Fractal-time local smoothing conjecture for the half-wave propagator

Let $E\subset[1,2]$, let $1<p\leq q<\infty$ with $q>p'$, and let $\nu_E^\sharp$ denote the Legendre-Assouad function of $E$. For each $j\geq1$, let $E_j$ be a $2^{-j}$-discretization of $E$, and let $P_j$ be the corresponding frequency-localization operator. Define

$$s_E(p,q)=\frac{d+1}{2}\left(\frac1p-\frac1q\right)+\frac1q\nu_E^\sharp\left(\frac{q(d-1)}2\left(1-\frac1p-\frac1q\right)\right).$$

Fractal-time local smoothing conjecture. For every $s>s_E(p,q)$, there exists a constant $C_{s,p,q}>0$ such that

$$\left(\sum_{t\in E_j}\|e^{it\sqrt{-\Delta}}P_jf\|_q^q\right)^{1/q}\leq C_{s,p,q}2^{js}\|f\|_p$$

holds for all $j\geq1$, all $2^{-j}$-discretizations $E_j$ of $E$, and all $f\in L^p(\mathbb R^d)$. This conjecture extends the paper's fractal-time $L^2\to L^q$ square-function estimates to general $L^p\to L^q$ bounds; its status is not resolved in the supplied text.