Harmonic representations of distributions

Abstract

This paper is concerned with a characterization of functions which are regular harmonic in the xy plane except on the line y = 0, and have distributional. limits as y -+ 0+ and as y -+ O-. It is shown that such functions satisfy a generalized Green’s formula and may be represented as potentials of singleand double-layer distributions on the line y = 0. This generalizes well-known classical results. Furthermore, it turns out that this class of harmonic functions is identical with the class of functions which are restrictions, to y # 0, of distributions in W(P), whose Laplacians have their support on the x axis. Notation and terminology used here fohow that of Gel’fand and Shilov ES] and Bremermann [2]. Let U be an open subset of R”. The space of test functions on U is the set b(U) of all infinitely differentiable functions with compact support in U. It is given the usual topology [S]. The space of continuous linear functionals on D(U) is denoted by B’(U) and is given the weak topology. Its elements are called (Schwartz) distributions. For the fundamental properties of test functions and distributions, see [2, 5, 10, 121, for example.