Fatou's boundary theorem for analytic Hardy spaces

For f identically zero take the zero boundary function; every assertion is immediate, including the unique zero density measure by [F6]. Otherwise [F1] supplies the finite nontangential boundary values and the $L^p$ bound in (a) under CC. For finite p put $h=\mathrm{\mid }{f}^{\ast }{\mathrm{\mid }}^p\in L^1$. The Poisson logarithmic inequality and [F2] give $\mathrm{\mid }f(z){\mathrm{\mid }}^p\le P[h](z)$. For p infinity [F1] already gives the Poisson representation and equality of infinity norms.

Uniform integrability for finite p. For $M\ge 1$ write $h=h_M+t_M$, where $h_M=\min (h,M)$ and $t_M=(h-M{)}_{+}$. Since h is integrable, $\mathrm{\parallel }{t}_M{\mathrm{\parallel }}_1\to 0$ by [F4]. For every Borel E and every radius r, positivity and [F3] give ${\int }_E\mathrm{\mid }{f}_r{\mathrm{\mid }}^{p}dm\le {\int }_E{P}_{r}hdm\le Mm(E)+\mathrm{\parallel }{t}_M{\mathrm{\parallel }}_1\mathrm{.}$ The same truncation bounds ${\int }_{E}hdm$. Therefore both $\mathrm{\mid }{f}_r{\mathrm{\mid }}^p$ uniformly in r and h have arbitrarily small integrals on sets of sufficiently small Haar measure. The inequality $\mathrm{\mid }a-b{\mathrm{\mid }}^p\le c_p(\mathrm{\mid }a{\mathrm{\mid }}^p+\mathrm{\mid }b{\mathrm{\mid }}^p)$, with $c_p=1$ for $p\le 1$ and $c_p=2^{p-1}$ for $p\ge 1$, gives the same uniform integrability for $e_r=\mathrm{\mid }{f}_r-f^{\ast }{\mathrm{\mid }}^p$.

The infinity case. The equal infinity norms and Poisson representation are in step 1.1. For any a in $L^1$, $a(f_r-f^{\ast })$ tends to zero almost everywhere and is bounded by $2\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\mid }a\mathrm{\mid }$, an integrable function. Dominated convergence [F4] gives $\int a{f}_{r}dm\to \int a{f}^{\ast }dm$, exactly the stated weak-star convergence.

Strong convergence for finite p. The radial limits in step 1.1 give $e_r\to 0$ almost everywhere. For each eta>0 the indicators of $\{e_r>\eta \}$ tend to zero almost everywhere, so [F4] gives $m\{e_r>\eta \}\to 0$. Given epsilon>0, choose delta>0 by step 2.1 so that ${\int }_E{e}_{r}dm<\epsilon \mathrm{/}2$ for every r whenever $m(E)<\delta$. Take eta=epsilon/2. For r sufficiently near one the exceptional superlevel has measure below delta, so $\int e_{r}dm\le \eta +{\int }_{\{e_r>\eta \}}{e}_{r}dm<\epsilon \mathrm{.}$ This argument along every sequence r tending to one proves $\mathrm{\parallel }{f}_r-f^{\ast }{\mathrm{\parallel }}_p\to 0$, including p<1. Also step 1.1 and unit kernel mass give $\int \mathrm{\mid }{f}_r{\mathrm{\mid }}^{p}dm\le \int hdm$ for every r by Tonelli; the supremum and the reverse inequality in (a) yield $\mathrm{\parallel }f{\mathrm{\parallel }}_{H^p}=\mathrm{\parallel }{f}^{\ast }{\mathrm{\parallel }}_p$.

Poisson representation for finite $p\ge 1$. Holder [F4] and step 3.1 give $L^1$ convergence of $f_r$ to $f^{\ast }$. Write $f(z)={\sum }_{n\ge 0}{c}_n{z}^n$ by [F5]. At each r>0 the Fourier coefficients of $f_r$ are $c_n{r}^n$ for nonnegative n and zero for negative n, by uniform Taylor convergence and orthogonality. $L^1$ convergence passes every Fourier coefficient to the limit, so $\widehat{f^{\ast }}(n)=c_n$ for n nonnegative and zero for n negative. Integrate the uniformly absolutely convergent kernel expansion of [F5] against the $L^1$ datum $f^{\ast }$; the uniform error is bounded in integral by its supremum times $\mathrm{\parallel }{f}^{\ast }{\mathrm{\parallel }}_1$. The result is $P[f^{\ast }](z)={\sum }_{n\ge 0}{c}_n{z}^n=f(z)$, proving (c). For p=1, [F6] makes $f^{\ast }m$ a finite complex representing measure; any other such measure has the same Poisson integral and is equal to it by [F6]. Its total variation is $\mathrm{\parallel }{f}^{\ast }{\mathrm{\parallel }}_1=\mathrm{\parallel }f{\mathrm{\parallel }}_{H^1}$.

Steps 1.1, 3.1, 2.2 and 4.1 prove every clause (a)–(c). All prior AC instances remain covered by the stronger CC conclusion. CC is used by the boundary and Fourier/measure uniqueness suppliers; the uniform integrability and convergence deduction is completely supplied in steps 2.1–3.1.

Ancillary maximal bound in the included source. Fix A>1. The superlevel set of $N_{A}f$ at t is the union, over interior z with $\mathrm{\mid }f(z)\mathrm{\mid }>t$, of the open circle sets $\{\zeta :\mathrm{\mid }z-\zeta \mathrm{\mid }<A(1-\mathrm{\mid }z\mathrm{\mid })\}$, so it is Borel measurable. First let $u\ge 0$ be in $L^2$ on the circle, hence in $L^1$ by [F4]. For t>0 split $u=\min (u,t\mathrm{/}2)+(u-t\mathrm{/}2{)}_{+}$. The first summand has maximal function at most t/2, while subadditivity of averages gives $\{M_{\mathbb{T}}u>t\}\subseteq \{M_{\mathbb{T}}(u-t\mathrm{/}2{)}_{+}>t\mathrm{/}2\}$. The weak bound in [F7] therefore gives $m\{M_{\mathbb{T}}u>t\}\le \frac{6}{t}{\int }_{\{u>t\mathrm{/}2\}}udm\mathrm{.}$ Apply layer-cake to $\min (M_{\mathbb{T}}u,K)$, then Tonelli to this nonnegative bound, and finally let K increase to infinity by [F7]. This yields $\int (M_{\mathbb{T}}u{)}^{2}dm\le 12{\int }_0^{\mathrm{\infty }}{\int }_{\{u>t\mathrm{/}2\}}udmdt=24\int u^{2}dm\mathrm{.}$ For finite p>0 put q=p/2 and $u=\mathrm{\mid }{f}^{\ast }{\mathrm{\mid }}^q\in L^2$. The same Poisson-Jensen and convexity argument as step 1.1 gives $\mathrm{\mid }f(z){\mathrm{\mid }}^q\le P[u](z)$; applying [F7] gives $(N_{A}f{)}^q\le (A+1{)}^2{M}_{\mathbb{T}}u$. Since p/q=2, $\mathrm{\parallel }{N}_{A}f{\mathrm{\parallel }}_p^p\le 24(A+1{)}^4\int \mathrm{\mid }{f}^{\ast }{\mathrm{\mid }}^{p}dm=24(A+1{)}^4\mathrm{\parallel }f{\mathrm{\parallel }}_{H^p}^p\mathrm{.}$ Thus $\mathrm{\parallel }{N}_{A}f{\mathrm{\parallel }}_p\le {24}^{1\mathrm{/}p}(A+1{)}^{4\mathrm{/}p}\mathrm{\parallel }f{\mathrm{\parallel }}_{H^p}$. For p infinity, $N_{A}f\le \mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}$ everywhere. All these estimates include f zero and imply the maximal function is finite almost everywhere for finite p.