. By density of in such an extension is possible if is continuous with respect to the -norm. The proof of this, i.e. that there exists (depending on and ) such that for all is the central ingredient in the construction of the trace operator. A local variant of this estimate for -functions is first proven for a locally flat boundary using the divergence theorem. By transformation, a general -boundary can be locally straightened to reduce to this case, where the -regularity of the transformation requires that the local estimate holds for -functions. With this continuity of the trace operator in an extension to exists by abstract arguments and for can be characterized as follows. Let be a sequence approximating by density. By the proven continuity of in the sequence is a Cauchy sequence in and with limit taken in . The extension property holds for by construction, but for any there exists a sequence which converges uniformly on to , verifying the extension property on the larger set . The case p = ∞ If is bounded and has a -boundary then by Morrey's inequality there exists a continuous embedding , where denotes the space of Lipschitz continuous functions. In particular, any function has a classical trace and there holds Functions with trace zero The Sobolev spaces for are defined as the closure of the set of compactly supported test functions with respect to the -norm. The following alternative characterization holds: where is the kernel of , i.e. is the subspace of functions in with trace zero. Image of the trace operator For p > 1 The trace operator is not surjective onto if , i.e. not every function in is the trace of a function in . As elaborated below the image consists of functions which satisfy a -version of Hölder continuity. Abstract characterization An abstract characterization of the image of can be derived as follows. By the isomorphism theorems there holds where denotes the quotient space of the Banach space by the subspace and the last identity follows from the characterization of from above. Equipping the quotient space with the quotient norm defined by the trace operator is then a surjective, bounded linear operator . Characterization using Sobolev–Slobodeckij spaces A more concrete representation of the image of can be given using Sobolev-Slobodeckij spaces which generalize the concept of Hölder continuous functions to the -setting. Since is a