Sobolev extension domains and extension operators

Assume Countable Choice. Fix $k\in {\mathbb{N}}_0$, $1\le p\le \mathrm{\infty }$, an open set $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ with $n\ge 1$, and $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$.

Restriction. Let $F\in W^{k,p}({\mathbb{R}}^n;\mathbb{K})$ and let $\alpha \in {\mathbb{N}}_0^n$ with $\mathrm{\mid }\alpha \mathrm{\mid }\le k$. By the restriction and locality clause of Linearity, locality, and commutation of weak derivatives, applied with $V=\mathrm{\Omega }$, the class $F{\mathrm{\mid }}_{\mathrm{\Omega }}$ lies in $W^{k,p}(\mathrm{\Omega };\mathbb{K})$ and $D^{\alpha }(F{\mathrm{\mid }}_{\mathrm{\Omega }})=(D^{\alpha }F){\mathrm{\mid }}_{\mathrm{\Omega }}$ almost everywhere on $\mathrm{\Omega }$; by the uniqueness of weak derivatives the derivative class of the restriction is determined by the class of $F$ alone. So restriction is a well-defined operation on Sobolev classes, and the norm formula of Integer-order Sobolev spaces and their norms gives $\mathrm{\parallel }F{\mathrm{\mid }}_{\mathrm{\Omega }}{\mathrm{\parallel }}_{W^{k,p}(\mathrm{\Omega })}\le \mathrm{\parallel }F{\mathrm{\parallel }}_{W^{k,p}({\mathbb{R}}^n)}$, since each restricted derivative has no larger $L^p$ norm.

Extension domain. For fixed $k,p,\mathrm{\Omega }$ and scalar field $\mathbb{K}$, call $\mathrm{\Omega }$ a $W^{k,p}$-extension domain if there is a bounded linear operator $E:W^{k,p}(\mathrm{\Omega };\mathbb{K})⟶W^{k,p}({\mathbb{R}}^n;\mathbb{K})$ such that, for every $u\in W^{k,p}(\mathrm{\Omega };\mathbb{K})$, $(Eu){\mathrm{\mid }}_{\mathrm{\Omega }}=u\text{as an almost-everywhere class on }\mathrm{\Omega }\mathrm{.}$ Equivalently, the restriction map in the first paragraph possesses a bounded linear right inverse. Boundedness of $E$ is the finiteness of the operator norm $\mathrm{\parallel }E\mathrm{\parallel }=\sup \{\mathrm{\parallel }Eu{\mathrm{\parallel }}_{W^{k,p}({\mathbb{R}}^n)}:u\in W^{k,p}(\mathrm{\Omega };\mathbb{K}),\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{k,p}(\mathrm{\Omega })}\le 1\};$ the displayed right-inverse identity is an identity of $L^p(\mathrm{\Omega })$ classes, not of pointwise values.

The operator, and the numerical bound $\mathrm{\parallel }E\mathrm{\parallel }$, may depend on $k$, $p$, $\mathrm{\Omega }$ and $\mathbb{K}$. This definition asserts no common operator for all indices at once, no linearity of some canonically selected extension, and no control of pointwise values on $\mathrm{\partial }\mathrm{\Omega }$; in particular it does not define a trace operator. No claim is made here that zero extension is an extension operator for a general open set, and none that any particular open set fails to be an extension domain; both assertions belong to later items of this page and to its companion.

Kinnunen, Definition 3.42, printed p. 84, introduces the extension domain and the bounded right inverse of the restriction map; Theorem 3.43 there transfers whole-space Sobolev inequalities through such an operator. Laugesen, Theorem 3.12, printed pp. 60–62, constructs such an operator for bounded $C^1$ graph domains and finite $p$, and Corollary 3.13 extends the first-order result to $p=\mathrm{\infty }$ by direct local bounds.