We examine uniform estimates for oscillatory integrals related to the Fourier transform of smooth charges that are concentrated on specific hypersurfaces in this study. The leading part of the phase function order is used to express these estimates. When the distance between the origin and the Newton polyhedron is at least two and the leading part of the phase function has an isolated critical point at the origin, uniform estimates are optimal. In addition, an analog of the estimate by A. Iosevich and E. Sauer for arbitrary convex analytic hypersurfaces is obtained.
Khasanov et al. (2025) studied this question.