I've been reading Duoandikoetxea's book, and, in chapter 3, he proves a weak (1,1) bound for the Hilbert transform. To set my question up, I'll outline the argument, and then point out where I'm having trouble (disclaimer: constants may change from line to line).
Starting with a nonnegative function $f \in L^1(\mathbb{R}) \cap L^\infty (\mathbb{R})$, we fix $\lambda > 0$ and let $\{I_j\}_j$ be the dyadic intervals from the Calderon-Zygmund decomposition of $f$ at height $\lambda$. Set $\Omega = \bigcup_j I_j$, and let $f = g + b$ where $g$ and $b$ are the good and bad parts of $f$ respectively ($g(x) = f(x)$ if $x \notin \Omega$ and $g(x) = \frac{1}{|I_j|} \int_{I_j} f(y) dy$ if $x \in I_j$, $b(x) = \sum_j b_j(x)$ where $b_j(x) = (f(x) - \frac{1}{|I_j|} \int_{I_j} f(y) dy) \chi_{I_j}(x)$ )
So $\int_{I_j} b_j(x) dx = 0$, $|g(x)| \le 2 \lambda$, $||g||_{L^1} = ||f||_{L^1}$. So it follows that $b \in L^2$ as well and we can talk about all of their Hilbert transforms.
Now $|\{|Hf| > \lambda \}| \le | \{|Hg| > \frac{\lambda}{2}\}| + |\{|Hb| > \frac{\lambda}{2}\}|$, and estimating that $| \{|Hg| > \frac{\lambda}{2}\}| \le \frac{C}{\lambda} ||f||_{L^1}$ is easy.
To estimate the second term, let $2I_j$ be the interval with the same center as $I_j$ and twice the length. Set $\Omega^* = \bigcup_j 2I_j$. Then it's easy to see that $|\{|Hb| > \frac{\lambda}{2}\}| \le |\Omega^*| + |\{ x \notin \Omega^*: |Hb| > \frac{\lambda}{2}\}|$. Since $|\Omega^*| \le \frac{C}{\lambda} ||f||_{L^1}$, all that's left is to estimate the second term in the sum there.
Now $|\{ x \notin \Omega^*: |Hb(x)| > \frac{\lambda}{2} \}| \le \frac{2}{\lambda} \int_{\mathbb{R} \setminus \Omega^*} |Hb(x)| dx$.
Here is where I'm stuck:
The author then notes that, since $|Hb(x)| \le \sum_j |Hb_j(x)|$ almost everywhere, it's enough to know that $\sum_j \int_{\mathbb{R} \setminus 2I_j} |Hb_j(x)| dx \le C ||f||_{L^1}$. While I can certainly see that
$\int_{\mathbb{R} \setminus \Omega^*} |Hb(x)| dx \le \sum_j \int_{\mathbb{R} \setminus 2I_j} |Hb(x)| dx$
and
$\int_{\mathbb{R} \setminus \Omega^*} |Hb(x)| dx \le \sum_j \int_{\mathbb{R} \setminus \Omega^*} |Hb_j(x)| dx$,
I do not see why the author's stated estimate holds. I was hoping someone could clarify this for me.
Nevermind, I'm an idiot.
You estimate
$\int_{\mathbb{R} \setminus \Omega^*} |Hb(x)| dx \le \sum_j \int_{\mathbb{R} \setminus \Omega^*} |Hb_j(x)| dx \le \sum_j \int_{\mathbb{R} \setminus 2I_j} |Hb_j(x)| dx$
because $\mathbb{R} \setminus 2I_j$ is a bigger set.