# Kakeya and Nikodym maximal functions

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 $1/2 < x_{n} \leq 1$?

Yes, the argument is dyadic decomposition followed by rescaling. I think I forgot to mention in the paper one initial reduction, which is to only consider the portion of the Nikodym maximal function coming from tubes which make an angle of at most 1/10 (say) with the basis vector $e_n$; note that one can reduce to this case by a finite partition of unity of the unit sphere. Once one restricts to such nearly-vertical tubes, the scaling argument should go through (averaging on $\delta$-tubes centred at a point $(\underline{x},0)$ applied to a function supported on the slab $\{ 2^{-k-1} < x_n \leq 2^{-k} \}$ will rescale to an average on $2^k \delta$-tubes centred at $(2^k \underline{x},0)$ supported on the slab $\{ 1/2 < x_n \leq 1 \}$). (It may be helpful to first consider the extreme case $2^{-k} = \delta$, which is an easy case of the Nikodym conjecture; the remaining cases are an interpolation between this easy case and the most difficult case $k=0$, which is intuitively why the rescaling is guaranteed to be favorable.)