$$L^p-L^q$$ Local Smoothing Estimates for the Wave Equation via k-Broad Fourier Restriction

Abstract

We explore the connection between k-broad Fourier restriction estimates and sharp regularity $$L^p-L^q$$ L p - L q local smoothing estimates for the solutions of the wave equation in $$\mathbb {R}^{n}\times \mathbb {R}$$ R n × R for all $$n \ge 3$$ n ≥ 3 via a Bourgain–Guth broad-narrow analysis. An interesting feature is that local smoothing estimates for $$e^{i t \sqrt{-\Delta }}$$ e i t - Δ are not invariant under Lorentz rescaling.