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.