**Theorem 1.** Let be differentiable with bounded and continuous derivatives and Then and

In particular if for all then and the chain rule holds provided

Let be open sets in We say is **compactly contained in** denoted if for any such that is compact. We now extend functions in to functions on

**Theorem 2.** Let be bounded, be a -manifold, and Then there exists a linear map such that for all we have

- where depends upon and

*Proof Outline*. Let and be the tangent hyperplane at –which satisfies Since is locally homeomorphic to approximately bisects a small ball around one which intersects with and one that does not. We will respectively call these and Suppose and define

Note that the two parts of the function agree on and that since if we let then

on (where .) Thus derivatives agree on Thus

then allows one to use partitions of unity on to write in local coordinates and establish the bounds in (3). (2) is then established by continuously sending the function to in or a slightly larger set. is then defined as elsewhere.

Above we assumed If we assume more generally that as in the hypotheses, then we can approximate (as seen in the last post) by functions in (a special case of which is on if is bounded and is a -manifold [see 5.3.2 and 5.3.3 in [2]]) which yields the result with consequence (1).

Lastly we begin with a function and attempt to extend it to the boundary.

**Theorem 3.** Let be bounded and be a -manifold. Then there exists a bounded linear operator such that

- if

The function is called the **trace** of on We also have

**Proposition 4.** Again let be bounded and be a -manifold. Let Then

This sort of seems intuitive since it’s like saying the summable extension is on the boundary iff it belongs to the class of functions whose derivatives vanish on the boundary. Recall elements of are functions whose support is a compact subset of or whose support extends to The proof is rather technical in the forward direction (see [2] for details). The backwards direction uses density of in

Regarding Theorem 3, at first glance this reminded me of Stokes’ theorem on manifolds, although Evans doesn’t mention anything about it (nor uses it in either of the two proofs). Recall the statement.

**Stokes’ Theorem**. Let be a an -dimensional, oriented, and compact manifold with boundary such that is an -form on Then

Since its derivatives were integrable (i.e. the left side), and under the trace is sent to a function which is integrable on the boundary.

[1] Lieb, Elliot and Michael Loss. *Analysis*. 2nd Edition. Graduate Studies in Mathematics. Vol. 14. American Mathematical Society. 2001.

[2] Evans, Lawrence. *Partial Differential Equations*. Graduate Studies in Mathematics. Vol. 19. American Mathematical Society. 1998.