On the Boundary Behaviour of Poisson Integrals of Finite Measures in (n + 1)-dimensional Half Space

It is well known that the harmonic functions u on the Euclidean upper (n + 1)-dimensional half-space E⁺ _n+1 = {(x, y) = (x ₁,…,x_n,y) ? E _n+1: y > 0} satisfying sup _y>0 ||u(·,y)||₁ < ∞ are precisely the Poisson-integrals u(x,y) = ∫ _En P(x - t,y)dμ(t) with respect to a measure μ of finite variation on E_n , and that (Fatou's theorem in E⁺ _n+1) in almost every point x ? E_n the non-tangential boundary limit of u exists and coincides with du/dλ. While this is a special case of a general assertion in potential theory, it is shown that the proof of Fatou's theorem for harmonic functions on a ball may readily be transferred to the given setup and that the influence of a singular component of μ on the boundary behaviour of u may also be established without recourse to the existence of the derivative dμ/dλ. Finally the C ₀-property of u is characterized by suitable conditions on μ.