Iosevich–Sawyer–Stein conjecture for maximal operators

Let $\rho>0$ and let $\rho$-Fourier decay mean that a measure $u$ satisfies the decay inequality

$$|\widehat{\nu}(\xi)|\lesssim (1+|\xi|)^{-\rho}.$$

Suppose $\Gamma$ is a smooth hypersurface and a measure $mu$ defined on $\Gamma$ satisfies this inequality for some $0<\rho\le 1/2$. Iosevich–Sawyer–Stein conjecture. The associated maximal operator is bounded on $L^p$ for $p>1/\rho$. This conjecture concerns the sharp relationship between Fourier decay and maximal-operator bounds; the source states that the general problem remains widely open, while several important cases are known.