Support and Locality

Support of a Distribution¶

A distribution $T$ is said to be supported in a closed set $K$ if $\langle T, \phi \rangle = 0$ for all $\phi \in \mathcal{C}_c^{\infty}(\mathbb{R}^d)$ with $\text{supp}(\phi) \cap K = \emptyset$ .

The support of $T$ , denoted $\text{supp}(T)$ , is the smallest set $K$ such that $T$ is supported in $K$ .

Example 1

The support of the Dirac delta is $\text{supp}(\delta) = \{ 0 \}$ .
If $f$ is a continuous function, $\text{supp}(T_f) = \overline{\{ x : f(x) \neq 0 \}}$ .

Local Structure¶

Two distributions $T_1$ and $T_2$ are equal on an open set $\Omega$ if $\langle T_1, \phi \rangle = \langle T_2, \phi \rangle$ for all $\phi$ .

If $T_f$ is a regular distribution and $T_f = 0$ on an open set $\Omega$ , then $f = 0$ almost everywhere on $\Omega$ .

Distributions with Point Support¶

A fundamental result: If $T$ is a distribution supported at a single point $x_0$ , then $T$ is a finite linear combination of derivatives of the delta function at $x_0$ :

T = \sum_{|\alpha| \leq m} c_{\alpha} D^\alpha \delta_{x_0}

(1)

where $m$ is the order of $T$ .

The Structure Theorem for Distributions¶

The results above characterize distributions with point support. The structure theorem generalizes this to all distributions: every distribution is, locally, a finite-order derivative of a continuous function.

Theorem 1 (Structure theorem for distributions)

Let $T \in \mathcal{D}'(\Omega)$ and let $K \subset \Omega$ be compact. Then there exist an integer $N \geq 0$ and continuous functions $f_\alpha \in C(K)$ , $|\alpha| \leq N$ , such that

T = \sum_{|\alpha| \leq N} D^\alpha f_\alpha \qquad \text{on a neighborhood of } K.

(2)

If $T$ has finite order $m$ globally (i.e., the seminorm bound $|\langle T, \varphi \rangle| \leq C_K \sum_{|\alpha| \leq m} \sup |D^\alpha \varphi|$ holds with the same $m$ for all compact $K$ ), then we can take $N = m$ uniformly.

Proof 1

We sketch the argument in one dimension for clarity; the general case is identical in structure.

Step 1: From the order bound to a norm estimate. On a compact set $K$ , the distribution $T$ has some finite order $m$ :

|\langle T, \varphi \rangle| \leq C_K \sum_{k=0}^{m} \|\varphi^{(k)}\|_\infty \qquad \text{for all } \varphi \in \mathcal{D}_K.

(3)

This means $T$ is a continuous linear functional on $\mathcal{D}_K$ equipped with the $C^m$ norm.

Step 2: Extension to a continuous linear functional. The space $\mathcal{D}_K$ is a subspace of $C^m(K)$ . By the Hahn-Banach theorem, $T$ extends to a continuous linear functional on $C^m(K)$ .

Step 3: Representation via antiderivatives. A continuous linear functional on $C^m(K)$ can be written as a finite sum of integrals against Radon measures $\mu_\alpha$ , $|\alpha| \leq m$ :

\langle T, \varphi \rangle = \sum_{|\alpha| \leq m} \int D^\alpha \varphi \, d\mu_\alpha.

(4)

Each Radon measure $\mu_\alpha$ on a compact subset of $\mathbb{R}^d$ has a continuous antiderivative: there exists $f_\alpha \in C(K)$ such that $\int \psi \, d\mu_\alpha = (-1)^{|\alpha|} \int f_\alpha \, D^\alpha \psi \, dx$ for test functions $\psi$ . (In one dimension, this is just the fact that the antiderivative of a measure is a function of bounded variation, and one more antiderivative gives a continuous function.) Substituting:

\langle T, \varphi \rangle = \sum_{|\alpha| \leq N} (-1)^{|\alpha|} \int f_\alpha \, D^\alpha \varphi \, dx = \sum_{|\alpha| \leq N} \langle D^\alpha f_\alpha, \varphi \rangle

(5)

where $N = m + d$ accounts for the additional antiderivatives needed in $d$ dimensions.

Remark 1 (Distributions are derivatives of continuous functions)

The structure theorem immediately implies the point support result stated above:

Corollary 1

If $T \in \mathcal{D}'(\mathbb{R}^d)$ with $\operatorname{supp}(T) = \{x_0\}$ , then $T = \sum_{|\alpha| \leq m} c_\alpha D^\alpha \delta_{x_0}$ .