On Instability of the Nikodym Maximal Function Bounds over Riemannian Manifolds

Abstract

We show that, for odd d, the $L^{\frac{d+2}{2}}$ bounds of Sogge (J Am Math Soc 12:1–31, 1999) and Xi (Trans Am Math Soc 369:6351–6372, 2017) for the Nikodym maximal function over manifolds of constant sectional curvature are unstable with respect to metric perturbation, in the spirit of the work of Minicozzi and Sogge (Math Res Lett 4:221–237, 1997). A direct consequence is the instability of the bounds for the corresponding oscillatory integral operator. Furthermore, we extend our construction to show that the same phenomenon appears for any d-dimensional Riemannian manifold with a local totally geodesic submanifold of dimension $\lceil \frac{d+1}{2}\rceil$ if $d\ge 3$. In contrast, Sogge’s $L^\frac{7}{3}$ bound for the Nikodym maximal function on 3-dimensional variably curved manifolds is stable with respect to metric perturbation.