The family of fractal Fourier restriction estimates

Let $n\geq 2$, $1\leq\alpha\leq n$, $R\geq 1$, and let $X$ be a union of lattice unit cubes in $\widetilde{B}(0,R)\subset\mathbb R^n$. Define

$$\gamma=\sup\frac{\#\{\widetilde{B}_k:\widetilde{B}_k\subset\widetilde{B}(x',r)\}}{r^\alpha},$$

where the supremum is over $(x',r)\in\mathbb R^n\times[1,\infty)$ such that $\widetilde{B}(x',r)\subset\widetilde{B}(0,R)$. Let $\beta$ satisfy $1/n\leq\beta\leq 2/n$, and define

$$p=2+\frac{n-\alpha}{n-1}\left(\frac{2}{n}-\beta\right).$$

The family of fractal Fourier restriction estimates. For every $\epsilon>0$ there is a constant $C_\epsilon$ such that

$$\int_X|Ef(x)|^p\,dx\leq C_\epsilon R^\epsilon\gamma^\beta R^{\alpha/n}\|f\|_{L^p(\mathbb B^{n-1})}^p$$

for all $f\in L^p(\mathbb B^{n-1})$.

This family interpolates between fractal restriction estimates and Kakeya-type bounds. The endpoint case $\beta=2/n$ is the previously known theorem of Du and Zhang, while the source notes that the statement is also a theorem when $n=2$; the general assertion is presented as a conjecture and is intended to yield sharp Kakeya consequences.