Subcritical boundary-weighted Fourier inequalities in dimensions at least three

Determine whether, for every \(1\leq q<2\), the \((q,q,d)\) boundary-weighted Fourier inequality for a ball \(B\subset\mathbb{R}^n\) holds for every subcritical exponent \(d<2-q\) when \(n\geq 3\).

Background

The paper proves that if the subcritical inequalities hold for every $d<2-q$, then every compact Kakeya set in $\mathbb{R}^n$ has full Minkowski dimension. It also proves that the critical inequality at $d=2-q$ always fails for balls.

For the reverse implication, the paper shows that the spherical restriction conjecture would imply the subcritical inequalities in the range $2n/(n+1)\leq q<2$. The authors do not know whether the full hypothesis is satisfied in dimensions $n\geq3$, making this question a problem intermediate between restriction theory and Kakeya phenomena.