# 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.