Poisson kernel

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 disk. The kernel can be understood as the derivative of the Green's function for the Laplace equation. It is named for Siméon Poisson.

Poisson kernels commonly find applications in control theory and two-dimensional problems in electrostatics. In practice, the definition of Poisson kernels are often extended to n-dimensional problems.

In the complex plane, the Poisson kernel for the unit disc is given by

This can be thought of in two ways: either as a function of r and θ, or as a family of functions of θ indexed by r.

is harmonic in D and has a radial limit that agrees with f almost everywhere on the boundary T of the disc.

That the boundary value of u is f can be argued using the fact that as r → 1, the functions Pr(θ) form an approximate unit in the convolution algebra L1(T). As linear operators, they tend to the Dirac delta function pointwise on Lp(T). By the maximum principle, u is the only such harmonic function on D.

Rearranging this absolutely convergent series shows that f is the boundary value of g + h, where g (resp. h) is a holomorphic (resp. antiholomorphic) function on D.

When one also asks for the harmonic extension to be holomorphic, then the solutions are elements of a Hardy space. This is true when the negative Fourier coefficients of f all vanish. In particular, the Poisson kernel is commonly used to demonstrate the equivalence of the Hardy spaces on the unit disk, and the unit circle.

The space of functions that are the limits on T of functions in Hp(z) may be called Hp(T). It is a closed subspace of Lp(T) (at least for p ≥ 1). Since Lp(T) is a Banach space (for 1 ≤ p ≤ ∞), so is Hp(T).

The unit disk may be conformally mapped to the upper half-plane by means of certain Möbius transformations. Since the conformal map of a harmonic function is also harmonic, the Poisson kernel carries over to the upper half-plane. In this case, the Poisson integral equation takes the form

Then, if u(x) is a continuous function defined on S, the corresponding Poisson integral is the function P[u](x) defined by

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

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 that as t → 0, P[u](t,x) → u(x) in a suitable sense.