Surface Integrals and Harmonic Functions

Wu showed in [4] that for a positive harmonic function in the unit disc, one has in most cases inequality, while equality occurs for functions whose boundary measures are absolutely continuous. She also showed that there exists a nonzero lower bound of the lim inf for this class of functions in the disc. The bound is achieved for functions whose boundary measures, for example, are purely singular. We generalize these results to higher dimensions. Let Ω be the open unit ball or upper half-space in Rn+1 and let S denote its boundary. Let u be a positive harmonic function on Ω, which, by Riesz’s theorem, is given by a Borel measure μ with the total measure ‖μ‖ on S. Definition 1.2. Let Γ be a piecewise C1-smooth hypersurface in Aδ = {q ∈Ω : d(q,S) < δ} separating the two boundaries of Aδ. The inferior mean of u is defined by