-independent operator for all spaces $$L^{p}_{k}(\Omega)$$ on a locally uniform domain $$\Omega$$.

Published version from JFA