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


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.


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