# Fractional Sobolev Spaces

Recall for a function for we define the **Fourier transform** of by

and the **inverse Fourier transform** by

If then a.e..

Now if and with we say is the **weak derivative** of provided

with and

The **Sobolev space** is then defined as the set of all functions such that for all These become Banach spaces under the norm

and Hilbert when whence we denote

For we can say that consists of such that

(using Fourier multipliers).

**Bessel Potential Spaces**:

We can generalize with and and define the **Bessel potential space** as all such that Note the fractional power of the Laplacian is defined by virtue of the fact that is dense in , so we say

These spaces satisfy our desire of being Banach (and Hilbert when ).

**Sobolev-Slobodeckij Spaces:**

There is an alternative approach. Recall that the **Holder space** is defined as all functions such that

That is, it is the set of functions on which are and whose -th partial derivatives are bounded and Holder continuous of degree These spaces are Banach under the above norm. We can generalize the Sobolev spaces to incorporate similar properties. Let us define the **Slobodeckij norm** for with and by

The corresponding **Sobolev-Slobodeckij space** is defined as all functions such that

where This becomes a Banach space under the norm

[1] http://en.wikipedia.org/wiki/Sobolev_space (unclear text references)

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

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

# Chain Rule, Extensions, and Trace in W^{1,p}

**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.

# Sobolev Spaces

Let and be the collection of functions (or if preferred) such that

It turns out are Banach spaces (see [2] for proof), called **Sobolev spaces**, under the above norm. In particular The spaces are Hilbert with inner product

We can also define the **local Sobolev spaces** as the subset of such that for We have

and where under dictionary ordering on provided has a finite Lebesgue measure. Thus in particular

is a -module by a simple boundedness argument following from: if and then is finite in Thus Correspondingly this gives us action on (which denotes the distributions on for us, although the action works on the dual space as well) defined by

**Theorem 1.** Let be bounded in and Then there exists a sequence of functions such that

We can also define (and similarly ) as the set of functions such that the following norm is finite

Thus

I’m wondering if we could consider attempting to define (and similarly ) as the collection of functions such that

is finite. This would require all derivatives to eventually get to in such a way that the sum converges. It would thus include polynomials if is bounded and hence be nonempty. In particular it would obey for all And for all

[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.