# 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.

# Distributions

Let and denote the set of smooth functions on with compact support.

**Definition 1.** Let and be a sequence in We say **converges to ** denoted if

- for all for a fixed compact
- uniformly on

By we mean

is a vector space, and the above notion of convergence allows us to define closed sets and hence turns into a topological vector space called the **space of test functions** of and we denote it

**Definition 2.** A **distribution** is a continuous linear functional on We will denote the set of distributions on by –noting that we are only considering continuous linear functionals.

We say a sequence of distributions **converges** to the distribution denoted if for all This gives us a topological vector space structure on as well.

**Definition 3.** Let be a distribution and We define the -th **distributional (weak) derivative** of by

where We may also write

We will omit if The **gradient of** is simply

Let us define the space as the space of functions such that

for all compact We define **strong convergence** on as convergence in and **weak convergence** on as weak (pointwise) convergence in for all compact

Let then clearly

Let and and define

This integral is finite since both functions are integrable (use integration by parts and establish bounds). Also if uniformly, then

where which gives us the continuity of So We also have

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