Wave Packet Density in Local Smoothing: Refinements and Limitations

Abstract: We study the role of wave packet density estimates in the local smoothing problem for the wave equation. Building on the framework of Gan, He, Li, and Wu, we incorporate an additional multilinear geometric estimate to obtain an improved local smoothing range in high even dimensions. We then investigate the limitations of the wave packet density method. By constructing a geometric configuration that nearly saturates the relevant density estimate, we show that the existing wave packet density bound is nearly sharp in certain regimes, even in the Euclidean setting.