Distributions

Chapter 5Operations on Distributions

This chapter is dedicated to the derivatives ∂_if of a distribution f ∈ ′(Ω; E), to its image Lf under a linear mapping L from E into another Neumann space F , to its product αf with a function α ∈ ^∞(Ω), and to its image f T after a change of variable T ∈ ^∞(Λ; Ω). For each of these four operations:

• — we verify that, when f ∈ (Ω; E), we obtain the usual operation;

• — we study its continuity and its “interactions” with the previous operations; for example, the interaction of the derivative with the product is the Leibniz formula ∂_i(αf ) = α∂_if + ∂_iα f .

We equally study positive real distributions, and prove that these are measures (Theorem 5.34).

We begin by introducing distributions fields, namely distributions with values in E^d, since gradient is such a field, and their separation into components (§ 5.1). In § 5.8, we more generally study distributions with values in a product space E₁ × · · · × E_I .

The results of this chapter are “natural” enough ...