Maximal averages and non-transversality

We investigate the $L^p$ mapping properties of maximal functions associated with analytic hypersurfaces in $\mathbb R^d$, with a particular emphasis on the role of transversality. Around points that are not transversal, we show that the associated maximal function is bounded on $L^p(\mathbb R^d)$ for all $p>2$, regardless of the decay of the Fourier transform of surface measures. In contrast, away from non-transversal points, we prove that $L^p$ bounds for the maximal operator imply that the Fourier transform of the surface measure decays at rate $1/q$ for $q>p$. Combining these two regimes, we demonstrate that the conjecture of Stein and Iosevich-Sawyer on maximal functions could be re-formulated, in the analytic setting, by restricting attention to transversal points. Moreover, our result completely settles the refined form of the conjecture for certain cases.