nLab support of a distribution

Idea

The concept of support of a distribution is an evident generalization of support of a continuous function as functions are generalized to distributions.

Of particular interest is the singular support of distributions, which is the subset of those points inside their support around which they are singular when regarded as generalized functions. This “local analysis” of singularities of distributions may be further refined to their microlocal analysis by considering not such the singular points, but also the directions of the propagation of the singularities at these points, which is the set of covectors over the singular support called the wave front set.

Definition

Definition

(support of a distribution)

Let $X$ be a smooth manifold and let $\phi \colon C^\infty_c(X) \longrightarrow \mathbb{R}$ be a distribution. Then the support of $\phi$ is the subset $supp(\phi) \subset X$ of all those points $x \in X$ such that for every open neighbourhood $U_x \subset X$ the restricted distribution $\phi\vert_{U_x}$ is not the zero-distribution

$$supp(\phi) \;\coloneqq\; \left\{ x \in X \;\vert\; \underset{ \text{nbhd}\, U_x }{\forall}\left( \phi\vert_{U_x} \neq 0 \right) \right\} \,.$$

(e.g. Hörmander 90, def. 2.2.2)

Remark

By construction, the support of a distribution $\phi$ according to def. is a closed subset. If it happens to be a compact subset, then the distribution is said to be a compactly supported distribution.

(e.g. Hörmander 90, section 2.3)

Definition

(singular support)

Let $X$ be a smooth manifold and let $\phi \colon C^\infty_c(X) \longrightarrow \mathbb{R}$ be a distribution. Then the singular support $supp_{sing}(\phi) \subset X$ is the subset of points such that for every open neighbourhood $U_x \subset X$ the restriction $\phi\vert_{U_x}$ is singular, hence not a non-singular distribution.

Remark

(wave front set)

A refinement of the information contained in the singular support of a distribution (def. ) is its wave front set, which over each point of the singular support records those covectors “along which the singularity propagates”, as made precise by the propagation of singularities theorem.

Properties

Remark

If a bump function $f \in C^\infty_c(X)$ has its (compact) support $supp(f)$ disjoint from the support of of a distribution $\phi$ (def. ), then the evaluation vanishes:

$$supp(f) \cap supp(\phi) = \emptyset \;\Rightarrow\; \phi(f) = 0 \,.$$

(e.g. Hörmander 90, theorem 2.2.1 and (2.2.1))

Examples

Related entries

• support of a function

• point-supported distribution

• order of a distribution

• scaling degree of a distribution

References

• Lars Hörmander, The analysis of linear partial differential operators, vol. I, Springer 1983, 1990

• Sergiu Klainerman, chapter 3, section 3.5 of Lecture notes in analysis, 2011 (pdf)