On the unit disc

In the complex plane, the Poisson kernel for the unit disc ^[1] is given by

$${\displaystyle P_{r}(\theta )=\sum _{n=-\infty }^{\infty }r^{|n|}e^{in\theta }={\frac {1-r^{2}}{1-2r\cos \theta +r^{2}}}=\operatorname {Re} \left({\frac {1+re^{i\theta }}{1-re^{i\theta }}}\right),\ \ \ 0\leq r<1.}$$

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.

If ${\displaystyle D=\{z:|z|<1\}}$ is the open unit disc in C, T is the boundary of the disc, and f a function on T that lies in L¹(T), then the function u given by

$${\displaystyle u(re^{i\theta })={\frac {1}{2\pi }}\int _{-\pi }^{\pi }P_{r}(\theta -t)f(e^{it})\,\mathrm {d} t,\quad 0\leq r<1}$$

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 P_r(θ) form an approximate unit in the convolution algebra L¹(T). As linear operators, they tend to the Dirac delta function pointwise on L^p(T). By the maximum principle, u is the only such harmonic function on D.

Convolutions with this approximate unit gives an example of a summability kernel for the Fourier series of a function in L¹(T) (Katznelson 1976). Let f ∈ L¹(T) have Fourier series {f_k}. After the Fourier transform, convolution with P_r(θ) becomes multiplication by the sequence {r^|k|} ∈ ℓ¹(Z). Taking the inverse Fourier transform of the resulting product {r^|k|f_k} gives the Abel means A_rf of f:

$${\displaystyle A_{r}f(e^{2\pi ix})=\sum _{k\in \mathbb {Z} }f_{k}r^{|k|}e^{2\pi ikx}.}$$

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 H^p(z) may be called H^p(T). It is a closed subspace of L^p(T) (at least for p ≥ 1). Since L^p(T) is a Banach space (for 1 ≤ p ≤ ∞), so is H^p(T).

On the upper half-plane

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

$${\displaystyle u(x+iy)=\int _{-\infty }^{\infty }P_{y}(x-t)f(t)\,dt,\qquad y>0.}$$

The kernel itself is given by

$${\displaystyle P_{y}(x)={\frac {1}{\pi }}{\frac {y}{x^{2}+y^{2}}}.}$$

Given a function ${\displaystyle f\in L^{p}(\mathbb {R} )}$, the L^p space of integrable functions on the real line, u can be understood as a harmonic extension of f into the upper half-plane. In analogy to the situation for the disk, when u is holomorphic in the upper half-plane, then u is an element of the Hardy space, ${\displaystyle H^{p},}$ and in particular,

$${\displaystyle \|u\|_{H^{p}}=\|f\|_{L^{p}}}$$

Thus, again, the Hardy space H^p on the upper half-plane is a Banach space, and, in particular, its restriction to the real axis is a closed subspace of ${\displaystyle L^{p}(\mathbb {R} ).}$ The situation is only analogous to the case for the unit disk; the Lebesgue measure for the unit circle is finite, whereas that for the real line is not.