On the unit disc In the
complex plane, the Poisson kernel for the unit disc is given by P_r(\theta) = \sum_{n=-\infty}^\infty r^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 \le r 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 D = \{z:|z| 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 u(re^{i\theta}) = \frac{1}{2\pi}\int_{-\pi}^\pi P_r(\theta-t)f(e^{it}) \, \mathrm{d}t, \quad 0 \le 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 , the functions 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. Convolutions with this approximate unit gives an example of a
summability kernel for the
Fourier series of a function in
L1(
T) . Let
f ∈
L1(
T) have Fourier series {
fk}. After the
Fourier transform, convolution with
Pr(
θ) becomes multiplication by the sequence {
r|k|} ∈
ℓ1(
Z). Taking the inverse Fourier transform of the resulting product {
r|k|fk} gives the
Abel means Arf of
f: A_r f(e^{2 \pi i x}) = \sum _{k \in \Z} f_k r^ e^{2 \pi i k x}. 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).
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 u(x+iy)=\int_{-\infty}^\infty P_y(x-t)f(t) \, dt, \qquad y > 0. The kernel itself is given by P_y(x)=\frac{1}{\pi}\frac {y}{x^2 + y^2}. Given a function f \in L^p(\R), the
Lp 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, H^p, and in particular, \|u\|_{H^p}=\|f\|_{L^p} Thus, again, the Hardy space
Hp on the upper half-plane is a
Banach space, and, in particular, its restriction to the real axis is a closed subspace of L^p(\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. ==On the ball==