Distributions

Chapter 9Potentials and Singular Functions

This chapter is dedicated to potentials, the most useful of which for the following is the localized potential γ, a solution of ∆γ + η = δ₀ in ^d with support in a ball B = B(0, r) as small as we like it, where the correction term η is a regular function also with support in B (Theorem 9.17). This will be used to decompose (in Theorem 12.1) any distribution as , on the domain of definition of f with an arbitrarily small neighborhood of its boundary removed.

We construct the localized potential γ in steps, as follows.

• — We associate a distribution with each continuous function which may diverge to infinity at the origin while growing less quickly than |x|^−λ, where λ < d (Theorem 9.5).

• — We construct the Newtonian potential ξ, a solution of ∆ξ = 0 on all of ^d (Theorem 9.10).

• — We localize ξ as γ = θξ, where θ has its support in B and is equal to 1 on a small ball (Theorem 9.17).

On the other hand, we decompose the Dirac mass into a sum of derivatives of ^m functions with support in B (Theorem 9.18). And we show that the weighting by a weight ...