2 6 A ug 2 00 6 Calculation and Estimation of the Poisson Kernel

We provide a simple method for obtain boundary asymptotics of the Poisson kernel on a domain in R . 0 Introduction Let Ω ⊆ R be a connected open set—called a domain. It is a matter of considerable interest to estimate the size of the Poisson kernel PΩ(x, t) = P (x, t) of Ω. Here, and throughout this paper, x ∈ Ω and t ∈ ∂Ω. In case Ω has a large group of symmetries, then it is often possible to calculate PΩ explicitly. For example, • The Poisson kernel of the disc D ⊆ R is PD(x, t) = 1 2π · 1− |x| |x− t|2 . • The Poisson kernel for the upper halfplane U = {(x1, x2) ∈ R 2 : x2 > 0} is given by PU2(x, t) = 1 π · x2 (x1 − t)2 + x2 . • The Poisson kernel for the unit ball B ⊆ R is given by PB(x, t) = Γ(N/2) 2πN/2 · 1− |x| |x− t|N . Here Γ is the classical gamma function. Author supported in part by NSF Grant DMS-9988854.