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.

我们证明了对具有真实解析边界的\（{{mathbb {R}} ^ n \）中有界开放集的指标函数的Fourier变换的精确估计，以及非平凡的格点差异结果。两者均来自对超表面量度的傅立叶变换的估计。讨论了与最大均值的关系，将Iosevich和Sawyer的两个猜想联系起来（Adv Math 132（1）：46–119，1997）。我们还证明了一个定理，该定理涉及在实际扰动函数的增长率接近零的函数扰动下的稳定性。在适当的意义上，该结果是清晰的。对于实际解析函数的负幂的局部可积性接近零而言，这意味着相应的稳定性结果。