Distributions

Chapter 6Restriction, Gluing and Support

We show here the local character of distributions: if two distributions are equal on various open subsets ω_i of their domains of definition, they are equal on the union of the ω_i. It is the gluing theorem for equalities (Theorem 6.10). Equal on ω_i meaning that their restrictions to ω_i are equal, we first study restriction (§ 6.1).

We also give the gluing property for distributions: if distributions f_i, respectively defined on open sets ω_i, satisfy f_i = f_j on ω_i ω_j for all i and j, there exists a unique distribution, defined on the union of the ω_i, which is equal to f_i on ω_i for each i. This is the gluing theorem for distributions (Theorem 6.16). The main tool for this is the localization–extension, which we study beforehand (§ 6.4).

We finally look at the support of a distribution (§ 6.6) and at the space ′_K (Ω; E) of distributions which have their support in a closed subset K of Ω (§ 6.8). We show that a distribution whose support is closed can be extended by 0 (Theorem 6.29).

In this chapter, as in the previous one, results are “natural” and proofs are simple.

6.1. Restriction

Let us define the restriction of a distribution.

Definition 6.1.– Let f ∈ ′(Ω; E), where Ω is an open subset of ^d and E is a Neumann space, and ω an open subset ...