Poisson kernel

Two-dimensional Poisson kernels

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^{|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 \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 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 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 P**r(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^{|k|} 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

For the ball of radius r, B_r \subset \R^n, the Poisson kernel takes the form P(x,\zeta) = \frac{r^2-|x|^2}{r\omega {n-1}|x-\zeta|^n} where x\in B_r, \zeta\in S (the surface of B{r}), and \omega _{n-1} is the surface area of the unit (n − 1)-sphere.

Then, if u(x) is a continuous function defined on S, the corresponding Poisson integral is the function Pu defined by Pu = \int_S u(\zeta)P(x,\zeta) , d\sigma(\zeta).

It can be shown that Pu is harmonic on the ball B_{r} and that Pu extends to a continuous function on the closed ball of radius r, and the boundary function coincides with the original function u.

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 \R^{n+1} by (t,x) = (t,x_1,\dots,x_n). The upper half-space is the set defined by H^{n+1} = \left{ (t;x) \in \R^{n+1} \mid t0 \right}. The Poisson kernel for H**n+1 is given by P(t,x) = c_n\frac{t}{\left(t^2 + \left|x\right|^2\right)^{(n+1)/2}} where c_n = \frac{\Gamma[(n+1)/2]}{\pi^{(n+1)/2}}.

The Poisson kernel for the upper half-space appears naturally as the Fourier transform of the Abel transform in which t assumes the role of an auxiliary parameter. To wit, P(t,x) = \mathcal{F}(K(t,\cdot))(x) = \int_{\R^n} e^{-2\pi t|\xi|} e^{-2\pi i \xi\cdot x},d\xi. In particular, it is clear from the properties of the Fourier transform that, at least formally, the convolution Pu = P(t,\cdot)*u is a solution of Laplace's equation in the upper half-plane. One can also show that as t → 0, Pu → u(x) in a suitable sense.

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ru:Ядро Пуассона

References

1. "complex analysis - Deriving the Poisson Integral Formula from the Cauchy Integral Formula".