3 problems

Bilinear Kakeya range conjecture. $K^*(q'\times q'\to p'/2)$ holds if and only if

Let $\Sigma\subset\mathbb{R}^d$ be a $C^2$ hypersurface with surface measure $\mathrm{d}\sigma$, let $f\in L^2(\Sigma;\mathrm{d}\sigma)$, and let $w$ be a nonnegative weight. Let…

For $0<\delta\ll1$, let $\mathbb{T}$ be a collection of rectangular $\delta$-tubes in $\mathbb{R}^d$, each with $d-1$ sides of length $\delta$ and one side of length $1$, whose ori…