It is well-known that [Cotlar-Stein's Lemma](https://en.wikipedia.org/wiki/Cotlar%E2%80%93Stein_lemma) can be used to prove the $L^2$ boundedness of the Hilbert transform. See e.g. <http://mathoverflow.net/questions/184316/l2-boundedness-of-the-hilbert-transform-via-cotlar-stein-lemma>. Then using the Euler formula $e^{ix}=\cos x+i\sin x$, we can easily show the uniform $L^2$ boundedness of the Dirichlet kernel: $||D_\lambda||_{L^2\to L^2}\le C$, where $$D_\lambda f(t)=\int \frac{\sin(\lambda(t-s))}{t-s}f(s)ds.$$

**My question** is: Can we prove the uniform $L^2$ boundedness of the Dirichlet kernel **directly** from the Cotlar-Stein Lemma, without using the $L^2$ boundedness of Hilbert transform?  It suffices to show $$\int\Big| \int \frac{\sin(\lambda(t-s))\sin(\lambda s)}{(t-s)s}1_{2^{-j}\le |s|\le 2^{1-j}}\  1_{2^{-k}\le |t-s|\le 2^{1-k}}\ ds\Big|dt\le C 2^{-|j-k|},$$
for $j,k\in \mathbb{Z}$. To get the bound on the RHS, it seems that one need to consider the cancellation in the integral with $ds$ and get a delicate estimate.