Is there an "easy" proof of the existence of radial limits of functions in the hardy spaces $H^p$?

Question

Let $HOL(\mathbb{D})$ be the analytic functions defined on the unit disk and for $1\leq p \leq \infty$, $$H^p = \{f\in HOL(\mathbb{D}) : \lim_{r\nearrow 1} \int_{rC}|f(z)|^p \frac{dz}{2\pi i} < \infty \} $$

We know that for $f\in H^p$ the radial limits, $\lim_{r\nearrow 1} f(re^{i\theta})$ exists almost everywhere. Is there an "easy" proof for this?

I think the steps are

1. $f\in H^p \implies f\in H^1$ and $f$ is harmonic (ok, no problem here).

2. There is a unique measure $\mu$ on the circle $C$ such that $f(re^{i\theta}) = \int_\theta P(r,\theta) d\mu$ where $P$ is the Poisson Kernel. (can someone help me with this?)

3. Write $\mu(\theta) = g(\theta) d\theta + d\mu_\perp $ and use this work to show that as $r\nearrow 1$, $f(re^{i\theta})$ converges to $g(\theta)$ on supp$(\mu_\perp)^c$.

Is this correct and is there an easier proof?

Answer 1

As far as I know - from reading several texts on Hardy spaces - all proofs of this fact utilize Fatou’s theorem (or a variation of).

Using weak/weak-* compactness of the unit ball in $L^p(C)$, one can show directly that “the boundary function” $f^*$ exists, i.e.

1. $f^*$ is in $L^p(C)$,

2. $f^*$ is a limit (in the norm for $1 \leq p < \infty$, and weak-* for $p = \infty$) of the functions $f_r(w) := f(rw); \; w \in C, \; 0 < r < 1$,

3. $\widehat{f^*}(n) = 0$, for all $n \in \mathbb{Z}_{< 0}$,

4. $\|f^*\|_{L^p(C)} = \|f\|_{H^p(\mathbb{D})}$, and

5. $f_r = f^* * P(r, \cdot)$.

See Theorem 2.2.2 from Hardy Spaces, by N. Nikolski for a proof of this fact. As Nikolski then mentions in the beginning of Section 2.5, this is not sufficient to obtain radial limits a.e. on $C$.

See also this math stackexchange discussion, and note that for $1 \leq p < \infty$ we get using $L^p(C)$ convergence that some subsequence $f_{r_k}$ converges to $f^*$ a.e. On $C$.

Of course, this is not to say that an “easier” proof cannot/does not exist.