Rough local smoothing conjecture for spherical averages

Let $E\subset [1,2]$ satisfy $\dim_M E=\dim_A E=\beta$. For each scale $j$, let $\mathcal{Z}_j(E)$ index a minimal collection of $\delta$-separated points $t_\nu$ whose intervals $[t_\nu,t_\nu+2^{-j}]$ cover $E$. Then, for $0\leq h\leq 2^{-j}$, Rough local smoothing conjecture.

\left\\|\mathcal{A}_{j}f(x,t_\nu+h)\right\\|_{L^{q_{\mathrm{LS},\beta}}_x(\mathbb{R}^d;\ell^{q_{\mathrm{LS},\beta}}_\nu(\mathcal{Z}_j(E)))}\lessapprox 2^{-\frac{j(d-1)}{2}}|\mathcal{Z}_j(E)|^{1/q_{\mathrm{LS},\beta}}\\|f\\|_{L^{q_{\mathrm{LS},\beta}}}.

This is a proposed local-smoothing estimate for rough time averages; the authors indicate that it will be explored further, and its status is open.