I have some problems reading *Pointwise convergence of Fourier series* by Fefferman https://www.jstor.org/stable/1970917

When I proceed to Lemma 2, Chapter 6, I could not verify either of the following:

"Trivial estimates show that $|T_{p'}T_p^*f(x)|\leq \delta^{10}/|I'^*|\int_{E(p)|f(y)|dy}$, if $\mathrm{distance}(\omega,\omega')>\delta^{-\epsilon/2}|\omega|$, "

"$T_{p'}T_p^*f(x)=0$ if $I\not\subseteq I'^*$."

For the second, I could only show that this is the case if $I^*\cap I'^*=\varnothing$, but I think it just needs some modification (any suggestions?) to be true.

However, for the first one, the only place that involves $\delta$ is the distance condition. However, I simply do not know how to use that in the trivial estimate. I tried using the decay of the Fourier transform, but it turns out I could only obtain a term involving $\delta^{\epsilon/2}$.

Thank you for spending time reading my question. If you happened to have read the paper before, could you help me understand what he meant?