Convolution

Summary

In Section 1.3 we defined the convolution u₁*u₂ of two continuous functions u₁ and u₂, one of which has compact support. The definition can be applied without change if u₁∈D’, (resp. D’,) and u₂∈C₀∞ (resp. C∞) we have u₁*u₂òC∞ then. Section 4.1 is devoted to such convolutions. As in the case of functions this is an efficient method to approximate distributions by C∞ functions. It can often be used to extend statements concerning smooth functions to distributions, particularly when translation invariant questions are concerned. As examples of this we give in Section 4.1 a discussion of convex, sub-harmonic and plurisubharmonic functions.

Convolution of two distributions u₁ u₂ one of which has compact support is defined in Section 4.2 so that the associativity (u₁ * u₂)* ϕ = u₁ *(u₂ * ϕ), ϕ∈C₀∞, is preserved. It is then elementary to see that supp(u₁ *u₂)⊂:suppu₁+suppu₂.

Section 4.3 is devoted to the proof of the theorem of supports which states that when u₁ and u₂ both have compact supports, then there is equality if one takes convex hulls of the supports. The standard proofs of this depend on analytic function theory, and we shall return to them later on. The reader might therefore prefer to wait for Section 7.3 rather than studying the end of the proof of Theorem 4.3.3.

Section 4.4 is intended to present the basic methods used to derive results on existence and smoothness of solutions of constant coefficient partial differential equations from the properties of a fundamental solution. This is an important application of the convolution. The final Section 4.5 is then devoted to L_p estimates for convolutions. In addition to estimates related to Hölder’s inequality we prove potential estimates of the Hardy-Littlewood-Sobolev type and derive from them relations between the L_p (or Hölder) classes of a function and its derivatives. These are basic particularly in the study of elliptic differential equations. They will be supplemented in Section 7.9 when we have Fourier analysis at our disposal.