# Fractional Sobolev Spaces

Recall for a function $f\in L^1(\Omega)$ for $\Omega\subseteq\mathbb{R}^n$ we define the Fourier transform of $f$ by

$\displaystyle\hat{f}(x)=F(f)(x)=\int_\Omega e^{-2\pi ix\cdot\xi}f(\xi)\,d\xi$

and the inverse Fourier transform by

$\displaystyle \check{f}(x)=F^{-1}(f)(x)=\int_\Omega e^{2\pi i x\cdot\xi}f(\xi)\,d\xi.$

If $f,\hat{f}\in L^1(\Omega),$ then $\check{\hat{f}}=\hat{\check{f}}=f$ a.e..

Now if $f\in L^1$ and $D^\alpha (f)\in L^1$ with $\alpha\in\mathbb{Z}^n,$ we say $D^\alpha f$ is the weak derivative of $f$ provided

$\displaystyle\int_\Omega f D^\alpha\phi\,dx=(-1)^{|\alpha|}\int_\Omega D^\alpha(f)\phi\,dx$

with $\phi\in C_C^\infty(\Omega),$ $|\alpha|=\sum_i\alpha_i,$ and

$\displaystyle D^\alpha\phi=\frac{\partial^{|\alpha|}}{\partial x_1^{\alpha_1}\cdots\partial x_n^{\alpha_n}}\phi.$

The Sobolev space $W^{k,p}(\Omega)$ is then defined as the set of all functions $f\in L^p(\Omega)$ such that $D^\alpha(f)\in L^p(\Omega)$ for all $|\alpha|\leq k.$  These become Banach spaces under the norm

$\displaystyle\|f\|_{k,p}=\left(\sum_{|\alpha|\leq k}\int_\Omega \left|D^\alpha(f)\right|^p\,dx\right)^{1/p}$

and Hilbert when $p=2,$ whence we denote $H^k(\Omega)=W^{k,2}(\Omega).$

For $1 we can say that $W^{k,p}(\mathbb{R}^n)$ consists of $f\in L^p(\mathbb{R}^n)$ such that

$\displaystyle F^{-1}\circ(1+|2\pi x|^2)^{k/2}F\circ f=(1-\Delta)^{k/2}f(x)\in L^p(\mathbb{R}^n)$

(using Fourier multipliers).

Bessel Potential Spaces:

We can generalize with $r\in\mathbb{R}$ and $1 and define the Bessel potential space $W^{r,p}(\mathbb{R}^n)$ as all $f\in L^p$ such that $\|f\|_{r,p}:=\|(1-\Delta)^{r/2}f\|_p<\infty.$  Note the fractional power of the Laplacian is defined by virtue of the fact that $L^p\cap L^1$ is dense in $L^p$, so we say

$\displaystyle(1-\Delta)^{r/2}(f):=(F^{-1}\circ (1+|2\pi x|^2)\circ F)(f).$

These spaces satisfy our desire of being Banach (and Hilbert when $p=2$).

Sobolev-Slobodeckij Spaces:

There is an alternative approach.  Recall that the Holder space $C^{k,\gamma}(\Omega)$ is defined as all functions $f\in C^k(\Omega)$ such that

$\displaystyle\|f\|_{C^{k,\gamma}}=\sum_{|\alpha|\leq k}\sup_{x\in\Omega}|D^\alpha|+\sum_{|\alpha|=k}\sup_{x,y\in\Omega}\frac{|D^\alpha f(x)-D^\alpha f(y)|}{|x-y|^\gamma}<\infty.$

That is, it is the set of functions on $\Omega$ which are $C^k$ and whose $k$-th partial derivatives are bounded and Holder continuous of degree $\gamma.$  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 $f\in L^p(\Omega)$ with $1\leq p<\infty$ and $\theta\in (0,1)$ by

$\displaystyle [f]_{\theta,p}=\int_\Omega\int_\Omega\frac{|f(x)-f(y)|^p}{|x-y|^{\theta p+n}}\,dxdy.$

The corresponding Sobolev-Slobodeckij space $W^{s,p}(\Omega)$ is defined as all functions $f\in W^{\lfloor s\rfloor,p}(\Omega)$ such that

$\displaystyle\sup_{|\alpha|=\lfloor s\rfloor}[D^\alpha f]_{\theta,p}<\infty$

where $\theta=s-\lfloor s\rfloor\in(0,1).$  This becomes a Banach space under the norm

$\displaystyle\|f\|_{s,p}=\|f\|_{W^{\lfloor s\rfloor,p}}+\sup_{|\alpha|=\lfloor s\rfloor}[D^\alpha f]_{\theta,p}.$

[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 $\Omega\subseteq\mathbb{R}^n$ and $C_c^\infty(\Omega)$ denote the set of smooth functions on $\Omega$ with compact support.

Definition 1.  Let $\alpha\in\mathbb{N}^n$ and $\{\phi_k\}$ be a sequence in $C_c^\infty(\Omega).$  We say $\{\phi_k\}$ converges to $\phi\in C_c^\infty(\Omega),$ denoted $\phi_k\to\phi,$ if

1. $\mbox{supp~}\phi_k-\phi\subseteq K$ for all $k$ for a fixed compact $K\subseteq\Omega,$
2. $D^\alpha\phi_m\to D^\alpha\phi$ uniformly on $K.$

By $D^\alpha\phi$ we mean

$\displaystyle\frac{\partial^{\alpha_1}}{\partial x_1^{\alpha_1}}\cdots\frac{\partial^{\alpha_n}}{\partial x_n^{\alpha_n}}\,\phi.$

$C_c^\infty(\Omega)$ is a vector space, and the above notion of convergence allows us to define closed sets and hence turns $C_c^\infty(\Omega)$ into a topological vector space called the space of test functions of $\Omega,$ and we denote it $D(\Omega).$

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

We say a sequence of distributions $\{T_k\}$ converges to the distribution $T,$ denoted $T_k\to T,$ if $T_k(\phi)\to T(\phi)$ for all $\phi\in D(\Omega).$  This gives us a topological vector space structure on $D(\Omega)^*$ as well.

Definition 3.  Let $T$ be a distribution and $\alpha\in\mathbb{N}^n.$  We define the $\alpha$-th distributional (weak) derivative of $T$ by

$D^\alpha(T)(\phi)=(-1)^{|\alpha|}T(D^\alpha\phi)$

where $|\alpha|=\sum\alpha_i.$  We may also write

$\displaystyle\partial_i^kT=(-1)^kT\left(\frac{\partial^k}{\partial x_i^k}\phi\right).$

We will omit $k$ if $k=1.$  The gradient of $T$ is simply

$\nabla T=(\partial_1T,...,\partial_nT).$

Let us define the space $L_{loc}^p(\Omega)$ as the space of functions $f:\Omega\to\mathbb{R}$ such that

$\displaystyle\|f\|_{L^p(K)}=\left(\int_K|f|^p\,dx\right)^{1/p}<\infty$

for all compact $K\subseteq\Omega.$  We define strong convergence on $L_{loc}^p(\Omega)$ as convergence in $L^p(\Omega)$ and weak convergence on $L_{loc}^p(\Omega)$ as weak (pointwise) convergence in $L^p(K)$ for all compact $K\subseteq\Omega.$

Let $1\leq p\leq q,$ then clearly

$L^q(\Omega)\subseteq L_{loc}^q(\Omega)\subseteq L_{loc}^p(\Omega).$

Let $f\in L_{loc}^1(\Omega)$ and $\phi\in D(\Omega)$ and define

$\displaystyle T_f(\phi)=\int_\Omega f\phi\,dx.$

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

$\begin{array}{lcl}|T_f(\phi)-T_f(\phi_k)|&=&\displaystyle\left|\int_\Omega(\phi(x)-\phi_k(x))f(x)\,dx\right|\\&\leq&\displaystyle\sup_{x\in K}|\phi(x)-\phi_k(x)|\int_K|f(x)|\,dx\\&<&\infty\end{array}$

where $K=\mbox{supp~}(\phi-\phi_k),$ which gives us the continuity of $T_f.$  So $T_f\in D(\Omega)^*.$  We also have

$\displaystyle (D^\alpha T_f)(\phi):=(-1)^{|\alpha|}\int_\Omega(D^\alpha\phi)f\,dx=\int_\Omega(D^\alpha f)\phi\,dx=T_{D^\alpha f}(\phi).$

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