Stein–Iosevich–Sawyer maximal estimate conjecture from Fourier decay

Let $S$ be a smooth hypersurface in ${\mathbb R}^d$ given by the graph of a smooth function, let $\sigma_1$ be the associated measure, and define

$$\mathcal Mf(x)=\sup_{t>0}|\mathcal Af(t,x)|,$$

where $\mathcal Af(t,x)=f\ast\sigma_t(x)$. Assume that, for some $C>0$,

$$|\hat\sigma_1(\xi)|\le C|\xi|^{-q}$$

with $0<q\le 1/2$. Stein–Iosevich–Sawyer conjecture. The maximal operator $\mathcal M$ is bounded on $L^p$ for every $p>1/q$. This conjecture seeks the sharp relation between Fourier decay and maximal $L^p$ bounds in the low-decay range; the statement is attributed in the source to Stein for $q=1/2$ and to Iosevich–Sawyer for $0<q<1/2$, and its resolution is not specified here.