(n-1)-dimensional Lipschitz manifold embedded into an explicit characterization of these spaces is technically involved. For simplicity consider first a planar domain . For define the (possibly infinite) norm which generalizes the Hölder condition . Then equipped with the previous norm is a Banach space (a general definition of for non-integer can be found in the article for Sobolev-Slobodeckij spaces). For the (n-1)-dimensional Lipschitz manifold define by locally straightening and proceeding as in the definition of . The space can then be identified as the image of the trace operator and there holds that is a surjective, bounded linear operator. For p = 1 For the image of the trace operator is and there holds that is a surjective, bounded linear operator. Right-inverse: trace extension operator The trace operator is not injective since multiple functions in can have the same trace (or equivalently, ). The trace operator has however a well-behaved right-inverse, which extends a function defined on the boundary to the whole domain. Specifically, for there exists a bounded, linear trace extension operator , using the Sobolev-Slobodeckij characterization of the trace operator's image from the previous section, such that for all and, by continuity, there exists with . Notable is not the mere existence but the linearity and continuity of the right inverse. This trace extension operator must not be confused with the whole-space extension operators which play a fundamental role in the theory of Sobolev spaces. Extension to other spaces Higher derivatives Many of the previous results can be extended to with higher differentiability if the domain is sufficiently regular. Let denote the exterior unit normal field on . Since can encode differentiability properties in tangential direction only the normal derivative is of additional interest for the trace theory for . Similar arguments apply to higher-order derivatives for . Let and be a bounded domain with -boundary. Then there exists a surjective, bounded linear higher-order trace operator with Sobolev-Slobodeckij spaces for non-integer defined on through transformation to the planar case for , whose definition is elaborated in the article on Sobolev-Slobodeckij spaces. The operator extends the classical normal traces in the sense that for all Furthermore, there exists a bounded, linear right-inverse of , a higher-order trace extension operator . Finally, the spaces , the completion of in the -norm, can be characterized as the kernel of , i.e. . Less regular spaces No trace in Lp There is