Some Properties of Positive Harmonic Functions

From a monotonicity property, we derive several results characterizing positive harmonic functions in the unit ball in R and positive measures on the unit sphere Sn−1. Let Bn = {x ∈ Rn : |x| < 1}, n ≥ 2 be the unit ball in Rn and Sn−1 = ∂Bn be the unit sphere. From a monotonicity property, we derive several results characterizing positive harmonic functions in Bn and positive measures on Sn−1. This paper has four sections. In the first section, we describe the monotonicity property for positive harmonic functions in the unit ball. The rest sections utilize the monotonicity result to characterize different aspects of positive harmonic functions. In Section 2 we derive a decomposition theorem for positive harmonic functions. Section 3 is on the properties of positive measures on the sphere. In Section 4, we obtain a precise asymptotic for spherical expansions of harmonic functions. 1. On monotonicity of positive harmonic functions It is known [1] that a positive harmonic function u in Bn can be uniquely represented by the Poisson kernel P (x, y) and a positive measure μ on Sn−1 as (1.1) u(x) = P [μ](x) = ∫ Sn−1 P (x, η)dμ(η) = ∫ Sn−1 1− |x|2 |x− η|n The following is the monotonicity theorem for positive harmonic functions. Its generalization to positive invariant harmonic functions can be found in [6]. Theorem 1.1. Let u be a positive harmonic function in Bn, ζ ∈ Sn−1. Then the function (1− r)n−1 1 + r u(rζ) is decreasing and the function (1 + r)n−1 1− r u(rζ) is increasing for 0 ≤ r < 1. Proof. Firstly, for x ∈ Bn, |x| = r, we claim that (1.2) − n− (n− 2)r |x− ζ|n ≤ ∂ ∂r ( 1− r2 |x− ζ|n ) ≤ n+ (n− 2)r |x− ζ|n . 1