Chapter V. 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 $C^{\infty}_{\mathrm{com}}(U)$ of smooth functions with compact support in an open set $U$, the space $C^{\infty}(U)$ of all smooth functions on $U$, and the space of Schwartz functions $\mathcal{S}(\mathbb{R}^{N})$ on $\mathbb{R}^{N}$. The corresponding spaces of continuous linear functionals are denoted by $\mathcal{D}^{\prime}(U)$, $\mathcal{E}^{\prime}(U)$, and $\mathcal{S}^{\prime}(\mathbb{R}^{N})$.

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 $\mathcal{E}^{\prime}(U)$ to $\mathcal{D}^{\prime}(U)$ the definition given earlier for support, it shows that the only distributions of compact support in $U$ are the ones that act continuously on $C^{\infty}(U)$, 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 $(\cdot)^{\vee}$ corresponding to $x \mapsto -x$. 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 $\mathbb{R}^{N}$ for $N > 2$ and concludes with an existence theorem for distribution solutions to $\Delta u = f$ when $f$ is any distribution of compact support.