Fefferman's article: Pointwise convergence of Fourier series

Question

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?

Answer 1

I have solved the problem by reading the following master's thesis:

http://diposit.ub.edu/dspace/bitstream/2445/107985/2/memoria.pdf

The idea is that we could iterate the process for $\sim \epsilon^{-1}$ times, so as to bootstrap the bounds on the right hand side from the $\delta^{\frac \epsilon 2}$ to $\delta^{10}$. The implicit constant here may become incredibly large, but it does no harm, since in Lemma 2, we tolerate a loss of $\eta$ on the power of $\delta$.

However, there seems also a problem in the thesis when I proceed to Chapter 6, Lemma 5.

Below is a link to another problem I raised.

Fefferman's article: Pointwise convergence of Fourier series, II

Also, I would like to correct a typo in the statement: the $\delta^{\frac 1 2-\eta}$ should be replaced by $\delta^{\frac 1 4-\eta}$.