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?



Fefferman's article: Pointwise convergence of Fourier series
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](https://mathoverflow.net/questions/272525/feffermans-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 $\delta^{\frac 1 4-\eta}$.


