The upper half-plane Poisson boundary density

Define the Cayley map $C(w):=(w-i)\mathrm{/}(w+i)$, which is holomorphic on $\mathbb{C}\setminus \{-i\}$ and in particular on $\mathbb{H}$, and satisfies $\mathrm{\mid }C(w)\mathrm{\mid }<1$ for $w\in \mathbb{H}$ because $\mathrm{\mid }w+i{\mathrm{\mid }}^2-\mathrm{\mid }w-i{\mathrm{\mid }}^2=4\mathrm{Im}w>0$. For real $t$ one computes $C(t)=\frac{(t-i{)}^2}{t^2+1}=\frac{t^2-1-2ti}{t^2+1}=e^{i\theta (t)},\theta (t):=2\arctan t-\pi \in (-2\pi ,0),$ so $\mathrm{\mid }C(t)\mathrm{\mid }=1$, $C(t)\mathrm{\ne }1$, and by the chain rule and [F3] $C^{\mathrm{\prime }}(t)=\frac{2i}{(t+i{)}^2},\mathrm{\mid }{C}^{\mathrm{\prime }}(t)\mathrm{\mid }=\frac{2}{t^2+1}={\theta }^{\mathrm{\prime }}(t)\mathrm{.}$

For every $R>0$ the substitution $s=(t-x)\mathrm{/}y$ on $[-R,R]$, which is admissible by [F4] with $\phi (t)=(t-x)\mathrm{/}y$ and $g(s)=1\mathrm{/}(\pi (1+s^2))$, gives ${\int }_{-R}^R{k}_z(t)dt=\frac{1}{\pi }(\arctan \frac{R-x}{y}-\arctan \frac{-R-x}{y}),$ and letting $R\to \mathrm{\infty }$ with [F3] yields the improper Riemann integral ${\int }_{\mathbb{R}}{k}_{z}dt=1$. Since $k_z$ is continuous and nonnegative, [F5] makes $k_z$ Lebesgue integrable with ${\int }_{\mathbb{R}}{k}_{z}d{\lambda }_1=1$.

For $w\in \mathbb{H}$ and real $t$, direct expansion gives $1-\mathrm{\mid }C(w){\mathrm{\mid }}^2=\frac{4\mathrm{Im}w}{\mathrm{\mid }w+i{\mathrm{\mid }}^2},\mathrm{\mid }C(t)-C(w)\mathrm{\mid }=\frac{2\mathrm{\mid }t-w\mathrm{\mid }}{\mathrm{\mid }t+i\mathrm{\mid }\mathrm{\mid }w+i\mathrm{\mid }},$ the second identity because $(t-i)(w+i)-(w-i)(t+i)=2i(t-w)$. Multiplying, and using [F1], [F3] and step 1.1, $\frac{P(C(w),C(t))\mathrm{\mid }{C}^{\mathrm{\prime }}(t)\mathrm{\mid }}{2\pi }=\frac{4y}{\mathrm{\mid }w+i{\mathrm{\mid }}^2}\cdot \frac{\mathrm{\mid }t+i{\mathrm{\mid }}^2\mathrm{\mid }w+i{\mathrm{\mid }}^2}{4\mathrm{\mid }t-w{\mathrm{\mid }}^2}\cdot \frac{2}{2\pi (t^2+1)}=\frac{y}{\pi ((t-x{)}^2+y^2)}=k_z(t)$ whenever $w=z=x+iy$.

By [F5] the set function ${\omega }^z(A):={\int }_A{k}_{z}d{\lambda }_1$, defined on the Lebesgue sigma-algebra and hence on the Borel sets of $\mathbb{R}$, is a measure, and ${\omega }^z(\mathbb{R})=1$ by step 1.2. So $d{\omega }_{\mathbb{H}}^z=k_{z}d{\lambda }_1$ is a Borel probability measure on $\mathbb{R}$.

Representation for compactly supported data. Let $f$ be continuous with compact support and define $\psi$ on $\mathrm{\partial }\mathbb{D}$ by $\psi (\zeta ):=f(i(1+\zeta )\mathrm{/}(1-\zeta ))$ for $\zeta \mathrm{\ne }1$ and $\psi (1):=0$; this is well defined with $C$-values on the unit circle, and it is continuous: near $\zeta =1$ one has $\mathrm{\mid }1+\zeta \mathrm{\mid }\ge 2-\mathrm{\mid }1-\zeta \mathrm{\mid }$, so the parameter $i(1+\zeta )\mathrm{/}(1-\zeta )$ tends to infinity and $\psi$ vanishes there because $f$ does, while continuity away from $1$ follows from continuity of $f$ and of $\zeta \mapsto i(1+\zeta )\mathrm{/}(1-\zeta )$. Let $u_D$ be the Poisson integral of $\psi$ and $u:=u_D\circ C$ on $\mathbb{H}$. Then $u$ is harmonic by [F2], and for $z\in \mathbb{H}$ the definition of the Poisson integral, the identity $\psi (C(t))=f(t)$ and step 2.1 give $u(z)=\frac{1}{2\pi }{\int }_0^{2\pi }\psi (e^{i\theta })P(C(z),e^{i\theta })d\theta =\frac{1}{2\pi }{\int }_{\mathbb{R}}f(t)P(C(z),C(t))\mathrm{\mid }{C}^{\mathrm{\prime }}(t)\mathrm{\mid }dt={\int }_{\mathbb{R}}fd{\omega }_{\mathbb{H}}^z,$ the middle equality being the substitution $\theta =\theta (t)$ of step 1.1 on a compact parameter interval, legitimate by [F4] and the vanishing of the integrand near the ends $0$ and $-2\pi$ of the period interval, together with $2\pi$-periodicity.

Boundary limits for compactly supported data. If $z_j\to t_0$ with $t_0\in \mathbb{R}$ and $z_j\in \mathbb{H}$, then $C(z_j)\to C(t_0)\in \mathrm{\partial }\mathbb{D}\setminus \{1\}$ by continuity of $C$ at $t_0$, and $u_D$ is continuous on $\bar{\mathbb{D}}$ with boundary values $\psi$ by [F1]; hence $u(z_j)=u_D(C(z_j))\to \psi (C(t_0))=f(t_0)$ by step 3.1.

General bounded continuous data. Let $f$ be bounded and continuous and let ${\chi }_N:\mathbb{R}\to [0,1]$ be continuous with ${\chi }_N=1$ on $[-N,N]$ and ${\chi }_N=0$ outside $[-N-1,N+1]$ (for instance ${\chi }_N(t)=\max \{0,\min \{1,N+1-\mathrm{\mid }t\mathrm{\mid }\}\}$). Each $f_N:=f{\chi }_N$ is continuous with compact support, so $u_{f_N}$ is harmonic on $\mathbb{H}$ by step 3.1. If $K\subseteq \mathbb{H}$ is compact and $R$ is such that $\mathrm{\mid }w\mathrm{\mid }\le R$ on $K$, then for $N>2R$ and $w=x+iy\in K$ the tail bound ${\int }_{\mathrm{\mid }t\mathrm{\mid }>N}{k}_w(t)dt\le \frac{2R}{\pi }{\int }_N^{\mathrm{\infty }}\frac{4}{t^2}dt=\frac{8R}{\pi N}$ holds, because $\mathrm{\mid }t-x\mathrm{\mid }\ge \mathrm{\mid }t\mathrm{\mid }-R\ge \mathrm{\mid }t\mathrm{\mid }\mathrm{/}2$ and $(t-x{)}^2+y^2\ge (t-x{)}^2$ there; hence $\mathrm{\mid }{u}_f-u_{f_N}\mathrm{\mid }\le \mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}\cdot 8R\mathrm{/}(\pi N)$ on $K$ and $u_{f_N}\to u_f$ locally uniformly on $\mathbb{H}$. Therefore $u_f$ is harmonic on $\mathbb{H}$ by [F6].

Boundary limits for general data. Fix $t_0\in \mathbb{R}$ and $\eta >0$. Continuity of $f$ at $t_0$ gives $\delta >0$ with $\mathrm{\mid }f(t)-f(t_0)\mathrm{\mid }<\eta$ for $\mathrm{\mid }t-t_0\mathrm{\mid }<\delta$. Splitting the defining integral at $\mathrm{\mid }t-t_0\mathrm{\mid }=\delta$ and using ${\int }_{\mathbb{R}}{k}_z=1$ from step 1.2, $\mid u_f(z)-f(t_0)\mid \le \eta +2\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}{\int }_{\mathrm{\mid }t-t_0\mathrm{\mid }\ge \delta }{k}_z(t)dt,$ and the last integral equals $\frac{1}{\pi }(\pi -\arctan \frac{t_0+\delta -x}{y}+\arctan \frac{t_0-\delta -x}{y})$, which tends to $0$ as $x\to t_0$, $y\downarrow 0$ by [F3], since then $(t_0\pm \delta -x)\mathrm{/}y\to \pm \mathrm{\infty }$. Hence $u_f(z)\to f(t_0)$ as $z\to t_0$ inside $\mathbb{H}$.

The density $k_z$ therefore defines a Borel probability measure of total mass one on $\mathbb{R}$ (step 2.2) whose integrals produce, for every bounded continuous boundary function, a harmonic function with the prescribed limit at every finite boundary point of $\mathbb{H}$ (steps 4.2 and 5.1). Steps 1.2 and 2.2 use ${\mathrm{A}\mathrm{C}}_{\omega }$ through the measure and integral interface of [F5], as does the integral interpretation in step 3.1; step 4.2 also uses it through the locally uniform harmonic-limit theorem [F6], while steps 1.1, 2.1 and the arctangent limits are choice-free calculations.