Distributions

Abstract

This chapter makes a detailed study of distributions, which are continuous linear functionals on vector spaces of smooth scalar-valued functions. The three spaces of smooth functions that are studied are the space C ^∞_com (U) of smooth functions with compact support in an open set U, the space C^∞(U) of all smooth functions on U, and the space of Schwartz functions S(ℝ^N) on ℝ^N. The corresponding spaces of continuous linear functionals are denoted by D′(U), ε′(U), and S′(ℝ^N).

Section 1 examines the inclusions among the spaces of smooth functions and obtains the conclusion that the corresponding restriction mappings on distributions are one-one. It extends from ε′(U) to D′(U) the definition given earlier for support, it shows that the only distributions of compact support in U are the ones that act continuously on C^∞(U), it gives a formula for these in terms of derivatives and compactly supported complex Borel measures, and it concludes with a discussion of operations on smooth functions.

Sections 2–3 introduce operations on distributions and study properties of these operations. Section 2 briefly discusses distributions given by functions, and it goes on to work with multiplications by smooth functions, iterated partial derivatives, linear partial differential operators with smooth coefficients, and the operation (·)∨ corresponding to x ↦ −x. Section 3 discusses convolution at length. Three techniques are used—the realization of distributions of compact support in terms of derivatives of complex measures, an interchange-of-limits result for differentiation in one variable and integration in another, and a device for localizing general distributions to distributions of compact support.

Section 4 reviews the operation of the Fourier transform on tempered distributions; this was introduced in Chapter III. The two main results are that the Fourier transform of a distribution of compact support is a smooth function whose derivatives have at most polynomial growth and that the convolution of a distribution of compact support and a tempered distribution is a tempered distribution whose Fourier transform is the product of the two Fourier transforms.

Section 5 establishes a fundamental solution for the Laplacian in ℝ^N for N > 2 and concludes with an existence theorem for distribution solutions to Δu = f when f is any distribution of compact support.