Support of a distribution - Imported Knowlpedia Content

Definition

Let $T$ be a distribution on an open set $\Omega\subseteq\mathbb R^n$ . The distribution vanishes on an open subset $U\subseteq\Omega$ if $\langle T,\varphi\rangle=0$ for every smooth test function supported in $U$ . The support of $T$ is

\operatorname{supp}T =\Omega\setminus\bigcup\{U\subseteq\Omega:U\text{ is open and }T|_U=0\}.

It is therefore a closed set relative to $\Omega$ , and it is the smallest relatively closed set outside which $T$ vanishes. This definition depends only on the local action of $T$ on test functions.

Equivalent local criterion

A point $x\in\Omega$ lies outside $\operatorname{supp}T$ exactly when it has an open neighborhood $U$ such that $T(\varphi)=0$ for every test function $\varphi$ supported in $U$ . Consequently, $T=0$ if and only if its support is empty. A partition-of-unity argument supplies the passage from local vanishing to vanishing on arbitrary test functions supported in the union.

Functions, measures, and delta distributions

For a locally integrable function $f$ , the support of its regular distribution is the complement of the largest open set on which $f=0$ almost everywhere; it can be smaller than the pointwise closure of $\{x:f(x)\ne0\}$ for a particular representative. The support of $\delta_a$ and of each derivative of $\delta_a$ is the singleton $\{a\}$ .

Behavior under operations

Distributional differentiation cannot enlarge support:

\operatorname{supp}(\partial^\alpha T)\subseteq\operatorname{supp}T.

Multiplication by a smooth function $a$ gives $\operatorname{supp}(aT)\subseteq\operatorname{supp}a\cap \operatorname{supp}T$ . Compactly supported distributions are precisely those whose distributional support is compact Hörmander, §2.3.

References

Lars Hörmander, The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, 2nd ed., Springer, 2003. DOI record. Relevant: §2.3 on support and singular support.
François Trèves, Topological Vector Spaces, Distributions and Kernels, Academic Press, 1967. Publisher record. Relevant: Chapter 24 on distributions and their support.