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.
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