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