Distributions

Chapter 16Banach Space Valued Distributions

This chapter is dedicated to the specifics of distributions with values in a Banach space.

1. — Such a distribution is a sum of derivatives of continuous functions g_n with compact support which, on each compact subset K of Ω, are all zero from a finite rank n_K (Theorem 16.9). Locally, it is even the derivative of a single continuous function with compact support (Theorem 16.11).
2. — Such a distribution on an open set Ω is extendable to all of ^d if and only if, on every bounded subset ω of Ω, it is the derivative ∂^βg of a uniformly continuous function g (Theorem 16.16).
3. — These properties are no longer true in some Fréchet spaces (Theorems 16.12 and 16.17).

These properties follow from the fact that every distribution with values in a Banach space is locally of finite order, a property that we begin by studying, in § 16.1 to 16.4.

16.1. Finite order distributions

Let us define finite order distributions¹.

Definition 16.1.– Let f ∈ ′(Ω; E), where Ω is an open subset of ^d and E is a Neumann space, and let {‖ ‖_E;ν : ν ∈ _E} be the family of semi-norms ...