The maximal Kakeya conjecture

For $0<\delta<1$, let $K_\delta f(\omega)$ be the Kakeya maximal function, defined using averages over the $\delta$-neighborhoods of unit line segments in direction $\omega\in\mathbb{S}^{d-1}$. Let $p'\!$ be the Hölder conjugate of $p$ and set $q=(d-1)p'$. Maximal Kakeya conjecture. For every $\epsilon>0$, there is a constant $C_\epsilon>0$ such that

$$\|K_\delta f\|_q\leq C_\epsilon\delta^{1-\frac{d}{p}-\epsilon}\|f\|_p,$$

for $1\leq p\leq d$. This is a central analytic formulation of the Kakeya problem and remains open in the full stated range.