Poisson Kernel - On The Upper Half-space

On The Upper Half-space

An expression for the Poisson kernel of an upper half-space can also be obtained. Denote the standard Cartesian coordinates of Rn+1 by

The upper half-space is the set defined by

The Poisson kernel for Hn+1 is given by

where

The Poisson kernel for the upper half-space appears naturally as the Fourier transform of the Abel kernel

in which t assumes the role of an auxiliary parameter. To wit,

In particular, it is clear from the properties of the Fourier transform that, at least formally, the convolution

is a solution of Laplace's equation in the upper half-plane. One can also show easily that as t → 0, P(t,x) → u(x) in a weak sense.