Distributions and Weak Derivatives

Distributions

Let $\Omega \sub \mathbb{R}^n$. The support of a function $\phi:\Omega \to \mathbb{R}$ is the set

$$\operatorname{supp}(\phi) = \overline{\{x \in \Omega : \phi(x) \neq 0\}}.$$

Let $\mathcal{D}(\Omega)$ be the set of all functions $\phi:\Omega \to \mathbb{R}$ that satisfy:

1. $\phi \in C^\infty(\Omega)$,
2. Compact support: $\operatorname{supp}(\phi)$ is compact,
3. $\operatorname{supp}(\phi) \sub \Omega.$

A function $\phi$ with these properties is called a test function.

Definition of Distributions

Let $\Omega \sub \mathbb{R}^n$. A distribution in $\Omega$ is a linear transformation $T: \mathcal{D}(\Omega) \to \mathbb{R}$ such that:

1. Linearity: For all $\alpha, \beta \in \mathbb{R}$ and $\phi, \psi \in \mathcal{D}(\Omega)$,

$$T(\alpha \phi + \beta \psi) = \alpha T(\phi) + \beta T(\psi).$$

2. Continuity: If $\phi_k \to \phi$ in $\mathcal{D}(\Omega)$, then

$$\lim_{k \to \infty} T(\phi_k) = T(\phi).$$

From a functional analysis perspective, the set of distributions in $\Omega$ is exactly the topological dual of $\mathcal{D}(\Omega)$, that is, $\mathcal{D}'(\Omega)$ This means that a distribution is simply a continuous linear transformation on $\mathcal{D}(\Omega)$.

Locally Integrable Functions

Let $\Omega \sub \mathbb{R}^n$. A function $f: \omega\to\mathbb{R}$ is said to be localley integrable on $\Omega$ provided $f\in L^1(A)$ for every open $A$ such that $\overline{A}\sub\Omega$ and $\overline{A}$ is compact. In this case write $f \in L¹_{loc}(\Omega)$.

Now, every $f \in L^1_{loc}(\Omega)$ is a disttribution $T_f\in\mathcal{D}'(\Omega)$ define by,

$$T_f(\phi)=\int_\Omega f\phi d\mu,$$

for all $\phi\in\mathcal{D}(\Omega).$ Evidently $T_f$ is a linear transformation on $\mathcal{D}(\Omega).$

Derivatives of Distributions

Let $\Omega \sub \mathbb{R}^n$. Let $f\in C^1(\Omega)$ and $\phi\in \mathcal{D}(\Omega).$ We see from the integration by parts in variable $x_j$

$$\int_\Omega \frac{\partial f}{\partial x_j}\phi d\mu = -\int_\Omega f \frac{\partial\phi}{\partial x_j}d\mu$$

More generally now, if $f\in C^{|\alpha|}(\Omega)$, then

$$\int_\Omega D^{\alpha}f\phi d\mu = -1^{|\alpha|}\int_\Omega f D^{\alpha}\phi d\mu.$$

This motivate the following definition of the derivative $D^\alpha T$ of a distribution $T\in\mathcal{D}'(\Omega)$

$$(D^\alpha T)(\phi)=-1^{|\alpha|}T(D^\alpha\phi)$$

Weak Derivative

Let $f \in L^1_{loc}(\Omega)$. We say that $g_\alpha\in L^1_{loc}$ is the weak derivative of $f$, and is denoted by $D^\alpha f.$

Thus $D^\alpha f=g_\alpha$ in the weak sense provided,

$$\int_\Omega fD^{\alpha}\phi d\mu = -1^{|\alpha|}\int_\Omega g_\alpha\phi d\mu.$$

for all test functions $\phi\in\mathcal{D}(\Omega).$