Maximal estimates for perturbations of the Schr\"odinger operator on $\mathbb{T}^d$

Abstract

We study $L^p_x L^\infty_t$ maximal estimates for exponential sums associated to $C^2$ graph hypersurfaces, motivated by Schr\"odinger maximal estimates on $\mathbb{T}^d$. We show that the conjectured maximal estimate for the periodic Schr\"odinger equation fails when one allows small perturbations of the paraboloid, which can be viewed as a higher-dimensional extension of the phenomenon proved by Fu, Ren, and Wang. Our approach uses new lower bounds for incidence estimates originally proven by Cairo and Zhang, for which we provide an alternative proof based on homogeneous dynamics. Moreover the estimates are essentially sharp at the decoupling endpoint for the paraboloid $p = \frac{2(d+2)}{d}$.