# 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.

# Chain Rule, Extensions, and Trace in W^{1,p}

Theorem 1.  Let $f:\mathbb{R}^n\to\mathbb{C}$ be differentiable with bounded and continuous derivatives and $g\in W_{loc}^{1,p}(\Omega).$  Then $h=f\circ g\in W_{loc}^{1,p}(\Omega)$ and

$\displaystyle\partial_i h=\sum_j\partial_j f(g)\,\partial_i g_j.$

In particular if $g_j\in W^{1,p}(\Omega)$ for all $j,$ then $h\in W^{1,p}(\Omega)$ and the chain rule holds provided $f(0)=0.$

Let $U,V$ be open sets in $\mathbb{R}^n.$  We say $U$ is compactly contained in $V,$ denoted $U\sqsubset V$ if for any $U'$ such that $U\subset U'\subset V,$ $U'$ is compact.  We now extend functions in $W^{1,p}(\Omega)$ to functions on $W^{1,p}(\mathbb{R}^n).$

Theorem 2.  Let $\Omega$ be bounded, $\partial\Omega$ be a $C^1$-manifold, and $\Omega\sqsubset V.$  Then there exists a linear map $E:W^{1,p}(\Omega)\to W^{1,p}(\mathbb{R}^n)$ such that for all $f\in W^{1,p}(\Omega)$ we have

1. $Ef=f\mbox{~a.e. in~}U,$
2. $\mbox{supp~}E\subseteq V,$
3. $\|Ef\|_{W^{1,p}(\mathbb{R^n})}\leq C\|f\|_{W^{1,p}(\Omega)}$ where $C$ depends upon $p,\Omega,$ and $V.$

Proof Outline.  Let $x_0\in\partial\Omega$ and $\Pi$ be the tangent hyperplane at $x_0$–which satisfies $x_n=0.$  Since $\partial\Omega$ is locally homeomorphic to $\mathbb{R}^{n-1},$ $\Pi$ approximately bisects a small ball around $x_0:$ one which intersects with $\Omega$ and one that does not.  We will respectively call these $B^{+}$ and $B^{-}.$  Suppose $f\in C^1(\overline{\Omega})$ and define

$\displaystyle\overline{f}(x)=\left\{\begin{array}{lcl}\displaystyle f(x)&\mbox{if}&x\in B^{+}\\\displaystyle -3f(x_1,...,x_{n-1},-x_n)+4f(x_1,...,x_{n-1},\frac{-x_n}{2})&\mbox{if}&x\in B^{-}\end{array}\right.$

Note that the two parts of the function agree on $\partial\Omega$ and that $\overline{f}\in C^1(B)$ since if we let $f^{-}=\overline{f}|_{B^{-}},$ then

$\displaystyle\partial_nf^{-}(x)=3\frac{\partial f}{\partial x_n}(x_1,...,-x_n)-2\frac{\partial f}{\partial x_n}(x_1,...,\frac{-x_n}{2})=\partial_n f$

on $\partial\Omega$ (where $x_n=0$.)  Thus derivatives agree on $\Pi.$  Thus $\overline{f}\in C^1(B).$

$\Omega\sqsubset V$ then allows one to use partitions of unity on $\partial\Omega$ to write $f$ in local coordinates and establish the bounds in (3).  (2) is then established by continuously sending the function to $0$ in $B^{-},$ or a slightly larger set.  $\overline{f}$ is then defined as $0$ elsewhere.

Above we assumed $f\in C^1(\overline{\Omega}).$  If we assume more generally that $f\in W^{1,p}(\Omega)$ as in the hypotheses, then we can approximate (as seen in the last post) by functions in $C^1(\Omega)$ (a special case of which is on $C^1(\overline{\Omega})$ if $\Omega$ is bounded and $\partial\Omega$ is a $C^1$-manifold [see 5.3.2 and 5.3.3 in [2]]) which yields the result with consequence (1).

Lastly we begin with a function $f\in W^{1,p}(\Omega)$ and attempt to extend it to the boundary.

Theorem 3.  Let $\Omega$ be bounded and $\partial\Omega$ be a $C^1$-manifold.  Then there exists a bounded linear operator $T:W^{1,p}(\Omega)\to L^p(\partial\Omega)$ such that

1. $Tf=f|_{\partial\Omega}$ if $f\in W^{1,p}(\Omega)\cap C(\overline{\Omega}),$
2. $\|Tf\|_{L^p(\partial\Omega)}\leq C\|f\|_{W^{1,p}(\Omega)}.$

The function $T$ is called the trace of $f$ on $\partial\Omega.$  We also have

Proposition 4.  Again let $\Omega$ be bounded and $\partial\Omega$ be a $C^1$-manifold.  Let $f\in W^{1,p}(\Omega).$  Then

$\displaystyle Tf=0\mbox{~on~}\partial\Omega\Leftrightarrow f\in W_0^{1,p}(\Omega).$

This sort of seems intuitive since it’s like saying the summable extension is $0$ on the boundary iff it belongs to the class of functions whose derivatives vanish on the boundary.  Recall elements of $W_0^{1,p}(\Omega)$ are functions whose support is a compact subset of $\Omega$ or whose support extends to $\partial\Omega.$  The proof is rather technical in the forward direction (see [2] for details).  The backwards direction uses density of $C_c^\infty(\Omega)$ in $W^{1,p}(\Omega).$

Regarding Theorem 3, at first glance this reminded me of Stokes’ theorem on manifolds, although Evans doesn’t mention anything about it (nor uses it in either of the two proofs).  Recall the statement.

Stokes’ Theorem.  Let $M$ be a an $n$-dimensional, $C^k,$ oriented, and compact manifold with boundary such that $\omega$ is an $(n-1)$-form on $M.$  Then

$\displaystyle\int_M\,d\omega=\int_{\partial M}\omega.$

Since $f\in W^{1,p}(\Omega),$ its derivatives were integrable (i.e. the left side), and under the trace $f$ is sent to a function which is integrable on the boundary.

[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.

# 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.