I've been working through part of Terry Tao's 1999 article "The Bochner-Riesz Conjecture Implies the Restriction Conjecture." (It appeared in the Duke Mathematical Journal.) A little more specifically, I care about the proof of Theorem 4.10, see below.
Here's the necessary background. For $0 < \delta \ll 1$, and $f$ a compactly supported function defined on $\mathbb{R}^{n}$, the Kakeya and Nikodym maximal functions of $f$, denoted $f_{\delta}^{*}$ and $f_{\delta}^{**}$, are defined as follows. First, $f_{\delta}^{*}$ is the function on $S^{n - 1}$ given by $$ f_{\delta}^{*}(\omega) = \sup_{T} \frac{1}{|T|} \int_{T} | f(y) | \, dy, $$ where the supremum is taken over all right cylindrical tubes $T$ having length $1$, radius $\delta$, and axis parallel to $\omega$. Next, $f_{\delta}^{**}$ is the function on $\mathbb{R}^{n}$ given by $$ f_{\delta}^{**}(x) = \sup_{T} \frac{1}{|T|} \int_{T} | f(y) | \, dy, $$ where this time the supremum is over all right cylindrical tubes $T$ having length $1$, radius $\delta$, and which contain $x$. (In both definitions, $|T|$ denotes the Lebesgue measure of $T$.)
Having defined these maximal functions (or maximal operators), what we're interested in are the size of the $L^{p}$ bounds for them. Tao uses $K(p, \alpha)$ to denote the estimate $$ \| f_{\delta}^{*} \|_{p} \leq C_{n, p, \alpha} \delta^{- (n/p - 1) - \alpha} \| f \|_{p}, $$ and likewise he uses $N(p, \alpha)$ to denote the estimate $$ \| f_{\delta}^{**} \|_{p} \leq C_{n, p, \alpha} \delta^{- (n/p - 1) - \alpha} \| f \|_{p}. $$ It's conjectured that $K(p, \alpha)$ and $N(p, \alpha)$ hold for all $1 \leq p \leq n$ and all $\alpha > 0$, although this is not presently known. The idea here is that $K(p, \alpha)$ and $N(p, \alpha)$ give estimates for the norms of the two maximal operators as $\delta \to 0$, i.e. as we consider thinner and thinner tubes.
Now on to what I'm specifically interested in. Part of Tao's Theorem 4.10 is that $K(p, \alpha)$ implies $N(p, \alpha)$, and this is what I'm struggling with. He first makes a reduction to showing a "frozen" estimate for the Nikodym maximal function, namely $$ \| f_{\delta}^{**}(\underline{x}, 0) \|_{p} \leq C_{n, p, \alpha} \delta^{- (n/p - 1) - \alpha} \| f \|_{p}, $$ where $\mathbb{R}^{n}$ has been parameterized by $x = (\underline{x}, x_{n})$. I'm okay with this reduction, as well as with another reduction whereby we assume that $f$ is supported in the "slab" $0 < x_{n} \leq 1$. Tao makes one further reduction which I do not understand the argument for: that we may assume $f$ is supported in the slab $1/2 < x_{n} \leq 1$. He says, "the condition $\alpha < (n + 1) / p$ and scaling considerations ensure that the other contributions are more favourable than this main term." That is, if $f$ is supported in $0 < x_{n} \leq 1$, the contribution to $f_{\delta}^{**}$ from the portion of $f$ supported in $0 < x_{n} \leq 1/2$ is in some less than the contribution from the portion of $f$ supported in $1/2 < x_{n} \leq 1$.
I'm guessing that the argument for this would involve a dyadic decomposition of $f$ into the slabs $2^{-k - 1} < x_{n} \leq 2^{-k}$ and some sort of re-scaling to take these thinner slabs to the slab $1/2 < x_{n} \leq 1$, but I've tried a couple different ideas and neither of them worked for me. So what, more precisely, is the argument reducing a function on $0 < x_{n} \leq 1$ to a function $1/2 < x_{n} \leq 1$?
