2024/02/26 by Greenblatt, Michael · 1 citation
#11K38 #11P21 #26E05 #42B10 #42B20 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2402.16636
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. In such situations, for a compact surface the exponent provided is sharp. This result has implications for lattice point discrepancy problems, which we describe.