Compactly supported distribution - Imported Knowlpedia Content

Definition

Let $\Omega\subseteq\mathbb R^n$ be open. A distribution $T\in\mathcal D'(\Omega)$ is a compactly supported distribution if its distributional support $\operatorname{supp}T$ is compact in $\Omega$ . The space of all such distributions is denoted $\mathcal E'(\Omega)$ . If $\chi\in C_c^\infty(\Omega)$ is a smooth cutoff that equals $1$ on a neighborhood of $\operatorname{supp}T$ , then

T(f):=T(\chi f),\qquad f\in C^\infty(\Omega),

is independent of the choice of $\chi$ . Hence $T$ extends canonically from test functions to all smooth functions.

Dual characterization

Equip $\mathcal E(\Omega)=C^\infty(\Omega)$ with the topology of uniform convergence of every derivative on compact subsets. Its continuous dual is $\mathcal E'(\Omega)$ : continuity on all smooth functions is equivalent to being a distribution with compact support. This explains the notation and the cutoff construction in the core Trèves, Chapter 27.

Operations and examples

Every compactly supported distribution has finite order. Dirac distributions and all their derivatives are compactly supported, with singleton support. A nonzero constant function on $\mathbb R^n$ , regarded as a regular distribution, is not compactly supported.

For distributions on $\mathbb R^n$ , convolution of two arbitrary distributions need not be defined, but it is defined when at least one factor has compact support; the resulting support is contained in the Minkowski sum $\operatorname{supp}S+\operatorname{supp}T =\{x+y:x\in\operatorname{supp}S,\ y\in\operatorname{supp}T\}$ Hörmander, §4.1.

Conventions and scope

Compactness is taken inside $\Omega$ : the support must be a compact subset that stays away from the boundary. The symbol $\mathcal E'$ is not a second notation for all distributions; it is the continuous dual of the smooth-function space $\mathcal E$ , and therefore selects precisely the compactly supported ones.

References

Lars Hörmander, The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, 2nd ed., Springer, 2003. Springer DOI record. Relevant: §§2.3 and 4.1, compact support, extension, and convolution.
François Trèves, Topological Vector Spaces, Distributions and Kernels, Academic Press, 1967. Elsevier publisher record. Relevant: Chapter 27, compactly supported distributions and the dual of $C^\infty$ .