Tag Archives: distributions

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.

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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.