Omega-Kakeya maximal conjecture

Let $n\geq 3$, let $\mu$ be a probability measure on $\mathbb{S}^{n-1}$, and define

$$[\mu]_d:=\sup_{e\in\mathbb{S}^{n-1},\,r>0}\mu(B_{e,r})r^{-d}.$$

For $\delta>0$, let $K_\delta$ be the Kakeya maximal operator and let $\|K_\delta\|_{\mu,n}$ be its operator norm from $L^n(\mathbb{R}^n)$ to $L^n(\mathbb{S}^{n-1},\mu)$.

$\Omega$-Kakeya maximal conjecture. Fix any probability measure $\mu$ on $\mathbb{S}^{n-1}$ satisfying, for some $d\in[0,n-1]$,

$$[\mu]_d\leq 1.$$

Then, for every $\epsilon>0$,

$$\|K_\delta\|_{\mu,n}\lesssim_{n,d,\epsilon}\delta^{\frac{d+1}{n}-(1+\epsilon)}.$$

This is the maximal analogue of the direction-restricted Kakeya conjecture. The paper proves that the ordinary Kakeya maximal conjecture implies this statement, but does not establish the conjecture itself.