Reformulated Iosevich–Sawyer–Stein conjecture for analytic hypersurfaces

Let $p_{cr}\ge 2$ and let $\Gamma$ be a compact analytic hypersurface. Let $M_\Gamma$ denote its associated maximal operator, let $\mu$ be the measure on $\Gamma$, and let $\mathcal F$ denote the Fourier transform. Reformulated Iosevich–Sawyer–Stein conjecture. The following statements are equivalent:

1. $M_\Gamma$ is bounded on $L^p(\mathbb R^d)$ for all $p>p_{cr}$.
2. For every $\sigma\in C^\infty(\Gamma)$ and every $p>p_{cr}$, there exists a constant $C>0$ such that

$$|\mathcal F[\sigma\,\mathrm d\mu](\xi)|\le C|\xi|^{-1/p},$$

provided that $\sigma$ vanishes on an open set containing all non-transversal points of $\Gamma$. The source presents this as a reformulation designed to isolate the role of non-transversality; it also states that the reformulated conjecture is proved for $\rho=1/2$, but does not establish the full equivalence in the displayed generality.