I have read S.Petermichl's paper Dyadic shifts and a logarithmic estimate for Hankel operators with matrix symbol many days and got trouble in the proof of Lemma 2.1 in this paper. Let us see Lemma 2.1:
Lemma 2.1. The convergence of sum $$ K^{\alpha,r}(t,x)=\sum_{I\in D^{\alpha,r}} h_{I}(t)(h_{I_{-}}(x)-h_{I_{+}}(x)) $$ is uniform for $|x-t|\geq \delta$ for every $\delta>0$. For $x\neq t$ let $$ K(t,x)=\underset{L\to\infty}{\lim}\frac{1}{2\log L}\int^{L}_{1/L} \underset{R\to\infty}{\lim} \frac{1}{2R}\int^{R}_{-R} K^{\alpha,r}(t,x) d\alpha \frac{dr}{r}. $$ The limits exist pointwise and the convergence is bounded for $|x-t|\geq \delta$ for every $\delta>0$ and $K(t,x)= \frac{c_0}{t-x}$ for some $c_0>0$.
Note that $h_{I}(x)$ is the Haar function with $I \in D^{\alpha,r}.$ And $D^{\alpha,r}$ is the set of all dyadic intervals on $\mathbb{R}$ originated at $\alpha$ and with interval length $r2^n$, $n\in \mathbb{Z}$. Interval $I_{-}$ is the left half of $I$, and the same as $I_{+}$.
I have verified almost all part of the proof of Lemma 2.1 but finding a proof of the existence of the limit $$ \underset{L\to \infty}{\lim} \frac{1}{2\log L}\int^{L}_{1/L} K^r(t,x)\frac{dr}{r} \tag{1} $$ is much more difficulty, where $K^r(t,x)=\underset{R\to\infty}{\lim} \frac{1}{2R}\int^{R}_{-R} K^{\alpha,r}(t,x) d\alpha $.
By using the property of the sum $$\sum_{I\in D^{\alpha,r}} |h_{I}(t)(h_{I_{-}}(x)-h_{I_{+}}(x))|$$ converges absolutely and uniformly for $|t-x|\geq \delta$ for every $\delta>0$, I have verified that this limit $$\underset{R\to\infty}{\lim} \frac{1}{2R}\int^{R}_{-R} K^{\alpha,r}(t,x) d\alpha$$ is really exist. However, the proof is not easy. At present, I have done much for seeking out a proof of the existence of limit $(1)$, never get through yet.
My question is: why is limit $(1)$ exist ? Thanks for your help.