# 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.

# 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.