# Sobolev Spaces

Let $\Omega\subseteq\mathbb{R}^n$ and $W^{k,p}(\Omega)$ be the collection of functions $f:\Omega\to\mathbb{R}$ (or $\mathbb{C}$ if preferred) such that

$\displaystyle\|f\|_{k,p}=\left(\sum_{|\alpha|\leq k}\left\|D^\alpha f\right\|_p^p\right)^{1/p}<\infty.$

It turns out $W^{k,p}(\Omega)$ are Banach spaces (see [2] for proof), called Sobolev spaces, under the above norm.  In particular $W^{0,p}(\Omega)=L^p(\Omega).$  The spaces $H^k(\Omega)=W^{k,2}(\Omega)$ are Hilbert with inner product

$\displaystyle\langle f,g\rangle_{H^k}=\sum_{|\alpha|\leq k}\int_\Omega\overline{D^\alpha f}D^\alpha g\,dx$

We can also define the local Sobolev spaces $W_{loc}^{k,p}(\Omega)$ as the subset of $L_{loc}^p(\Omega)$ such that $D^\alpha f\in L_{loc}^p(\Omega)$ for $|\alpha|\leq k.$  We have

$W_{loc}^{r,s}(\Omega)\subseteq W_{loc}^{k,p}(\Omega)\subseteq L_{loc}^p(\Omega),$

$W^{r,s}(\Omega)\subseteq W^{k,p}(\Omega)\subseteq W^{0,p}(\Omega)=L^p(\Omega)\subseteq L_{loc}^p(\Omega),$

and $W^{r,s}(\Omega)\subseteq W_{loc}^{r,s}(\Omega)$  where $(k,p)\leq (r,s)$ under dictionary ordering on $\mathbb{N}^2$ provided $\Omega$ has a finite Lebesgue measure.  Thus in particular

$W^{r,s}(\Omega)\subseteq W_{loc}^{r,s}(\Omega)\cap W^{k,p}(\Omega).$

$C_c^\infty(\Omega)$ is a $C^\infty(\Omega)$-module by a simple boundedness argument following from:  if $f\in C^\infty(\Omega)$ and $\phi\in C_c^\infty(\Omega),$ then $\sup f$ is finite in $\mbox{supp~}\phi.$  Thus $f\phi\in C_c^\infty(\Omega).$  Correspondingly this gives us $C^\infty(\Omega)$ action on $D(\Omega)^*$ (which denotes the distributions on $\Omega$ for us, although the action works on the dual space as well) defined by

$(fT)(\phi):=T(f\phi).$

Theorem 1.  Let $\Omega$ be bounded in $\mathbb{R}^n$ and $f\in W^{k,p}(\Omega).$  Then there exists a sequence of functions $f_m\in C^\infty(\Omega)\cap W^{k,p}(\Omega)$ such that

$\|f-f_m\|_{k,p}\to 0.$

We can also define $W^{k,\infty}(\Omega)$ (and similarly $W_{loc}^{k,\infty}(\Omega)$) as the set of functions $f:\Omega\to\mathbb{R}$ such that the following norm is finite

$\displaystyle\|f\|_{k,\infty}=\sum_{|\alpha|\leq k}\mbox{ess}\,\sup|D^\alpha f|.$

Thus $W^{0,\infty}(\Omega)=L^\infty(\Omega).$

I’m wondering if we could consider attempting to define $W^{\infty,p}(\Omega)$ (and similarly $W^{\infty,\infty}(\Omega)$) as the collection of functions $f:\Omega\to\mathbb{R}$ such that

$\displaystyle\|f\|_{\infty,p}=\left(\sum_{\alpha\in\mathbb{N}^n}\|D^\alpha f\|_p^p\right)^{1/p}$

is finite.  This would require all derivatives to eventually get to $0$ in such a way that the sum converges.  It would thus include polynomials if $\Omega$ is bounded and hence be nonempty.  In particular it would obey $W^{\infty,p}(\Omega)\subseteq W^{k,p}(\Omega)$ for all $k.$  And $W^{\infty,\infty}(\Omega)\subseteq W^{k,p}(\Omega)$ for all $k,p.$

[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 $\Omega\subseteq\mathbb{R}^n$ 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.