An Application of Hardy-littlewood Tauberian Theorem to Harmonic Expansion of a Complex Measure on the Sphere

We apply Hardy-Littlewood’s Tauberian theorem to obtain an estimate on the harmonic expansion of a complex measure on the unit sphere, using a monotonicity property for positive harmonic functions. 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 obtain a precise asymptotic for the spherical harmonic expansion of a complex measure on Sn−1 by applying the Tauberian theorem of Hardy and Littlewood. 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) u(x) = P [μ](x) = ∫ Sn−1 P (x, η)dμ(η) = ∫ Sn−1 1− |x|2 |x− η|n dμ(η). In the following we state a monotonicity property for positive harmonic functions as a theorem (Theorem 1), which is the special case δ = 0 of Theorem 1.1 in [5]. A corollary (Corollary 2) on asymptotic results follow. Then we apply the monotonicity and the asymptotic property to obtain an estimate on the spherical harmonic expansion of a complex measure on Sn−1 (Theorem 3) by applying Hardy-Littlewood’s Tauberian Theorem. Two corollaries follow. Theorem 1. (Theorem 1.1 in [5]) 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. The following is needed to prove our main result in Theorem 3.