# On Instability of the Nikodym Maximal Function bounds over Riemannian Manifolds

## TL;DR

This paper demonstrates that certain bounds for the Nikodym maximal function on Riemannian manifolds are unstable under metric perturbations, especially in odd dimensions, highlighting the delicate nature of these bounds.

## Contribution

It establishes the instability of specific Nikodym maximal function bounds under metric perturbations for odd-dimensional manifolds with constant curvature, extending to manifolds with certain submanifolds.

## Key findings

- Bounds are unstable for odd dimensions with constant curvature.
- Stability of Sogge's $L^{7/3}$ bound on 3D variably curved manifolds.
- Instability extends to manifolds with totally geodesic submanifolds.

## Abstract

We show that, for odd $d$, the $L^{\frac{d+2}2}$ bounds of Sogge and Xi 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 Sogge and Minicozzi. 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 $L^\frac73$ bound for the Nikodym maximal function on 3-dimensional variably curved manifolds is stable with respect to metric perturbation.