Use of Stein's maximal principle in Bourgain's paper on Besicovitch sets

Question

I'm trying to understand Bourgain's paper "Besicovitch type maximal operators and applications to Fourier analysis". Let $\xi\in S^2\subset\mathbb{R}^3$ be a unit vector and $\delta>0$, by a $(\xi,\delta)$ tube in $\mathbb{R}^3$, we mean a cylinder $\tau$ of unit length in direction $\xi$ and of radius $\delta$. For $f$ a locally integrable function on $\mathbb{R}^3$, we define $$f_\delta^\ast(\xi)=\sup_\tau\frac{1}{|\tau|}\int_\tau f(x)\,dx$$ where the supremum refers to all $(\xi,\delta)$ tubes $\tau$ and $|\tau|$ stands for the measure of $\tau$. Bourgain proves the following bound

$$||f_\delta^\ast||_{L^p(S^2)}\leq C_\epsilon (1/\delta)^\epsilon ||f||_{L^P(\mathbb{R}^3)}$$

On page 152 of the paper, Bourgain says we only need to check that for every $A\subset B(0,1)$ and $\sigma\in[0,1]$, we have $$|A|\geq c\delta^{2/3+\epsilon}\sigma^{7/3}|(\chi_A)^\ast_\delta>\sigma\}|$$

Bourgain says this is due to Stein's maximal principle, as in the paper "Limtis of sequences of operators" by Stein. Also found here https://terrytao.wordpress.com/2011/05/12/steins-maximal-principle/ But why? Stein's maximal principle gives weak Lp type bound of maximal functions, Bourgain is saying weak Lp type estimates give strong Lp type estimates. What are we taking to be the sequence of operators in Stein's paper here?

Answer 1

This may be a slight misattribution. It is the implication $(1.22) \implies (1.21)$ which is essentially in Stein's paper; the implication $(1.21) \implies (1.9)$ is much simpler, following from Marcinkiewicz interpolation. Roughly speaking, if one has a weak-type $L^p$ estimate on the Kakeya maximal operator, one can interpolate it with the $L^2$ estimate (1.20) to get a strong-type $L^{p-\varepsilon}$ estimate, and then interpolate again with the trivial $L^\infty$ estimate to get back a strong $L^p$ estimate, losing some factors of $\delta^{-O(\varepsilon)}$ in the process.

[A more general principle here is that, in situations where one is willing to concede epsilon losses in the exponents, strong type and weak type estimates are equivalent; also, singular integrals are all basically bounded, rendering a large part of Calderon-Zygmund theory trivial in this regime.]