Can anyone outline Cotlar's original proof of the $L^2$ boundedness of the Hilbert transform. I cannot locate the original paper on the web. I know the Cotlar-Stein lemma but I don't see how to make the estimates needed for the conclusion of the lemma to hold. A reference would also be helpful if I can access it.



$L^2$ boundedness of the Hilbert transform via Cotlar-Stein Lemma
(Sketch) Recall that the Hilbert transform is given (in a principal value sense) by the convolution with the function 1/t. That is the Hilbert transform of $f$ is defined (up to normalization) by


$$Tf(x) = \int\_{-\infty}^{\infty} \frac{f(x-t)}{t} dt .$$ 


To apply the Coltar-Stein lemma we wish to decompose T as the infinite sum of "almost orthogonal" operators:


$$Tf = \sum\_{i \in \mathbb{Z}} T\_i f.$$


The trick is to let each $T\_i$ denote the contribution of the kernel $\frac{1}{t}$ at a given dyadic frequency scale. More precisely, $T\_i$ should correspond to convolution with the $i$-th component of a Littlewood-Paley decomposition of the function 1/t.


