Let \(f:\Delta\to\mathbb C\) be holomorphic and suppose \[ \int_\Delta |f'(z)|^2\,|dz|^2<\infty. \] Prove that for almost every \(\zeta\in S^1\), the radial limit \[ F(\zeta)=\lim_{r\to1^-} f(r\zeta) \] exists and is finite.
Solution
Write points of \(S^1\) as \(\zeta=e^{i\theta}\). Fubini’s theorem shows that for almost every \(\theta\), the radial restriction of \(f'\) is square-integrable near \(r=1\). Cauchy–Schwarz upgrades this to integrability of the radial derivative of \(f\), which makes the radial values Cauchy as \(r\to1^-\).
1. In polar coordinates, the hypothesis is\[ \int_0^{2\pi}\int_0^1 |f'(re^{i\theta})|^2 r\,dr\,d\theta<\infty. \]
Since \(|dz|^2=dA\) and \(dA=r\,dr\,d\theta\) in polar coordinates, \[ \int_\Delta |f'(z)|^2\,dA(z) =\int_0^{2\pi}\int_0^1 |f'(re^{i\theta})|^2r\,dr\,d\theta. \] The left side is finite by hypothesis.
2. For almost every \(\theta\in[0,2\pi)\),\[ \int_{1/2}^1|f'(re^{i\theta})|^2\,dr<\infty. \]
By 1 and Fubini’s theorem, for almost every \(\theta\), \[ \int_0^1|f'(re^{i\theta})|^2r\,dr<\infty. \] For \(r\in[1/2,1)\) we have \(r\ge1/2\), so \[ \int_{1/2}^1|f'(re^{i\theta})|^2\,dr \le 2\int_{1/2}^1|f'(re^{i\theta})|^2r\,dr<\infty. \]
3. For every \(\theta\) satisfying 2,\[ \int_{1/2}^1|f'(re^{i\theta})|\,dr<\infty. \]
By Cauchy–Schwarz, \[ \begin{aligned} \int_{1/2}^1|f'(re^{i\theta})|\,dr &\le \left(\int_{1/2}^1|f'(re^{i\theta})|^2\,dr\right)^{1/2} \left(\int_{1/2}^1 1\,dr\right)^{1/2}\\ &<\infty \end{aligned} \] by 2.
4. For such a \(\theta\), the radial path\[ r\longmapsto f(re^{i\theta}) \]is Cauchy as \(r\to1^-\).
Its derivative is \[ \frac d{dr}f(re^{i\theta}) =e^{i\theta}f'(re^{i\theta}). \] Hence, for \(1/2<r<s<1\), \[ \begin{aligned} |f(se^{i\theta})-f(re^{i\theta})| &\le\int_r^s|f'(te^{i\theta})|\,dt. \end{aligned} \] By 3, the function \(t\mapsto|f'(te^{i\theta})|\) is integrable on \([1/2,1)\). Therefore the right-hand side tends to \(0\) uniformly as \(r,s\to1^-\). Thus the radial values form a Cauchy family in \(\mathbb C\).
5. For almost every \(\zeta\in S^1\), the radial limit exists and isfinite.
By 2, the conclusions of 3 and 4 hold for almost every angle \(\theta\). Since \(\mathbb C\) is complete, the Cauchy family \[ f(re^{i\theta}) \] has a finite limit as \(r\to1^-\). Writing \(\zeta=e^{i\theta}\) gives \[ F(\zeta)=\lim_{r\to1^-}f(r\zeta)\in\mathbb C \] for almost every \(\zeta\in S^1\).