On local smoothing estimates for wave equations
Shengwen Gan, Danqing He, Xiaochun Li, Shukun Wu
TL;DR
This work develops sharp local smoothing estimates for wave equations on compact manifolds in odd dimensions and improves bounds in even dimensions by translating the problem into sharp L^p bounds for Fourier integral operators satisfying natural geometric conditions. Central to the approach is a novel mixed-norm framework that uses a Lorentz-respecting wave-packet density $\mathcal{W}(f,B_R^{n+1})$, combined with a broad-narrow decomposition and a refined cone decoupling bound. The authors introduce a broad & two-ends algorithm to enforce non-concentration properties, and employ incidence geometry (hairbrush) and transversality-based trilinear estimates to obtain improved L^2 and $L^p$ bounds, culminating in a complete resolution of the odd-dimensional case and near-optimal bounds in even dimensions. The results advance the understanding of local smoothing on manifolds and have implications for the study of dispersive PDEs via FIO methods and decoupling techniques, connecting geometric measure theory with harmonic analysis tools. Specifically, the paper proves that conjectures for local smoothing hold for $n\ge 3$ and $p\ge p(n)$, with explicit exponents $p(n)=\begin{cases} p_n^+,& n\text{ odd},\\ 2+\frac{8}{3n-2-\omega(n)},& n\text{ even},\end{cases}$ where $p_n^+=2+\frac{8}{3n-3}$ for odd $n$ and $\omega(n)=\frac{4(7n-2)}{3n^2+5n+14}$, capturing the main contributions of the paper.
Abstract
We prove sharp local smoothing estimates for wave equations on compact Riemannian manifolds in $n+1$ dimensions for odd $n$ and obtain improved estimates in even dimensions. This is achieved by deriving local smoothing estimates for certain Fourier integral operators. We also obtain improved local smoothing estimates for wave equations in Euclidean spaces.