Inner, singular inner and outer functions

Assume countable choice. Let $\mathbb{T}$ be the one-dimensional torus with its normalized Haar measure $m$, identified with the Euclidean unit circle through $\phi ([t])=e^{2\pi it}$ (The one-dimensional torus and its normalized Haar integral), and let $\mathbb{D}$ be the unit disc (The unit disc, the upper half-plane, and Blaschke factors).

The Cauchy kernel. For $\zeta \in \mathbb{T}$ (regarded as a point of the unit circle) and $z\in \mathbb{D}$ put $K(z,\zeta ):=\frac{\zeta +z}{\zeta -z}\mathrm{.}$ For fixed $z$ the function $\zeta \mapsto K(z,\zeta )$ is continuous on the compact torus with $\mathrm{\mid }K(z,\zeta )\mathrm{\mid }\le \frac{1+\mathrm{\mid }z\mathrm{\mid }}{1-\mathrm{\mid }z\mathrm{\mid }},\mathrm{Re}K(z,\zeta )=\frac{1-\mathrm{\mid }z{\mathrm{\mid }}^2}{\mathrm{\mid }\zeta -z{\mathrm{\mid }}^2}=P(z,\zeta )>0,$ where $P$ is the Poisson kernel of The Poisson kernel on the unit disc under the identification of The Poisson integral of a finite complex boundary measure; for fixed $\zeta$ the function $z\mapsto K(z,\zeta )$ is holomorphic on $\mathbb{C}\setminus \{\zeta \}$.

(a) Inner functions. A holomorphic $\theta :\mathbb{D}\to \mathbb{C}$ is inner if $\theta \in H^{\mathrm{\infty }}(\mathbb{D})$ and $\mathrm{\mid }{\theta }^{\ast }(\zeta )\mathrm{\mid }=1$ for $m$-almost every $\zeta \in \mathbb{T}$, where ${\theta }^{\ast }$ is its nontangential boundary function (Analytic Hardy spaces on the unit disc). Every Blaschke product is inner, and a Blaschke product has $\mathrm{\mid }B\mathrm{\mid }\le 1$ on $\mathbb{D}$ (Boundary values and zeros of a Blaschke product). An inner function also satisfies $\mathrm{\mid }\theta \mathrm{\mid }\le 1$ on $\mathbb{D}$; this is the case $p=\mathrm{\infty }$ of the bounded-holomorphic boundary-norm identity under countable choice proved in Bounded holomorphic disc functions have Poisson boundary data and Fatou limits under countable choice, and no inner function is assumed here to have any particular product form.

(b) Singular inner functions. For a finite positive Borel measure $\mu$ on $\mathbb{T}$ define $S_{\mu }(z):=\exp (-{\int }_{\mathbb{T}}K(z,\zeta )d\mu (\zeta ))(z\in \mathbb{D})\mathrm{.}$ If $\mu$ is singular with respect to $m$ (written $\mu \perp m$), $S_{\mu }$ is called a singular inner function. The function $S_{\mu }$ is well defined and holomorphic on $\mathbb{D}$, has no zeros, satisfies $\mathrm{\mid }{S}_{\mu }(z)\mathrm{\mid }=e^{-P[\mu ](z)}\le 1$ and $S_{\mu }(0)=e^{-\mu (\mathbb{T})}>0$; these properties and the boundary behaviour are proved in Properties of the singular functions $S_{\mu }$ ↗.

(c) Outer functions. For a nonnegative measurable $h:\mathbb{T}\to [0,+\mathrm{\infty }]$ with $\log h\in L^1(\mathbb{T},m)$ (Complex Lp classes and Euclidean test-function conventions) define the outer function $[h](z):=\exp ({\int }_{\mathbb{T}}K(z,\zeta )\log h(\zeta )dm(\zeta ))(z\in \mathbb{D}),$ where $\log h$ is extended by $-\mathrm{\infty }$ where $h=0$. The integral is absolutely convergent because $\mathrm{\mid }K(z,\zeta )\mathrm{\mid }\le \frac{1+\mathrm{\mid }z\mathrm{\mid }}{1-\mathrm{\mid }z\mathrm{\mid }}$ and $\mathrm{\mid }\log h\mathrm{\mid }\in L^1$. To see holomorphy without any $L^p$ hypothesis on $h$, expand $K(z,\zeta )=1+2{\sum }_{n\ge 1}{z}^n{\zeta }^{-n}$: its geometric tail is uniformly bounded on $\mathrm{\mid }z\mathrm{\mid }\le r<1$, so termwise integration against $\log h$ gives a power series whose coefficients have modulus at most $2\mathrm{\parallel }\log h{\mathrm{\parallel }}_1$. It is holomorphic on $\mathbb{D}$ (A complex power series converges absolutely and uniformly on every closed subdisc strictly inside its disc of convergence, A complex function is holomorphic if and only if it is analytic). Its exponential $[h]$ is holomorphic and zero-free, with $\log \mathrm{\mid }[h](z)\mathrm{\mid }=P[\log h](z),[h](0)=\exp ({\int }_{\mathbb{T}}\log hdm)>0,$ by The complex exponential is entire and its complex derivative is itself and $\exp (x+iy)=e^x(\cos y+i\sin y)$, $\mathrm{\mid }\exp (x+iy)\mathrm{\mid }=e^x$, and $e^{i\pi }+1=0$, and depends only on the $m$-class of $h$. If $h\in L^p(\mathbb{T},m)$ for some $0<p\le \mathrm{\infty }$, then $[h]\in H^p(\mathbb{D})$ and $\mathrm{\mid }[h{]}^{\ast }\mathrm{\mid }=h$ $m$-almost everywhere, with $\log \mathrm{\mid }[h{]}^{\ast }\mathrm{\mid }=\log h$ a.e.; these additional properties are proved in Properties of outer functions ↗.

A holomorphic $f$ on $\mathbb{D}$ with finite nontangential boundary values $f^{\ast }$ almost everywhere and $\log \mathrm{\mid }{f}^{\ast }\mathrm{\mid }\in L^1(\mathbb{T},m)$ is called outer if $f=e^{i\gamma }[\mathrm{\mid }{f}^{\ast }\mathrm{\mid }]$ for some $\gamma \in \mathbb{R}$. Equivalently, $\log \mathrm{\mid }f(z)\mathrm{\mid }=P[\log \mathrm{\mid }{f}^{\ast }\mathrm{\mid }](z)$ for every $z\in \mathbb{D}$: the representation implies the equality by the displayed identity; conversely, the equality makes the holomorphic quotient $f\mathrm{/}[\mathrm{\mid }{f}^{\ast }\mathrm{\mid }]$ have modulus one throughout the disc, hence it is a unimodular constant by Local maximum modulus principle. Thus the outer function with the prescribed modulus and this interior equality is determined up to a unimodular constant, and $[h](0)>0$ fixes its normalization.