User:Silly rabbit/Sobolev spaces on the unit circle

We start by introducing Sobolev spaces in the simplest settings, the one-dimensional case on the unit circle. In this case the Sobolev space $W^{k,p}$ is defined to be the subset of L^p such that f and its weak derivatives up to some order k have a finite L^p norm, for given p ≥ 1. Some care must be taken to define derivatives in the proper sense. In the one-dimensional problem it is enough to assume $f^{(k-1)}$ is differentiable almost everywhere and is equal almost everywhere to the Lebesgue integral of its derivative (this gets rid of examples such as Cantor's function which are irrelevant to what the definition is trying to accomplish).

With this definition, the Sobolev spaces admit a natural norm,

\|f\|_{k,p}={\Big (}\sum _{i=0}^{k}\|f^{(i)}\|_{p}^{p}{\Big )}^{1/p}={\Big (}\sum _{i=0}^{k}\int |f^{(i)}(t)|^{p}\,dt{\Big )}^{1/p}.

$W^{k,p}$ equipped with the norm $\|\cdot \|_{k,p}$ is a Banach space. It turns out that it is enough to take only the first and last in the sequence, i.e., the norm defined by

\|f^{(k)}\|_{p}+\|f\|_{p}

is equivalent to the norm above.

The case p = 2

Sobolev spaces with p = 2 are especially important because of their connection with Fourier series and because they form a Hilbert space. A special notation has arisen to cover this case:

\,H^{k}=W^{k,2}.

The space $H^{k}$ can be defined naturally in terms of Fourier series, namely,

H^{k}({\mathbb {T} })={\Big \{}f\in L^{2}({\mathbb {T} }):\sum _{n=-\infty }^{\infty }(1+n^{2}+n^{4}+\dotsb +n^{2k})|{\widehat {f}}(n)|^{2}<\infty {\Big \}}

where ${\widehat {f}}$ is the Fourier series of $f$ . As above, one can use the equivalent norm

\|f\|^{2}=\sum _{n=-\infty }^{\infty }(1+n^{2k})|{\widehat {f}}(n)|^{2}.

Both representations follow easily from Parseval's theorem and the fact that differentiation is equivalent to multiplying the Fourier coefficient by in.

Furthermore, the space H^k admits an inner product, like the space H⁰ = L². In fact, the H^k inner product is defined in terms of the L² inner product:

\langle u,v\rangle _{H^{k}}=\sum _{i=0}^{k}\langle D^{i}u,D^{i}v\rangle _{L_{2}}.

The space H^k becomes a Hilbert space with this inner product.

Other examples

Some other Sobolev spaces permit a simpler description. For example, $W^{1,1}(0,1)$ is the space of absolutely continuous functions on $(0,1)$ , while W^1,∞(I) is the space of Lipschitz functions on $I$ , for every interval $I$ . All spaces W^k,∞ are (normed) algebras, i.e. the product of two elements is once again a function of this Sobolev space, which is not the case for p < ∞. (E.g., functions behaving like |x|^−1/3 at the origin are in L², but the product of two such functions is not in L²).

Sobolev spaces with non-integer k

To prevent confusion, when talking about k which is not integer we will usually denote it by s, i.e. $W^{s,p}$ or $H^{s}.$

The case p = 2

The case p = 2 is the easiest since the Fourier description is straightforward to generalize. We define the norm

\|f\|_{s,2}^{2}=\sum (1+n^{2})^{s}|{\widehat {f}}(n)|^{2}

and the Sobolev space $H^{s}$ as the space of all functions with finite norm.

Fractional order differentiation

A similar approach can be used if p is different from 2. In this case Parseval's theorem no longer holds, but differentiation still corresponds to multiplication in the Fourier domain and can be generalized to non-integer orders. Therefore we define an operator of fractional order differentiation of order s by

F^{s}(f)=\sum _{n=-\infty }^{\infty }(in)^{s}{\widehat {f}}(n)e^{int}

or in other words, taking Fourier transform, multiplying by $(in)^{s}$ and then taking inverse Fourier transform (operators defined by Fourier-multiplication-inverse Fourier are called multipliers and are a topic of research in their own right). This allows to define the Sobolev norm of $s,p$ by

\|f\|_{s,p}=\|f\|_{p}+\|F^{s}(f)\|_{p}

and, as usual, the Sobolev space is the space of functions with finite Sobolev norm.

Complex interpolation

Another way of obtaining the "fractional Sobolev spaces" is given by complex interpolation. Complex interpolation is a general technique: for any 0 ≤ t ≤ 1 and X and Y Banach spaces that are continuously included in some larger Banach space we may create "intermediate space" denoted [X,Y]_t. (below we discuss a different method, the so-called real interpolation method, which is essential in the Sobolev theory for the characterization of traces).

Such spaces X and Y are called interpolation pairs.

We mention a couple of useful theorems about complex interpolation:

Theorem (reinterpolation): [ [X,Y]_a , [X,Y]_b ]_c = [X,Y]_cb+(1-c)a.

Theorem (interpolation of operators): if {X,Y} and {A,B} are interpolation pairs, and if T is a linear map defined on X+Y into A+B so that T is continuous from X to A and from Y to B then T is continuous from [X,Y]_t to [A,B]_t. and we have the interpolation inequality:

$\|T\|_{[X,Y]_{t}\to [A,B]_{t}}\leq C\|T\|_{X\to A}^{1-t}\|T\|_{Y\to B}^{t}.$