Properties of outer functions

Holomorphy, zero-freeness and the value at the origin. By [L2] the function $L$ is holomorphic on $\mathbb{D}$ with $L(0)=\int \log hdm\in \mathbb{R}$, so $[h]=e^L$ is holomorphic and zero-free with $[h](0)=e^{\int \log hdm}>0$ (finite because $\log h\in L^1$).

The pointwise bound. Let $p<\mathrm{\infty }$ and fix $z\in \mathbb{D}$. Since $\mathrm{Re}L(z)=\int P(z,\zeta )\log h(\zeta )dm(\zeta )$ by [L1], put $t:=p\log h$, choosing a finite real representative on the null exceptional set. Both $t$ and $e^t=h^p$ are integrable against $P(z,\zeta )dm(\zeta )$ because $\log h\in L^1$, $h\in L^p$, and the fixed kernel is bounded by [L1]. Jensen's inequality [L3] therefore gives $\mathrm{\mid }[h](z){\mathrm{\mid }}^p=e^{p\mathrm{Re}L(z)}=e^{\int P(z,\zeta )p\log h(\zeta )dm(\zeta )}\le {\int }_{\mathbb{T}}P(z,\zeta )h(\zeta {)}^{p}dm(\zeta )=P[h^p](z)\mathrm{.}$ For $p=\mathrm{\infty }$, $\log h\le \log \mathrm{\parallel }h{\mathrm{\parallel }}_{\mathrm{\infty }}$ $m$-almost everywhere (with $\mathrm{\parallel }h{\mathrm{\parallel }}_{\mathrm{\infty }}>0$ because $\log h\in L^1$), so $\mathrm{Re}L(z)=\int P\log hdm\le \log \mathrm{\parallel }h{\mathrm{\parallel }}_{\mathrm{\infty }}$ and $\mathrm{\mid }[h](z)\mathrm{\mid }\le \mathrm{\parallel }h{\mathrm{\parallel }}_{\mathrm{\infty }}$.

Uniqueness up to a unimodular constant. Assume $g\in H^p$, $g\equiv ̸0$, $\mathrm{\mid }{g}^{\ast }\mathrm{\mid }=h$ a.e. and $\log \mathrm{\mid }g(z)\mathrm{\mid }=P[\log h](z)$ for all $z$. Then $\log \mathrm{\mid }g\mathrm{\mid }=\log \mathrm{\mid }[h]\mathrm{\mid }$ on $\mathbb{D}$ by step 1.1 and [L1], so the holomorphic zero-free function $g\mathrm{/}[h]$ has constant modulus $1$; by [L6] it is a constant of modulus one, that is, $g=e^{i\gamma }[h]$ for some $\gamma \in \mathbb{R}$.

Membership in $H^p$. For $p<\mathrm{\infty }$ and $0<r<1$, integrating the bound of step 2.1 over the circle and applying Tonelli's theorem [L4] to the nonnegative integrand gives ${\int }_{\mathbb{T}}\mathrm{\mid }[h](r\zeta ){\mathrm{\mid }}^{p}dm(\zeta )\le {\int }_{\mathbb{T}}({\int }_{\mathbb{T}}P(r\zeta ,\eta )dm(\zeta ))h(\eta {)}^{p}dm(\eta )=\mathrm{\parallel }h{\mathrm{\parallel }}_p^p,$ because the inner integral equals the unit mass of the kernel by translation invariance. Taking the supremum over $r$ gives $[h]\in H^p(\mathbb{D})$ with $\mathrm{\parallel }[h]{\mathrm{\parallel }}_{H^p}\le \mathrm{\parallel }h{\mathrm{\parallel }}_p$; for $p=\mathrm{\infty }$ step 2.1 gives $[h]\in H^{\mathrm{\infty }}$ with $\mathrm{\parallel }[h]{\mathrm{\parallel }}_{\mathrm{\infty }}\le \mathrm{\parallel }h{\mathrm{\parallel }}_{\mathrm{\infty }}$.

Boundary modulus. Since $\log \mathrm{\mid }[h]\mathrm{\mid }=\mathrm{Re}L=P[\log h]$ by [L1], the harmonic Fatou theorem for the $L^1$ datum $\log h$ gives $\log \mathrm{\mid }[h](z)\mathrm{\mid }\to \log h(\zeta )$ as $z\to \zeta$ within every cone, at $m$-almost every $\zeta$. By step 3.1, $[h]\in H^p$, so by [L5] it has nontangential limits $[h{]}^{\ast }$ $m$-almost everywhere; at every point where both statements hold, taking moduli gives $\mathrm{\mid }[h{]}^{\ast }(\zeta )\mathrm{\mid }=h(\zeta )$. This proves (ii), and (iii) is immediate from the definition of $[h]$ as an integral against $dm$.

Assembly. Steps 1.1, 2.1 and 3.1 give (i), step 4.1 gives (ii) and (iii), and step 2.2 gives (iv). All four clauses are proved under the stated hypotheses.