Bounded holomorphic disc functions have Poisson boundary data and Fatou limits under countable choice

For $0<r<1$, [F1] identifies the Fourier coefficients of $f_r$ as $c_n{r}^n$ for $n\ge 0$ and zero for $n<0$, by uniform termwise integration and character orthogonality. By [F2], for each $N$, ${\sum }_{n=0}^N\mathrm{\mid }{c}_n{\mathrm{\mid }}^2{r}^{2n}\le {\int }_{\mathbb{T}}\mathrm{\mid }f(r\zeta ){\mathrm{\mid }}^{2}dm\le M^2\mathrm{.}$ Let $r\uparrow 1$ at this fixed finite $N$, then take the supremum in $N$: ${\sum }_{n\ge 0}\mathrm{\mid }{c}_n{\mathrm{\mid }}^2\le M^2$. Riesz-Fischer in [F2] gives a unique $\phi \in L^2$ with coefficients $c_n$ for $n\ge 0$ and zero for $n<0$.

This datum is in $L^1$ by [F3]. The geometric identity, for $\mathrm{\mid }z\mathrm{\mid }<1$ and $\mathrm{\mid }\zeta \mathrm{\mid }=1$, gives $P(z,\zeta )=1+{\sum }_{n\ge 1}(z^n{\zeta }^{-n}+{\bar{z}}^n{\zeta }^n)\mathrm{.}$ The series converges absolutely uniformly in $\zeta$ at each fixed $z$, with total absolute bound $1+2{\sum }_{n\ge 1}\mathrm{\mid }z{\mathrm{\mid }}^n<\mathrm{\infty }$. Its integral against $\phi$ therefore converges termwise, since the error is bounded by its uniform norm times $\mathrm{\parallel }\phi {\mathrm{\parallel }}_1$. Using its prescribed Fourier coefficients yields $P[\phi ](z)={\sum }_{n\ge 0}{c}_n{z}^n=f(z)$.

By [F3], $f=P[\phi ]$ has the asserted nontangential limits. Since $\mathrm{\mid }f(z)\mathrm{\mid }\le M$, passing to these limits gives $\mathrm{\mid }\phi \mathrm{\mid }\le M$ almost everywhere, so $\phi \in L^{\mathrm{\infty }}$. Conversely positivity and unit mass of the kernel give $\mathrm{\mid }f(z)\mathrm{\mid }\le \mathrm{\parallel }\phi {\mathrm{\parallel }}_{\mathrm{\infty }}$, hence $M=\mathrm{\parallel }\phi {\mathrm{\parallel }}_{\mathrm{\infty }}$. If another bounded datum $\psi$ has $P[\psi ]=f$, its nontangential limits are $\psi$ by [F3]; the same limits give $\psi =\phi$ almost everywhere. If $M=0$ then $f=0$ and every coefficient and the unique datum are zero, so all claims remain valid. Only countable choice, in [F2] and [F3], has been used.