In practice, the definition of Poisson kernels are often extended to n-dimensional problems.

An expression for the Poisson kernel of an upper half-space can also be obtained.

Similar expressions are available for the expansion of the Poisson kernel in a ball .

The Poisson kernel is also closely related to the Cauchy distribution.

The proof given uses the Poisson kernel and the existence of boundary values for the Hardy space H.

The integrand is known as the Poisson kernel; this solution follows from the Green's function in two dimensions:

The Radon-Nikodym derivative is called the Poisson kernel.

This harmonic function is obtained from "f" by taking a convolution with the "'conjugate Poisson kernel"'

The proof utilizes the symmetry of the Poisson kernel using the Hardy-Littlewood maximal function for the circle.

In particular, the Poisson kernel is commonly used to demonstrate the equivalence of the Hardy spaces on the unit disk, and the unit circle.

Since the conformal map of a harmonic function is also harmonic, the Poisson kernel carries over to the upper half-plane.

See also McCullagh's parametrization of the Cauchy distributions and Poisson kernel for related concepts.

Kellogg's method of proof analyzes the representation of harmonic functions provided by the Poisson kernel, applied to an interior tangent sphere.

In mathematics, it is closely related to the Poisson kernel, which is the fundamental solution for the Laplace equation in the upper half-plane.

In potential theory, the Poisson kernel is an integral kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit disc.

When L is particularly simple, this problem can often be resolved using the Fourier transform directly (as in the case of the Poisson kernel and heat kernel already mentioned).

The Poisson kernel is important in complex analysis because its integral against a function defined on the unit circle - the Poisson integral - gives the extension of a function defined on the unit circle to a harmonic function on the unit disk.