Distributions as Boundary Values of Analytic Functions

Abstract

Denote by \(H_{\pm}\) the upper respectively lower half plane of the complex plane \(\mathbb C\). For functions \(F_{\pm}\) which are analytic in \(H_{\pm}\) we introduce the concept of boundary value in the sense of distributions and show under which conditions (growth restrictions near the boundary) such an analytic function has a distributional boundary value. For distributions T with compact support \(K \subset \mathbb{R}\) it is straightforward to show that there is a function F analytic in \(\mathbb C \backslash K\) such that T is the difference of boundary values \(F(x +\,\textrm{i}o) - F(x -\,\textrm{i}o)\) of F from \(H_{\pm}\). Without proof we state the result that every distribution T on the real line is the difference of the boundary values of two functions \(F_{\pm}\) analytic in \(H_{\pm}\): \(T(x)= F_+(x +\,\textrm{i}o)-F_-(x-\,\textrm{i}o)\). This result can be extended to the case of distributions in more than one variable. We indicate briefly the main difficulties and the solution by Martineau.