Convexity, Fourier transforms, and lattice point discrepancy

In a well-known paper by Bruna, Nagel and Wainger BNW, Fourier transform decay estimates were proved for smooth hypersurfaces of finite line type bounding a convex domain. In this paper, we generalize their results in the following ways. First, for a surface that is locally the graph of a convex real analytic function, we show that a natural analogue holds even when the surface in question is not of finite line type. Secondly, we show a result for a general surface that is locally the graph of a convex C² function, or a piece of such a surface defined through real analytic equations, that implies an analogous Fourier transform decay theorem in situations where the oscillatory index is less than 1. This result has implications for lattice point discrepancy problems, which we describe.