While reading the paper "A geometric proof of the strong maximal theorem", by A. Cordoba and R. Fefferman -Annals of Mathematics Vol 102 no. 1, I got stuck trying to understand a main step in the proof.

The goal is to prove a inequality for the strong maximal operator: $$ Mf(x) = \sup_R \frac{1}{R}\int_R |f|, $$ where $R$ ranges over all rectangles with sides parallel to the coordinate axes which contain $x$.

In particular they state $$ \bigl|\{x:Mf(x)>\lambda\}\bigr| \leq A\int\frac{|f(x)|}{\lambda}\biggl( 1 + \log^+\frac{|f(x)|}{\lambda} \biggr)\,dx. $$ (for dimension 2).

To achieve this they prove that given a collection of rectangles $\{R_i\}$ there is a subcollection $\{\tilde{R}_j\}$ such that

i) $|\cup_i R_i| \leq C |\cup_j \tilde{R}_j|$

ii) $\|\exp(\sum \chi_{\tilde{R}_j})\| \leq C |\cup_i R_i|$

Then, they assert that the strong maximal theorem follows from this and their proof of the fact that the strong maximal operator if weak-type $(p,p)$ if and only if you have the $V_q$ property where $q$ is the dual exponent of $p$: Given any collection of rectangles $\{R\}$ there exists a subcollection $\{\tilde{R}\}$ such that

$(i)$ $|\cup_i R_i| \leq C |\cup_j \tilde{R}_j|$

$(ii)_q$ $\|\sum \chi_{\tilde{R}_j}\|_{q} \leq C |\cup_i R_i|^{1/q}$

I don't understand how their proof, which is based on Holder's inequality, extends to the case $p=1.$ The main obstacle for me is the fact that the right hand side of $(ii)_\infty$ should be just $1$ ($\text{something}^{1/\infty}$) but the exponent in their main result is instead $1$.

I tried using the inequality

$$ \int fg \leq C \|f\|_{L\log L} \|g\|_{e^L} $$ but I end up with $|\{x: Mf(x)>\lambda\}|$ multiplied on both sides of the equation...

How is this proved?