Disjoint trilinear dual Kakeya maximal operator conjecture in three dimensions

Let $0<\delta\leq\nu<1$. Three families $\mathbb{T}_{1},\mathbb{T}_{2},\mathbb{T}_{3}$ of $\delta$-tubes are $\nu$-disjoint when their orientations lie in spherical patches of diameter comparable to $\nu$, with distinct patches separated by at least $\nu$. For $0<\varepsilon,\nu<1$, let $\mathcal{K}_{\operatorname*{disj}\nu}^{\ast}\left(\otimes_{3}L^{\infty}\rightarrow L^{\frac{1}{2}};\varepsilon\right)$ denote the assertion that there is a positive constant $C_{\varepsilon,\nu}$ such that

$$\left\Vert \prod_{k=1}^{3}\left(\sum_{T_{k}\in\mathbb{T}_{k}}\mathbf{1}_{T_{k}}\right)\right\Vert_{L^{\frac{1}{2}}\left(\mathbb{R}^{3}\right)}\leq C_{\varepsilon,\nu}\delta^{-\varepsilon}$$

for all $\nu$-disjoint families of $\delta$-separated $\delta$-tubes in $\mathbb{R}^{3}$ and $0<\delta\leq\nu$. The disjoint trilinear dual Kakeya conjecture. The statement $\mathcal{K}_{\operatorname*{disj}\nu}^{\ast}\left(\otimes_{3}L^{\infty}\rightarrow L^{\frac{1}{2}};\varepsilon\right)$ holds for all $0<\varepsilon,\nu<1$. It is the trilinear analogue of the dual Kakeya maximal-operator estimate and is part of the paper’s proposed route to an equivalence between linear and trilinear Kakeya inequalities.