Fourier transforms of indicator functions, lattice point discrepancy, and the stability of integrals

Abstract

We prove sharp estimates for Fourier transforms of indicator functions of bounded open sets in \({\mathbb {R}}^n\) with real analytic boundary, as well as nontrivial lattice point discrepancy results. Both are derived from estimates on Fourier transforms of hypersurface measures. Relations with maximal averages are discussed, connecting two conjectures of Iosevich and Sawyer (Adv Math 132(1):46–119, 1997). We also prove a theorem concerning the stability under function perturbations of the growth rate of a real analytic function near a zero. This result is sharp in an appropriate sense. It implies a corresponding stability result for the local integrablity of negative powers of a real analytic function near a zero.