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

I got stuck in Chapter 6, Lemma 5. In the proof he split the $\mathcal P'$ into three subcollections $\mathcal P'_k$, $\mathcal P''_k$, $\mathcal P'''_k$. The first and third subcollections were estimated. However, for the second, he imposed a new assumption:

$\varphi_k(\xi_0')=0$, where $\xi'_0$ is the midpoint of $\omega_0'$.

This is weird to me, since when we constructed the functions $\varphi_k$, we assumed that it was supported in $\{|x|\leq \delta^{\frac 1 2}d_k\}$, which is very small. Since $|\xi'_0|$ is obviously large, this assumption seems to be a typo or something else.

On the other hand, the reason he imposed this condition is that we would like to prove (23) $$ |T^{\mathcal P_k''}(\varphi_k*F)(x)|\leq C_M \delta^M \Phi_k *|F|(x), $$ for any good $F$ supported in $I_0^k$.

However, according to our definition, I think that $\varphi_k$ is roughly a $\delta$-function behaving like $\frac 1{\delta^{1/2}d_k}\phi(\frac x {\delta^{1/2}d_k})$ for some universal bump function $\phi$ with support in $[-1,1]$, hence the left hand side behaves like $$ |T^{\mathcal P_k''}(F)(x)|, $$ in which case it is impossible to have rapid decay (the $C_M\delta^M$ term). Could anyone who read it through tell me what was wrong? Thanks!

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